Research ArticleAsymptotic Behavior of Ruin Probabilities in an Insurance RiskModel with Quasi-Asymptotically Independent or BivariateRegularly Varying-Tailed Main Claim and By-Claim

Yang Yang Xinzhi Wang Xiaonan Su and Aili Zhang

Department of Statistics Nanjing Audit University Nanjing 211815 China

Correspondence should be addressed to Yang Yang yangyangmath163com

Received 6 April 2019 Accepted 24 June 2019 Published 20 October 2019

Academic Editor Dimitri Volchenkov

Copyright copy 2019 Yang Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper considers a by-claim risk model under the asymptotical independence or asymptotical dependence structure betweeneach main claim and its by-claim In the presence of heavy-tailed main claims and by-claims we derive some asymptotic behaviorfor ruin probabilities

1 Introduction

Consider a continuous-time risk model with no interest ratein which each severe accident leads to two kinds of claimsone is caused by the accident immediately called the mainclaim and the other is caused as a compensation occurringafter a period of time called the by-claimThe surplus processcan be described as

119880 (119905) = 119909 + 119888119905 minus 119873(119905)sum119894=1

119883119894 minus infinsum119894=1

1198841198941120591119894+119863119894le119905 119905 ge 0 (1)

where 1119860 stands for the indicator function of a set 119860 Inrelation (1) 119909 ge 0 is interpreted as the initial reserve of aninsurance company 119888 ge 0 is the constant premium rate119883119894 isthe size of the 119894th main claim occurring at the arrival time120591119894 the counting process 119873(119905) = suminfin

119896=1 1120591119896le119905 represents thenumber of the accidents (equal to that of the main claims)before time 119905 ge 0 119884119894 is the size of the by-claim correspondingto the 119894th main claim and 119863119894 denotes the uncertain delaytime after the accident arrival time 120591119894 In this setup thefinite-time and infinite-time ruin probabilities can be definedrespectively as

120595 (119909 119905) = 119875 ( inf0le119904le119905

119880 (119904) lt 0 | 119880 (0) = 119909) 119905 ge 0 (2)

120595 (119909) = lim119905997888rarrinfin

120595 (119909 119905)= 119875 (inf

119904ge0119880 (119904) lt 0 | 119880 (0) = 119909) (3)

Such a by-claim risk model is initially studied by Yuen etal [1] andmay be of practical use in insurance For instance atraffic accident will cause different kinds of claims includingan immediate payoff for the car damage and another delayedmedical claim for injured drivers which may need a randomperiod of time to be settled

We are concerned with the asymptotic behavior for thefinite-time and infinite-time ruin probabilities since exceptfor few cases under ideal distributional assumptions a closed-form expression for the ruin probability 120595(119909 119905) or 120595(119909) ishardly availableThus themainstreamof the study focuses oncharacterizing the asymptotic behavior for ruin probabilitiesSome earlier works have been achieved with light-tailedand mutually independent claims that is both 119883119894 119894 isinN and 119884119894 119894 isin N are sequences of independent andidentically distributed (iid) and light-tailed nonnegativerandomvariables and they aremutually independent too seeYuen and Guo [2] Xiao and Guo [3] Wu and Li [4] and Liand Wu (2015)

Clearly modeling total surplus should address extremerisks which result from the marginal tails of and the taildependence between claims In the past decade more and

HindawiComplexityVolume 2019 Article ID 4582404 6 pageshttpsdoiorg10115520194582404

2 Complexity

more attention has been paid to heavy-tailed claims and thedependence between them In the presence of heavy-tailedclaim sizes Tang [5] Leipus and Siaulys [6 7] Yang et al [8]Wang et al [9] Liu et al [10] Yang and Yuen [11] Yang et al[12 13] andChen et al [14] investigated some independent ordependent riskmodels with no by-claims Li [15] considered adependent by-claim riskmodel with positive interest rate andextendedly varying tailed main claims and by-claims underthe pairwise quasi-asymptotical independence structure (seethe definition below) Fu and Li [16] further generalizedLirsquos result by allowing the insurance company to invest itssurplus into a portfolio consisting of risk-free and riskyassets Recently Li [17] studied a by-claim risk model withno interest rate under the setting that each pair of the mainclaim and by-claim follows the asymptotical independencestructure or possesses a bivariate regularly varying tail (hencefollows the asymptotical dependence structure)

In this paper we continue to consider the above by-claim risk model (1) Precisely speaking let (119883119894 119884119894) 119894 isinN be a sequence of nonnegative and iid random vectorsrepresenting the main claims and by-claims with genericrandom vector (119883 119884) having marginal distributions 119865119866 andfinite means 120583119865 120583119866 Assume that the counting process notnecessarily the renewal one 119873(119905) 119905 ge 0 is generated bythe identically distributed and nonnegative interarrival times120579119894 = 120591119894minus120591119894minus1 119894 isin N with finitemean function120582(119905) = 119864119873(119905)The delay times 119863119894 119894 isin N are a sequence of nonnegative(possibly degenerate at zero) random variables In additionwe assume that (119883119894 119884119894) 119894 isin N and 119873(119905) 119905 ge 0 aremutually independent

Inspired by the work of Li [17] our goal is to derivesharp asymptotics for the finite-time and infinite-time ruinprobabilities in some dependent by-claim risk models Wemake some meaningful adjustments and extensions on hismodel First we allow some certain dependence structureamong the interarrival times 120579119894 119894 isin N that is 119873(119905) 119905 ge0 is not necessarily a renewal counting process Secondwhen investigating the infinite-time ruin probability weextend both distributions 119865 and 119866 to be consistently varyingtailed in the case that the main claim 119883 and the by-claim119884 are asymptotically independent We also complementanother case that 119883 and 119884 are arbitrarily dependent when 119883dominates 119884

The rest of the paper is organized as follows Section 2prepares some preliminaries including some classes of heavy-tailed distributions and some concepts of dependence struc-tures Section 3 exhibits our main results The proofs of themain results as well as several lemmas needed for the proofsare relegated to Section 4

2 Preliminaries

Throughout the paper all limit relations are according to119909 997888rarr infin unless otherwise stated For two positive functions119891(119909) and 119892(119909) we write 119891(119909) sim 119892(119909) if lim(119891(119909)119892(119909)) = 1write 119891(119909) ≲ 119892(119909) or 119891(119909) ≳ 119892(119909) if lim sup(119891(119909)119892(119909)) le1 and write 119891(119909) = 119900(119892(119909)) if lim(119891(119909)119892(119909)) = 0Furthermore for two positive bivariate functions 119891(119909 119905) and119892(119909 119905) we write 119891(119909 119905) sim 119892(119909 119905) uniformly for 119905 isin 119860 if

lim sup119905isin119860|119891(119909 119905)119892(119909 119905) minus 1| = 0 For any 119909 119910 isin R we write119909 or 119910 = max119909 119910 and 119909 and 119910 = min119909 119910In this paper we shall restrict the claim distributions

to some classes of heavy-tailed distributions supported onR+ = [0infin) A commonly used class is the class C ofconsistently varying tailed distributions A distribution 119881belongs to the class C if lim119910darr1lim inf119909997888rarrinfin(119881(119909119910)119881(119909)) =1 where 119881(119909) = 1 minus 119881(119909) gt 0 for all 119909 ge 0 Closely relatedis a wider classL of long-tailed distributions A distribution119881 belongs to the class L if 119881(119909 + 119910) sim 119881(119909) for any 119910 gt0 An important subclass of C is the class R of regularlyvarying tailed specified by 119881(119909119910) sim 119910minus120572119881(119909) for any 119910 gt 0and some 120572 gt 0 denoted by 119881 isin Rminus120572 The reader isreferred to Embrechts et al [18] and Foss et al [19] for relateddiscussions on the properties of the subclasses of heavy-taileddistributions

Modeling a practical risk model must carefully addressasymptotical (in)dependence between each pair of the mainclaim and its corresponding by-claim or among the inter-arrival times Asymptotical dependence represents that theprobability of two components being simultaneously largecannot be negligible compared with one component beinglarge Generally two random variables 1205851 and 1205852 are said to beasymptotically dependent if they have a positive coefficient of(upper) tail dependence defined by

120581 = lim119875 (1205851 gt 119909 | 1205852 gt 119909) (4)

see eg McNeil et al [20]The following concept of bivariateregular variation exhibits the asymptotic dependence of tworandom variables A random pair (1205851 1205852) taking values inR2+ is said to follow a distribution with a bivariate regularly

varying (BRV) tail if there exist a distribution 119881 and anondegenerate (ie not identically 0) limit measure ] suchthat the following vague convergence holds

1119881 (119909)119875((1205851 1205852)119909 isin sdot) V997888rarr ] (sdot) on [0infin] 0 (5)

Necessarily 119881 is regularly varying Assume that 119881 isin Rminus120572 forsome 120572 gt 0 for which case we write (1205851 1205852) isin BRVminus120572(] 119881)By definition for a random pair (1205851 1205852) isin BRVminus120572(] 119881) if](1infin] gt 0 then its marginal tails satisfy

lim119875 (1205851 gt 119909)119881 (119909) = ] ((1 0) infin] gt 0

lim119875 (1205852 gt 119909)119881 (119909) = ] ((0 1) infin] gt 0

lim119875 (1205851 gt 119909 1205852 gt 119909)119881 (119909) = ] (1infin] gt 0

(6)

which imply 120581 = ](1infin]]((0 1)infin] gt 0 in (4) showingthat 1205851 and 1205852 are asymptotically dependent see eg Tang etal [21] and Tang and Yang [22] If 120581 = 0 in (4) then 1205851 and1205852 are asymptotically independent A natural extension is the

Complexity 3

concept of quasi-asymptotical independence (QAI) proposedby Chen and Yuen [23] and defined by

lim119875 (1205851 gt 119909 1205852 gt 119909)119875 (1205851 gt 119909) + 119875 (1205852 gt 119909) = 0 (7)

In our main results we shall use the structure of BRV orQAI to model the main claim and its corresponding by-claim and capture their tail dependence simultaneously Thefollowing concept of dependence structure is a special case ofasymptotical independence which will be used to describethe inter-arrival times A sequence of random variables120585119894 119894 isin N is said to be widely upper orthant dependent(WUOD) if there exists a finite real sequence 119892119880(119899) 119899 isin Nsatisfying for each 119899 isin N and for all 119909119894 isin R

119875( 119899⋂119894=1

(120585119894 gt 119909119894)) le 119892119880 (119899) 119899prod119894=1

119875 (120585119894 gt 119909119894) (8)

and it is said to be widely lower orthant dependent (WLOD)if there exists a finite real sequence 119892119871(119899) 119899 isin N satisfyingfor each 119899 isin N and for all 119909119894 isin R

119875( 119899⋂119894=1

(120585119894 le 119909119894)) le 119892119871 (119899) 119899prod119894=1

119875 (120585119894 le 119909119894) (9)

The sequence of 120585119894 119894 isin N is said to be widelyorthant dependent (WOD) if both (8) and (9) hold Here119892119880(119899) 119892119871(119899) 119899 isin N are called dominating coefficientsSuch dependence structures are introduced by Wang et al[24] Specially when 119892119880(119899) = 119892119871(119899) equiv 1 for any 119899 isin N in(8) and (9) the sequence of random variables 120585119894 119894 isin Nis said to be upper negatively dependent (UND) and lowernegatively dependent (LND) respectively The sequence issaid to be negatively dependent (ND) if it is both UNDand LND See Ebrahimi and Ghosh [25] Note that the NDstructure is weaker than the well-known negative associationsee Alam and Saxena [26] and Joag-Dev and Proschan [27]among others

3 Main Results

In this section we firstly introduce some assumptions onthe by-claim risk model (1) Throughout this paper let(119883119894 119884119894) 119894 isin N be a sequence of iid random pairs withgeneric random vector (119883 119884) having marginal distributions119865 and119866 both onR+ and finite means 120583119865 and 120583119866 respectivelylet 120579119894 119894 isin N be a sequence of nonnegative and dependentrandom variables with finite mean 120582minus1 which are indepen-dent of (119883119894 119884119894) 119894 isin N and let 119863119894 119894 isin N be a sequenceof nonnegative and upper bounded random variables that isthere exists a positive constant 119872 such that 119863119894 le 119872 for all119894 isin N In addition as usual in the risk model with no interestrate we require the safety loading condition

120588 fl120582119888 (120583119865 + 120583119866) lt 1 (10)

We remark that the reasonability for the upper bounded 119863119894rsquosis that by-claims should occur before the termination date due

to the insurance policy and the safety loading condition (10)excludes the trivial case 120595(119909) equiv 1

The following is our first main result which investigatesthe asymptotics for infinite-time ruin probability under threekinds of the dependence structures between119883 and 119884Theorem 1 Consider the by-claim risk model (1) with WLODinterarrival times 120579119894 119894 isin N satisfying

lim119899997888rarrinfin

119892119871 (119899) 119899minus119901 = 0 (11)

for some 119901 gt 0 Assume further that either of the following fourconditions is satisfied

Condition 1 120579119894 119894 isin N is a sequence of LND randomvariables

Condition 2 120579119894 119894 isin N is a sequence of WOD random vari-ables and there exists a positive and nondecreasing function119892(119909) such that 119892(119909) 997888rarr infin 119909minus119896119892(119909) dArr for some 0 lt 119896 lt 11198641205791119892(1205791) lt infin and 119892119880(119899) or 119892119871(119899) le 119892(119899) here 119909minus119896119892(119909) dArrmeans that there exists some 119862 gt 0 such that

119909minus1198961 119892 (1199091) ge 119862119909minus1198962 119892 (1199092) for all 1199092 gt 1199091 ge 0 (12)

Condition 3 120579119894 119894 isin N is a sequence of WOD randomvariables with 1198641205791199031 lt infin for some 119903 ge 2 and there exists aconstant 1199011 gt 0 such that

lim119899997888rarrinfin

(119892119880 (119899) or 119892119871 (119899)) 119899minus1199011 = 0 (13)

Condition 4 120579119894 119894 isin N is a sequence of WOD randomvariables with 1198641198901199041205791 lt infin for some 119904 gt 0 and for any 1199012 gt 0

lim119899997888rarrinfin

(119892119880 (119899) or 119892119871 (119899)) 119890minus1199012119899 = 0 (14)

(i) Assume that 119883 and 119884 are QAI If 119865 isin C and 119866 isin Cthen

120595 (119909) sim 120582119888 (1 minus 120588)intinfin

119909(119865 (119906) + 119866 (119906)) 119889119906 (15)

(ii) If (119883 119884) isin BRVminus120572(] 119867) for some 120572 gt 1 and](1infin] gt 0 then

120595 (119909) sim 120582] (119860)119888 (1 minus 120588)intinfin

119909119867(119906) 119889119906 (16)

where 119860 = (119909 119910) isin R2+ 119909 + 119910 gt 1

(iii) Assume that 119883 and 119884 are arbitrarily dependent If 119865 isinC and 119866(119909) = 119900(119865(119909)) then

120595 (119909) sim 120582119888 (1 minus 120588)intinfin

119909119865 (119906) 119889119906 (17)

We remark that if Condition 1 is satisfied that is 120579119894 119894 isinN is a sequence of LND random variables then (11) holdsautomatically

4 Complexity

Our second result is concerned with the asymptoticbehavior of the finite-time ruin probability Comparing withTheorem 2 of Li [17] we allow 119873(119905) 119905 ge 0 to be aquasi-renewal counting process generated by the WOD andidentically distributed inter-arrival times 120579119894 119894 isin N whereasit is required to be a Poisson process in Li [17]

Theorem 2 Under the conditions of Theorem 1 assume that120579119894 119894 isin N is a sequence of WOD and identically distributednonnegative random variables such that (11) and Condition 2in Theorem 1 are satisfied Then for every 119905 gt 0

lim119909997888rarrinfin

120595 (119909 119909119905)120595 (119909) = 1 minus (1 + 119888 (1 minus 120588) 119905)minus120574+1 (18)

holds if either of the following holds

(i) 119883 and 119884 are QAI and 119865 isin Rminus120572 119866 isin Rminus120573 for some120572 and 120573 gt 1 In this case 120574 = 120572 and 120573(ii) (119883 119884) isin B119877119881minus120572(] 119867) for some 120572 gt 1 In this case120574 = 120572(iii) 119883 and 119884 are arbitrarily dependent 119865 isin Rminus120572 for some120572 gt 1 and 119866(119909) = 119900(119865(119909)) In this case 120574 = 120572As pointed by Li [17] Theorem 2 shows that given

that the ruin occurs the ruin time divided by 119909 convergesin distribution to a Pareto random variable of type II Inaddition if 120579119894 119894 isin N is a sequence of ND random variablesthen (11) and Condition 2 in Theorem 1 are both satisfiedautomatically

4 Proofs of Main Results

Before the proofs of our two main results we firstly cite aseries of lemmas Consider a nonstandard risk model

119880lowast (119905) = 119909 + 119888119905 minus 119873(119905)sum119894=1

119885119894 119905 ge 0 (19)

where 119885119894 = 119883119894+119884119894 119894 isin N is a sequence of iid nonnegativerandom variables with generic random variable119885 As (2) and(3) define the infinite-time and finite-time ruin probabilitiesof risk model (19) with119880(sdot) replaced by119880lowast(sdot) denoting themby120595lowast(119909 119905) and120595lowast(119909) respectivelyThe following first lemmacomes from Corollary of Wang et al [9] see some similarresults in Yang et al [8]

Lemma 3 Consider a nonstandard risk model (19) in whichthe claims 119885119894 119894 isin N form a sequence of iid nonnegativerandom variables with common distribution 119865119885 and finitemean 119864119885 the interarrival times 120579119894 119894 isin N are a sequenceof nonnegative NLOD and identically distributed randomvariables with finite positive mean 1198641205791 = 120582minus1 such that (11)is satisfied and 119885119894 119894 isin N and 120579119894 119894 isin N are mutuallyindependent Assume further that either of Conditions 1ndash4 inTheorem 1 is satisfied If 119865119885 isin C and (10) holds then for any119879 such that 120582(119879) = 119864119873(119879) gt 0

120595lowast (119909 119905) sim 1120583int119909+120583120582(119905)

119909119865119885 (119906) 119889119906 (20)

holds uniformly for all 119905 isin [119879infin) where 120583 = 119888120582minus1 minus 119864119885Weremark that in Lemma 3 the uniformity of (20) implies

120595lowast (119909) sim 1120583intinfin

119909119865119885 (119906) 119889119906 (21)

We point out that Corollary 21 of Wang et al [9] gavesome more situations when discussing the uniform asymp-totics for 120595lowast(119909 119905) in which they allowed the distribution 119865119885to be strongly subexponential

The second lemma can be found in Yang et al [12]

Lemma 4 Let (119883 119884) be a random vector with marginaldistributions 119865 and 119866 both on R+ respectively but 119883 and 119884are arbitrarily dependent If 119865 isin C and 119866(119909) = 119900(119865(119909)) then

119875 (119883 + 119884 gt 119909) sim 119865 (119909) (22)

The third lemma gives the elementary renewal theoremfor WOD random variables which is due to Theorem 14 ofWang and Cheng [28]

Lemma5 Let 120579119894 119894 isin N be a sequence of nonnegativeWODand identically distributed randomvariables with finite positivemean 1198641205791 = 120582minus1 which constitutes a quasi-renewal countingprocess119873(119905) with mean function 120582(119905) = 119864119873(119905) If Condition 2in Theorem 1 is satisfied then 120582(119905) sim 120582119905 as 119905 997888rarr infin

As pointed out by Tang [5] it is easy to see that underthe conditions of Lemma 3 by using Lemma 5 relation (20)holds with 120582(119905) replaced by 120582119905 but the range of the uniformityfor 119905 becomes smaller

Lemma 6 Under the conditions of Lemma 3 if in addition120579119894 119894 isin N is a sequence of nonnegative WOD and identicallydistributed random variables with finite positive mean 1198641205791 =120582minus1 such that (11) and Condition 2 in Theorem 1 are satisfiedthen

120595lowast (119909 119905) sim 1120583int119909+120583120582119905

119909119865119885 (119906) 119889119906 (23)

holds uniformly for all 119905 isin [119891(119909)infin) where 119891(119909) is anarbitrary infinitely increasing function and 120583 = 119888120582minus1 minus 119864119885Proof By Lemmas 3 and 5 for any 0 lt 120598 lt 12 and all 119905 isin[119891(119909)infin) when 119909 is sufficiently large we have

1120583int119909+(1minus120598)120583120582119905

119909119865119885 (119906) 119889119906 le 120595lowast (119909 119905)

le 1120583int119909+(1+120598)120583120582119905

119909119865119885 (119906) 119889119906

(24)

Complexity 5

For the upper bound by the right-hand side of (24) we havethat for sufficiently large 119909 and all 119905 isin [119891(119909)infin)

120595lowast (119909 119905) le 1120583int119909+120583120582119905

119909119865119885 (119906) 119889119906

sdot (1 + int119909+(1+120598)120583120582119905119909+120583120582119905

119865119885 (119906) 119889119906int119909+120583120582119905119909

119865119885 (119906) 119889119906 )le 1 + 120598120583 int119909+120583120582119905

119909119865119885 (119906) 119889119906

(25)

which concludes the upper bound by the arbitrariness of 120598 gt0 We next consider the lower bound Similarly by the left-hand side of (24) we have that for sufficiently large 119909 and all119905 isin [119891(119909)infin)

120595lowast (119909 119905) ge 1120583int119909+120583120582119905

119909119865119885 (119906) 119889119906

sdot (1 minus int119909+120583120582119905119909+(1minus120598)120583120582119905

119865119885 (119906) 119889119906int119909+120583120582119905119909

119865119885 (119906) 119889119906 )ge 1120583int119909+120583120582119905

119909119865119885 (119906) 119889119906

sdot (1 minus 120598 sdot 119865119885 (119909 + (1 minus 120598) 120583120582119905)119865119885 (119909 + 120583120582119905) )ge 1 minus 1198880120598120583 int119909+120583120582119905

119909119865119885 (119906) 119889119906

(26)

for some 1198880 gt 0 where the last step holds because by 0 lt 120598 lt12 and 119865119885 isin C

lim sup119909997888rarrinfin

sup119905ge119891(119909)

119865119885 (119909 + (1 minus 120598) 120583120582119905)119865119885 (119909 + 120583120582119905)le lim sup

119909997888rarrinfinsup119905ge119891(119909)

119865119885 ((12) (119909 + 120583120582119905))119865119885 (119909 + 120583120582119905) lt infin (27)

This completes the proof of the lemma

We are now ready for the proofs of the two main results

Proof of Theorem 1 We firstly consider the tail probability of119885 = 119883 + 119884 In case (i) since 119883 and 119884 are QAI and 119865 isin C119866 isin C by usingTheorem 31 of Chen and Yuen [23] we have

119875 (119885 gt 119909) sim 119865 (119909) + 119866 (119909) (28)

implying 119865119885 isin C In case (ii) by (119883 119884) isin BRVminus120572(] 119867) wehave

119875 (119885 gt 119909) = 119875((119883 119884)119909 isin 119860) sim ] (119860)119867 (119909) (29)

implying 119865119885 isin Rminus120572 sub C In case (iii) applying Lemma 4gives 119875 (119885 gt 119909) sim 119865 (119909) (30)

also implying 119865119885 isin C The above three relations togetherwith (21) lead to

120595lowast (119909) sim 120582119888 (1 minus 120588)intinfin

119909119865119885 (119906) 119889119906

sim

120582119888 (1 minus 120588)intinfin

119909(119865 (119906) + 119866 (119906)) 119889119906 in case (i)120582] (119860)119888 (1 minus 120588)intinfin

119909119867(119906) 119889119906 in case (ii)120582119888 (1 minus 120588)intinfin

119909119865 (119906) 119889119906 in case (iii)

(31)

By the first equivalence of (31) we have120595lowast(119909+119910) sim 120595lowast(119909) forany fixed 119910 gt 0 because of 119865119885 isin C sub L Then we can followthe same line of Li [17] to verify120595 (119909) sim 120595lowast (119909) (32)

Therefore the desired relations (15)ndash(17) hold from (31) and(32)

Proof of Theorem 2 Under the conditions of Theorem 2 by(28)ndash(30) we have 119865119885 isin Rminus120574 with 120574 gt 1 specified inTheorem 2 By Lemma 6 we have that for every 119905 gt 0

120595lowast (119909 119909119905) sim 120582119888 (1 minus 120588)int119909(1+119888(1minus120588)119905)

119909119865119885 (119906) 119889119906 (33)

by this and (31) according to Karamatarsquos theorem we derive

lim119909997888rarrinfin

120595lowast (119909 119909119905)120595lowast (119909) = lim119909997888rarrinfin

int119909(1+119888(1minus120588)119905)119909

119865119885 (119906) 119889119906intinfin119909119865119885 (119906) 119889119906

= 1 minus (1 + 119888 (1 minus 120588) 119905)minus120574+1 (34)

Similarly to (32) Li [17] proved120595 (119909 119909119905) sim 120595lowast (119909 119909119905) (35)

Thus the desired relation (18) follows from (32) (34) and(35) This completes the proof of the theorem

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by National Natural Science Foun-dation of China (NSFC 71671166)

6 Complexity

References

[1] K C Yuen J Guo andKWNg ldquoOnultimate ruin in a delayed-claims risk modelrdquo Journal of Applied Probability vol 42 no 1pp 163ndash174 2005

[2] K C Yuen and J Y Guo ldquoRuin probabilities for time-correlatedclaims in the compound binomial modelrdquo Insurance Mathe-matics and Economics vol 29 no 1 pp 47ndash57 2001

[3] Y Xiao and J Guo ldquoThe compound binomial risk model withtime-correlated claimsrdquo InsuranceMathematics and Economicsvol 41 no 1 pp 124ndash133 2007

[4] L Wu and S Li ldquoOn a discrete time risk model with time-delayed claims and a constant dividend barrierrdquo InsuranceMarkets and Companies Analyses and Actuarial Computationsvol 3 pp 50ndash57 2012

[5] Q Tang ldquoAsymptotics for the finite time ruin probability in therenewalmodel with consistent variationrdquo StochasticModels vol20 no 3 pp 281ndash297 2004

[6] R Leipus and J Siaulys ldquoAsymptotic behaviour of the finite-time ruin probability under subexponential claim sizesrdquo Insur-ance Mathematics and Economics vol 40 no 3 pp 498ndash5082007

[7] R Leipus and J Siaulys ldquoAsymptotic behaviour of the finite-time ruin probability in renewal riskmodelsrdquoApplied StochasticModels in Business and Industry vol 25 no 3 pp 309ndash321 2009

[8] Y Yang R Leipus J Siaulys and Y Cang ldquoUniform estimatesfor the finite-time ruin probability in the dependent renewalrisk modelrdquo Journal of Mathematical Analysis and Applicationsvol 383 no 1 pp 215ndash225 2011

[9] Y Wang Z Cui K Wang and X Ma ldquoUniform asymptoticsof the finite-time ruin probability for all timesrdquo Journal ofMathematical Analysis and Applications vol 390 no 1 pp 208ndash223 2012

[10] X Liu Q Gao and YWang ldquoA note on a dependent risk modelwith constant interest raterdquo Statistics amp Probability Letters vol82 no 4 pp 707ndash712 2012

[11] Y Yang and K C Yuen ldquoFinite-time and infinite-time ruinprobabilities in a two-dimensional delayed renewal risk modelwith Sarmanov dependent claimsrdquo Journal of MathematicalAnalysis and Applications vol 442 no 2 pp 600ndash626 2016

[12] Y Yang K C Yuen and J-F Liu ldquoAsymptotics for ruinprobabilities in Levy-driven risk models with heavy-tailedclaimsrdquo Journal of Industrial and Management Optimizationvol 14 no 1 pp 231ndash247 2018

[13] Y Yang K Wang J Liu and Z Zhang ldquoAsymptotics fora bidimensional risk model with two geometric Levy priceprocessesrdquo Journal of Industrial andManagement Optimizationvol 15 no 2 pp 481ndash505 2019

[14] Y Chen Y Yang and T Jiang ldquoUniform asymptotics for finite-time ruin probability of a bidimensional risk modelrdquo Journal ofMathematical Analysis and Applications vol 469 no 2 pp 525ndash536 2019

[15] J Li ldquoOn pairwise quasi-asymptotically independent randomvariables and their applicationsrdquo Statistics amp Probability Lettersvol 83 no 9 pp 2081ndash2087 2013

[16] K Fu and H Li ldquoAsymptotic ruin probability of a renewalrisk model with dependent by-claims and stochastic returnsrdquoJournal of Computational andAppliedMathematics vol 306 pp154ndash165 2016

[17] J Li ldquoA note on a by-claim risk model asymptotic resultsrdquoCommunications in StatisticsmdashTheory and Methods vol 46 no22 pp 11289ndash11295 2017

[18] P Embrechts C Kluppelberg and T Mikosch Extremal Eventsin Finance and Insurance Springer New York NY USA 1997

[19] S FossDKorshunov and S ZacharyAn Introduction toHeavy-Tailed and Subexponential Distributions Springer New YorkNY USA 2011

[20] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Concepts Techniques and Tools Princeton Seriesin Finance Princeton University Press Princeton NJ USA2015

[21] Q Tang Z Tang and Y Yang ldquoSharp asymptotics for largeportfolio losses under extreme risksrdquo European Journal ofOperational Research vol 276 no 2 pp 710ndash722 2019

[22] Q Tang and Y Yang ldquoInterplay of insurance and financial risksin a stochastic environmentrdquo Scandinavian Actuarial Journalvol 2019 no 5 pp 432ndash451 2019

[23] Y Chen andK C Yuen ldquoSums of pairwise quasi-asymptoticallyindependent random variables with consistent variationrdquoStochastic Models vol 25 no 1 pp 76ndash89 2009

[24] K Wang Y Wang and Q Gao ldquoUniform asymptotics for thefinite-time ruin probability of a dependent risk model with aconstant interest raterdquo Methodology and Computing in AppliedProbability vol 15 no 1 pp 109ndash124 2013

[25] N Ebrahimi and M Ghosh ldquoMultivariate negative depen-dencerdquoCommunications in StatisticsmdashTheory andMethods vol10 no 4 pp 307ndash337 1981

[26] K Alam and K M L Saxena ldquoPositive dependence in multi-variate distributionsrdquo Communications in Statistics-Theory andMethods vol 10 pp 1183ndash1196 1981

[27] K Joag-Dev and F Proschan ldquoNegative Association of RandomVariables with ApplicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983

[28] Y Wang and D Cheng ldquoBasic renewal theorems for randomwalkswithwidely dependent incrementsrdquo Journal ofMathemat-ical Analysis and Applications vol 384 no 2 pp 597ndash606 2011

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 3

concept of quasi-asymptotical independence (QAI) proposedby Chen and Yuen [23] and defined by

lim119875 (1205851 gt 119909 1205852 gt 119909)119875 (1205851 gt 119909) + 119875 (1205852 gt 119909) = 0 (7)

In our main results we shall use the structure of BRV orQAI to model the main claim and its corresponding by-claim and capture their tail dependence simultaneously Thefollowing concept of dependence structure is a special case ofasymptotical independence which will be used to describethe inter-arrival times A sequence of random variables120585119894 119894 isin N is said to be widely upper orthant dependent(WUOD) if there exists a finite real sequence 119892119880(119899) 119899 isin Nsatisfying for each 119899 isin N and for all 119909119894 isin R

119875( 119899⋂119894=1

(120585119894 gt 119909119894)) le 119892119880 (119899) 119899prod119894=1

119875 (120585119894 gt 119909119894) (8)

and it is said to be widely lower orthant dependent (WLOD)if there exists a finite real sequence 119892119871(119899) 119899 isin N satisfyingfor each 119899 isin N and for all 119909119894 isin R

119875( 119899⋂119894=1

(120585119894 le 119909119894)) le 119892119871 (119899) 119899prod119894=1

119875 (120585119894 le 119909119894) (9)

The sequence of 120585119894 119894 isin N is said to be widelyorthant dependent (WOD) if both (8) and (9) hold Here119892119880(119899) 119892119871(119899) 119899 isin N are called dominating coefficientsSuch dependence structures are introduced by Wang et al[24] Specially when 119892119880(119899) = 119892119871(119899) equiv 1 for any 119899 isin N in(8) and (9) the sequence of random variables 120585119894 119894 isin Nis said to be upper negatively dependent (UND) and lowernegatively dependent (LND) respectively The sequence issaid to be negatively dependent (ND) if it is both UNDand LND See Ebrahimi and Ghosh [25] Note that the NDstructure is weaker than the well-known negative associationsee Alam and Saxena [26] and Joag-Dev and Proschan [27]among others

3 Main Results

In this section we firstly introduce some assumptions onthe by-claim risk model (1) Throughout this paper let(119883119894 119884119894) 119894 isin N be a sequence of iid random pairs withgeneric random vector (119883 119884) having marginal distributions119865 and119866 both onR+ and finite means 120583119865 and 120583119866 respectivelylet 120579119894 119894 isin N be a sequence of nonnegative and dependentrandom variables with finite mean 120582minus1 which are indepen-dent of (119883119894 119884119894) 119894 isin N and let 119863119894 119894 isin N be a sequenceof nonnegative and upper bounded random variables that isthere exists a positive constant 119872 such that 119863119894 le 119872 for all119894 isin N In addition as usual in the risk model with no interestrate we require the safety loading condition

120588 fl120582119888 (120583119865 + 120583119866) lt 1 (10)

We remark that the reasonability for the upper bounded 119863119894rsquosis that by-claims should occur before the termination date due

to the insurance policy and the safety loading condition (10)excludes the trivial case 120595(119909) equiv 1

The following is our first main result which investigatesthe asymptotics for infinite-time ruin probability under threekinds of the dependence structures between119883 and 119884Theorem 1 Consider the by-claim risk model (1) with WLODinterarrival times 120579119894 119894 isin N satisfying

lim119899997888rarrinfin

119892119871 (119899) 119899minus119901 = 0 (11)

for some 119901 gt 0 Assume further that either of the following fourconditions is satisfied

Condition 1 120579119894 119894 isin N is a sequence of LND randomvariables

Condition 2 120579119894 119894 isin N is a sequence of WOD random vari-ables and there exists a positive and nondecreasing function119892(119909) such that 119892(119909) 997888rarr infin 119909minus119896119892(119909) dArr for some 0 lt 119896 lt 11198641205791119892(1205791) lt infin and 119892119880(119899) or 119892119871(119899) le 119892(119899) here 119909minus119896119892(119909) dArrmeans that there exists some 119862 gt 0 such that

119909minus1198961 119892 (1199091) ge 119862119909minus1198962 119892 (1199092) for all 1199092 gt 1199091 ge 0 (12)

Condition 3 120579119894 119894 isin N is a sequence of WOD randomvariables with 1198641205791199031 lt infin for some 119903 ge 2 and there exists aconstant 1199011 gt 0 such that

lim119899997888rarrinfin

(119892119880 (119899) or 119892119871 (119899)) 119899minus1199011 = 0 (13)

Condition 4 120579119894 119894 isin N is a sequence of WOD randomvariables with 1198641198901199041205791 lt infin for some 119904 gt 0 and for any 1199012 gt 0

lim119899997888rarrinfin

(119892119880 (119899) or 119892119871 (119899)) 119890minus1199012119899 = 0 (14)

(i) Assume that 119883 and 119884 are QAI If 119865 isin C and 119866 isin Cthen

120595 (119909) sim 120582119888 (1 minus 120588)intinfin

119909(119865 (119906) + 119866 (119906)) 119889119906 (15)

(ii) If (119883 119884) isin BRVminus120572(] 119867) for some 120572 gt 1 and](1infin] gt 0 then

120595 (119909) sim 120582] (119860)119888 (1 minus 120588)intinfin

119909119867(119906) 119889119906 (16)

where 119860 = (119909 119910) isin R2+ 119909 + 119910 gt 1

(iii) Assume that 119883 and 119884 are arbitrarily dependent If 119865 isinC and 119866(119909) = 119900(119865(119909)) then

120595 (119909) sim 120582119888 (1 minus 120588)intinfin

119909119865 (119906) 119889119906 (17)

We remark that if Condition 1 is satisfied that is 120579119894 119894 isinN is a sequence of LND random variables then (11) holdsautomatically

4 Complexity

Our second result is concerned with the asymptoticbehavior of the finite-time ruin probability Comparing withTheorem 2 of Li [17] we allow 119873(119905) 119905 ge 0 to be aquasi-renewal counting process generated by the WOD andidentically distributed inter-arrival times 120579119894 119894 isin N whereasit is required to be a Poisson process in Li [17]

Theorem 2 Under the conditions of Theorem 1 assume that120579119894 119894 isin N is a sequence of WOD and identically distributednonnegative random variables such that (11) and Condition 2in Theorem 1 are satisfied Then for every 119905 gt 0

lim119909997888rarrinfin

120595 (119909 119909119905)120595 (119909) = 1 minus (1 + 119888 (1 minus 120588) 119905)minus120574+1 (18)

holds if either of the following holds

(i) 119883 and 119884 are QAI and 119865 isin Rminus120572 119866 isin Rminus120573 for some120572 and 120573 gt 1 In this case 120574 = 120572 and 120573(ii) (119883 119884) isin B119877119881minus120572(] 119867) for some 120572 gt 1 In this case120574 = 120572(iii) 119883 and 119884 are arbitrarily dependent 119865 isin Rminus120572 for some120572 gt 1 and 119866(119909) = 119900(119865(119909)) In this case 120574 = 120572As pointed by Li [17] Theorem 2 shows that given

that the ruin occurs the ruin time divided by 119909 convergesin distribution to a Pareto random variable of type II Inaddition if 120579119894 119894 isin N is a sequence of ND random variablesthen (11) and Condition 2 in Theorem 1 are both satisfiedautomatically

4 Proofs of Main Results

Before the proofs of our two main results we firstly cite aseries of lemmas Consider a nonstandard risk model

119880lowast (119905) = 119909 + 119888119905 minus 119873(119905)sum119894=1

119885119894 119905 ge 0 (19)

where 119885119894 = 119883119894+119884119894 119894 isin N is a sequence of iid nonnegativerandom variables with generic random variable119885 As (2) and(3) define the infinite-time and finite-time ruin probabilitiesof risk model (19) with119880(sdot) replaced by119880lowast(sdot) denoting themby120595lowast(119909 119905) and120595lowast(119909) respectivelyThe following first lemmacomes from Corollary of Wang et al [9] see some similarresults in Yang et al [8]

Lemma 3 Consider a nonstandard risk model (19) in whichthe claims 119885119894 119894 isin N form a sequence of iid nonnegativerandom variables with common distribution 119865119885 and finitemean 119864119885 the interarrival times 120579119894 119894 isin N are a sequenceof nonnegative NLOD and identically distributed randomvariables with finite positive mean 1198641205791 = 120582minus1 such that (11)is satisfied and 119885119894 119894 isin N and 120579119894 119894 isin N are mutuallyindependent Assume further that either of Conditions 1ndash4 inTheorem 1 is satisfied If 119865119885 isin C and (10) holds then for any119879 such that 120582(119879) = 119864119873(119879) gt 0

120595lowast (119909 119905) sim 1120583int119909+120583120582(119905)

119909119865119885 (119906) 119889119906 (20)

holds uniformly for all 119905 isin [119879infin) where 120583 = 119888120582minus1 minus 119864119885Weremark that in Lemma 3 the uniformity of (20) implies

120595lowast (119909) sim 1120583intinfin

119909119865119885 (119906) 119889119906 (21)

We point out that Corollary 21 of Wang et al [9] gavesome more situations when discussing the uniform asymp-totics for 120595lowast(119909 119905) in which they allowed the distribution 119865119885to be strongly subexponential

The second lemma can be found in Yang et al [12]

Lemma 4 Let (119883 119884) be a random vector with marginaldistributions 119865 and 119866 both on R+ respectively but 119883 and 119884are arbitrarily dependent If 119865 isin C and 119866(119909) = 119900(119865(119909)) then

119875 (119883 + 119884 gt 119909) sim 119865 (119909) (22)

The third lemma gives the elementary renewal theoremfor WOD random variables which is due to Theorem 14 ofWang and Cheng [28]

Lemma5 Let 120579119894 119894 isin N be a sequence of nonnegativeWODand identically distributed randomvariables with finite positivemean 1198641205791 = 120582minus1 which constitutes a quasi-renewal countingprocess119873(119905) with mean function 120582(119905) = 119864119873(119905) If Condition 2in Theorem 1 is satisfied then 120582(119905) sim 120582119905 as 119905 997888rarr infin

As pointed out by Tang [5] it is easy to see that underthe conditions of Lemma 3 by using Lemma 5 relation (20)holds with 120582(119905) replaced by 120582119905 but the range of the uniformityfor 119905 becomes smaller

Lemma 6 Under the conditions of Lemma 3 if in addition120579119894 119894 isin N is a sequence of nonnegative WOD and identicallydistributed random variables with finite positive mean 1198641205791 =120582minus1 such that (11) and Condition 2 in Theorem 1 are satisfiedthen

120595lowast (119909 119905) sim 1120583int119909+120583120582119905

119909119865119885 (119906) 119889119906 (23)

holds uniformly for all 119905 isin [119891(119909)infin) where 119891(119909) is anarbitrary infinitely increasing function and 120583 = 119888120582minus1 minus 119864119885Proof By Lemmas 3 and 5 for any 0 lt 120598 lt 12 and all 119905 isin[119891(119909)infin) when 119909 is sufficiently large we have

1120583int119909+(1minus120598)120583120582119905

119909119865119885 (119906) 119889119906 le 120595lowast (119909 119905)

le 1120583int119909+(1+120598)120583120582119905

119909119865119885 (119906) 119889119906

(24)

Complexity 5

For the upper bound by the right-hand side of (24) we havethat for sufficiently large 119909 and all 119905 isin [119891(119909)infin)

120595lowast (119909 119905) le 1120583int119909+120583120582119905

119909119865119885 (119906) 119889119906

sdot (1 + int119909+(1+120598)120583120582119905119909+120583120582119905

119865119885 (119906) 119889119906int119909+120583120582119905119909

119865119885 (119906) 119889119906 )le 1 + 120598120583 int119909+120583120582119905

119909119865119885 (119906) 119889119906

(25)

which concludes the upper bound by the arbitrariness of 120598 gt0 We next consider the lower bound Similarly by the left-hand side of (24) we have that for sufficiently large 119909 and all119905 isin [119891(119909)infin)

120595lowast (119909 119905) ge 1120583int119909+120583120582119905

119909119865119885 (119906) 119889119906

sdot (1 minus int119909+120583120582119905119909+(1minus120598)120583120582119905

119865119885 (119906) 119889119906int119909+120583120582119905119909

119865119885 (119906) 119889119906 )ge 1120583int119909+120583120582119905

119909119865119885 (119906) 119889119906

sdot (1 minus 120598 sdot 119865119885 (119909 + (1 minus 120598) 120583120582119905)119865119885 (119909 + 120583120582119905) )ge 1 minus 1198880120598120583 int119909+120583120582119905

119909119865119885 (119906) 119889119906

(26)

for some 1198880 gt 0 where the last step holds because by 0 lt 120598 lt12 and 119865119885 isin C

lim sup119909997888rarrinfin

sup119905ge119891(119909)

119865119885 (119909 + (1 minus 120598) 120583120582119905)119865119885 (119909 + 120583120582119905)le lim sup

119909997888rarrinfinsup119905ge119891(119909)

119865119885 ((12) (119909 + 120583120582119905))119865119885 (119909 + 120583120582119905) lt infin (27)

This completes the proof of the lemma

We are now ready for the proofs of the two main results

Proof of Theorem 1 We firstly consider the tail probability of119885 = 119883 + 119884 In case (i) since 119883 and 119884 are QAI and 119865 isin C119866 isin C by usingTheorem 31 of Chen and Yuen [23] we have

119875 (119885 gt 119909) sim 119865 (119909) + 119866 (119909) (28)

implying 119865119885 isin C In case (ii) by (119883 119884) isin BRVminus120572(] 119867) wehave

119875 (119885 gt 119909) = 119875((119883 119884)119909 isin 119860) sim ] (119860)119867 (119909) (29)

implying 119865119885 isin Rminus120572 sub C In case (iii) applying Lemma 4gives 119875 (119885 gt 119909) sim 119865 (119909) (30)

also implying 119865119885 isin C The above three relations togetherwith (21) lead to

120595lowast (119909) sim 120582119888 (1 minus 120588)intinfin

119909119865119885 (119906) 119889119906

sim

120582119888 (1 minus 120588)intinfin

119909(119865 (119906) + 119866 (119906)) 119889119906 in case (i)120582] (119860)119888 (1 minus 120588)intinfin

119909119867(119906) 119889119906 in case (ii)120582119888 (1 minus 120588)intinfin

119909119865 (119906) 119889119906 in case (iii)

(31)

By the first equivalence of (31) we have120595lowast(119909+119910) sim 120595lowast(119909) forany fixed 119910 gt 0 because of 119865119885 isin C sub L Then we can followthe same line of Li [17] to verify120595 (119909) sim 120595lowast (119909) (32)

Therefore the desired relations (15)ndash(17) hold from (31) and(32)

Proof of Theorem 2 Under the conditions of Theorem 2 by(28)ndash(30) we have 119865119885 isin Rminus120574 with 120574 gt 1 specified inTheorem 2 By Lemma 6 we have that for every 119905 gt 0

120595lowast (119909 119909119905) sim 120582119888 (1 minus 120588)int119909(1+119888(1minus120588)119905)

119909119865119885 (119906) 119889119906 (33)

by this and (31) according to Karamatarsquos theorem we derive

lim119909997888rarrinfin

120595lowast (119909 119909119905)120595lowast (119909) = lim119909997888rarrinfin

int119909(1+119888(1minus120588)119905)119909

119865119885 (119906) 119889119906intinfin119909119865119885 (119906) 119889119906

= 1 minus (1 + 119888 (1 minus 120588) 119905)minus120574+1 (34)

Similarly to (32) Li [17] proved120595 (119909 119909119905) sim 120595lowast (119909 119909119905) (35)

Thus the desired relation (18) follows from (32) (34) and(35) This completes the proof of the theorem

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by National Natural Science Foun-dation of China (NSFC 71671166)

6 Complexity

References

[1] K C Yuen J Guo andKWNg ldquoOnultimate ruin in a delayed-claims risk modelrdquo Journal of Applied Probability vol 42 no 1pp 163ndash174 2005

[2] K C Yuen and J Y Guo ldquoRuin probabilities for time-correlatedclaims in the compound binomial modelrdquo Insurance Mathe-matics and Economics vol 29 no 1 pp 47ndash57 2001

[3] Y Xiao and J Guo ldquoThe compound binomial risk model withtime-correlated claimsrdquo InsuranceMathematics and Economicsvol 41 no 1 pp 124ndash133 2007

[4] L Wu and S Li ldquoOn a discrete time risk model with time-delayed claims and a constant dividend barrierrdquo InsuranceMarkets and Companies Analyses and Actuarial Computationsvol 3 pp 50ndash57 2012

[5] Q Tang ldquoAsymptotics for the finite time ruin probability in therenewalmodel with consistent variationrdquo StochasticModels vol20 no 3 pp 281ndash297 2004

[6] R Leipus and J Siaulys ldquoAsymptotic behaviour of the finite-time ruin probability under subexponential claim sizesrdquo Insur-ance Mathematics and Economics vol 40 no 3 pp 498ndash5082007

[7] R Leipus and J Siaulys ldquoAsymptotic behaviour of the finite-time ruin probability in renewal riskmodelsrdquoApplied StochasticModels in Business and Industry vol 25 no 3 pp 309ndash321 2009

[8] Y Yang R Leipus J Siaulys and Y Cang ldquoUniform estimatesfor the finite-time ruin probability in the dependent renewalrisk modelrdquo Journal of Mathematical Analysis and Applicationsvol 383 no 1 pp 215ndash225 2011

[9] Y Wang Z Cui K Wang and X Ma ldquoUniform asymptoticsof the finite-time ruin probability for all timesrdquo Journal ofMathematical Analysis and Applications vol 390 no 1 pp 208ndash223 2012

[10] X Liu Q Gao and YWang ldquoA note on a dependent risk modelwith constant interest raterdquo Statistics amp Probability Letters vol82 no 4 pp 707ndash712 2012

[11] Y Yang and K C Yuen ldquoFinite-time and infinite-time ruinprobabilities in a two-dimensional delayed renewal risk modelwith Sarmanov dependent claimsrdquo Journal of MathematicalAnalysis and Applications vol 442 no 2 pp 600ndash626 2016

[12] Y Yang K C Yuen and J-F Liu ldquoAsymptotics for ruinprobabilities in Levy-driven risk models with heavy-tailedclaimsrdquo Journal of Industrial and Management Optimizationvol 14 no 1 pp 231ndash247 2018

[13] Y Yang K Wang J Liu and Z Zhang ldquoAsymptotics fora bidimensional risk model with two geometric Levy priceprocessesrdquo Journal of Industrial andManagement Optimizationvol 15 no 2 pp 481ndash505 2019

[14] Y Chen Y Yang and T Jiang ldquoUniform asymptotics for finite-time ruin probability of a bidimensional risk modelrdquo Journal ofMathematical Analysis and Applications vol 469 no 2 pp 525ndash536 2019

[15] J Li ldquoOn pairwise quasi-asymptotically independent randomvariables and their applicationsrdquo Statistics amp Probability Lettersvol 83 no 9 pp 2081ndash2087 2013

[16] K Fu and H Li ldquoAsymptotic ruin probability of a renewalrisk model with dependent by-claims and stochastic returnsrdquoJournal of Computational andAppliedMathematics vol 306 pp154ndash165 2016

[17] J Li ldquoA note on a by-claim risk model asymptotic resultsrdquoCommunications in StatisticsmdashTheory and Methods vol 46 no22 pp 11289ndash11295 2017

[18] P Embrechts C Kluppelberg and T Mikosch Extremal Eventsin Finance and Insurance Springer New York NY USA 1997

[19] S FossDKorshunov and S ZacharyAn Introduction toHeavy-Tailed and Subexponential Distributions Springer New YorkNY USA 2011

[20] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Concepts Techniques and Tools Princeton Seriesin Finance Princeton University Press Princeton NJ USA2015

[21] Q Tang Z Tang and Y Yang ldquoSharp asymptotics for largeportfolio losses under extreme risksrdquo European Journal ofOperational Research vol 276 no 2 pp 710ndash722 2019

[22] Q Tang and Y Yang ldquoInterplay of insurance and financial risksin a stochastic environmentrdquo Scandinavian Actuarial Journalvol 2019 no 5 pp 432ndash451 2019

[23] Y Chen andK C Yuen ldquoSums of pairwise quasi-asymptoticallyindependent random variables with consistent variationrdquoStochastic Models vol 25 no 1 pp 76ndash89 2009

[24] K Wang Y Wang and Q Gao ldquoUniform asymptotics for thefinite-time ruin probability of a dependent risk model with aconstant interest raterdquo Methodology and Computing in AppliedProbability vol 15 no 1 pp 109ndash124 2013

[25] N Ebrahimi and M Ghosh ldquoMultivariate negative depen-dencerdquoCommunications in StatisticsmdashTheory andMethods vol10 no 4 pp 307ndash337 1981

[26] K Alam and K M L Saxena ldquoPositive dependence in multi-variate distributionsrdquo Communications in Statistics-Theory andMethods vol 10 pp 1183ndash1196 1981

[27] K Joag-Dev and F Proschan ldquoNegative Association of RandomVariables with ApplicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983

[28] Y Wang and D Cheng ldquoBasic renewal theorems for randomwalkswithwidely dependent incrementsrdquo Journal ofMathemat-ical Analysis and Applications vol 384 no 2 pp 597ndash606 2011

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

4 Complexity

Our second result is concerned with the asymptoticbehavior of the finite-time ruin probability Comparing withTheorem 2 of Li [17] we allow 119873(119905) 119905 ge 0 to be aquasi-renewal counting process generated by the WOD andidentically distributed inter-arrival times 120579119894 119894 isin N whereasit is required to be a Poisson process in Li [17]

Theorem 2 Under the conditions of Theorem 1 assume that120579119894 119894 isin N is a sequence of WOD and identically distributednonnegative random variables such that (11) and Condition 2in Theorem 1 are satisfied Then for every 119905 gt 0

lim119909997888rarrinfin

120595 (119909 119909119905)120595 (119909) = 1 minus (1 + 119888 (1 minus 120588) 119905)minus120574+1 (18)

holds if either of the following holds

(i) 119883 and 119884 are QAI and 119865 isin Rminus120572 119866 isin Rminus120573 for some120572 and 120573 gt 1 In this case 120574 = 120572 and 120573(ii) (119883 119884) isin B119877119881minus120572(] 119867) for some 120572 gt 1 In this case120574 = 120572(iii) 119883 and 119884 are arbitrarily dependent 119865 isin Rminus120572 for some120572 gt 1 and 119866(119909) = 119900(119865(119909)) In this case 120574 = 120572As pointed by Li [17] Theorem 2 shows that given

that the ruin occurs the ruin time divided by 119909 convergesin distribution to a Pareto random variable of type II Inaddition if 120579119894 119894 isin N is a sequence of ND random variablesthen (11) and Condition 2 in Theorem 1 are both satisfiedautomatically

4 Proofs of Main Results

Before the proofs of our two main results we firstly cite aseries of lemmas Consider a nonstandard risk model

119880lowast (119905) = 119909 + 119888119905 minus 119873(119905)sum119894=1

119885119894 119905 ge 0 (19)

where 119885119894 = 119883119894+119884119894 119894 isin N is a sequence of iid nonnegativerandom variables with generic random variable119885 As (2) and(3) define the infinite-time and finite-time ruin probabilitiesof risk model (19) with119880(sdot) replaced by119880lowast(sdot) denoting themby120595lowast(119909 119905) and120595lowast(119909) respectivelyThe following first lemmacomes from Corollary of Wang et al [9] see some similarresults in Yang et al [8]

Lemma 3 Consider a nonstandard risk model (19) in whichthe claims 119885119894 119894 isin N form a sequence of iid nonnegativerandom variables with common distribution 119865119885 and finitemean 119864119885 the interarrival times 120579119894 119894 isin N are a sequenceof nonnegative NLOD and identically distributed randomvariables with finite positive mean 1198641205791 = 120582minus1 such that (11)is satisfied and 119885119894 119894 isin N and 120579119894 119894 isin N are mutuallyindependent Assume further that either of Conditions 1ndash4 inTheorem 1 is satisfied If 119865119885 isin C and (10) holds then for any119879 such that 120582(119879) = 119864119873(119879) gt 0

120595lowast (119909 119905) sim 1120583int119909+120583120582(119905)

119909119865119885 (119906) 119889119906 (20)

holds uniformly for all 119905 isin [119879infin) where 120583 = 119888120582minus1 minus 119864119885Weremark that in Lemma 3 the uniformity of (20) implies

120595lowast (119909) sim 1120583intinfin

119909119865119885 (119906) 119889119906 (21)

We point out that Corollary 21 of Wang et al [9] gavesome more situations when discussing the uniform asymp-totics for 120595lowast(119909 119905) in which they allowed the distribution 119865119885to be strongly subexponential

The second lemma can be found in Yang et al [12]

Lemma 4 Let (119883 119884) be a random vector with marginaldistributions 119865 and 119866 both on R+ respectively but 119883 and 119884are arbitrarily dependent If 119865 isin C and 119866(119909) = 119900(119865(119909)) then

119875 (119883 + 119884 gt 119909) sim 119865 (119909) (22)

The third lemma gives the elementary renewal theoremfor WOD random variables which is due to Theorem 14 ofWang and Cheng [28]

Lemma5 Let 120579119894 119894 isin N be a sequence of nonnegativeWODand identically distributed randomvariables with finite positivemean 1198641205791 = 120582minus1 which constitutes a quasi-renewal countingprocess119873(119905) with mean function 120582(119905) = 119864119873(119905) If Condition 2in Theorem 1 is satisfied then 120582(119905) sim 120582119905 as 119905 997888rarr infin

As pointed out by Tang [5] it is easy to see that underthe conditions of Lemma 3 by using Lemma 5 relation (20)holds with 120582(119905) replaced by 120582119905 but the range of the uniformityfor 119905 becomes smaller

Lemma 6 Under the conditions of Lemma 3 if in addition120579119894 119894 isin N is a sequence of nonnegative WOD and identicallydistributed random variables with finite positive mean 1198641205791 =120582minus1 such that (11) and Condition 2 in Theorem 1 are satisfiedthen

120595lowast (119909 119905) sim 1120583int119909+120583120582119905

119909119865119885 (119906) 119889119906 (23)

holds uniformly for all 119905 isin [119891(119909)infin) where 119891(119909) is anarbitrary infinitely increasing function and 120583 = 119888120582minus1 minus 119864119885Proof By Lemmas 3 and 5 for any 0 lt 120598 lt 12 and all 119905 isin[119891(119909)infin) when 119909 is sufficiently large we have

1120583int119909+(1minus120598)120583120582119905

119909119865119885 (119906) 119889119906 le 120595lowast (119909 119905)

le 1120583int119909+(1+120598)120583120582119905

119909119865119885 (119906) 119889119906

(24)

Complexity 5

For the upper bound by the right-hand side of (24) we havethat for sufficiently large 119909 and all 119905 isin [119891(119909)infin)

120595lowast (119909 119905) le 1120583int119909+120583120582119905

119909119865119885 (119906) 119889119906

sdot (1 + int119909+(1+120598)120583120582119905119909+120583120582119905

119865119885 (119906) 119889119906int119909+120583120582119905119909

119865119885 (119906) 119889119906 )le 1 + 120598120583 int119909+120583120582119905

119909119865119885 (119906) 119889119906

(25)

which concludes the upper bound by the arbitrariness of 120598 gt0 We next consider the lower bound Similarly by the left-hand side of (24) we have that for sufficiently large 119909 and all119905 isin [119891(119909)infin)

120595lowast (119909 119905) ge 1120583int119909+120583120582119905

119909119865119885 (119906) 119889119906

sdot (1 minus int119909+120583120582119905119909+(1minus120598)120583120582119905

119865119885 (119906) 119889119906int119909+120583120582119905119909

119865119885 (119906) 119889119906 )ge 1120583int119909+120583120582119905

119909119865119885 (119906) 119889119906

sdot (1 minus 120598 sdot 119865119885 (119909 + (1 minus 120598) 120583120582119905)119865119885 (119909 + 120583120582119905) )ge 1 minus 1198880120598120583 int119909+120583120582119905

119909119865119885 (119906) 119889119906

(26)

for some 1198880 gt 0 where the last step holds because by 0 lt 120598 lt12 and 119865119885 isin C

lim sup119909997888rarrinfin

sup119905ge119891(119909)

119865119885 (119909 + (1 minus 120598) 120583120582119905)119865119885 (119909 + 120583120582119905)le lim sup

119909997888rarrinfinsup119905ge119891(119909)

119865119885 ((12) (119909 + 120583120582119905))119865119885 (119909 + 120583120582119905) lt infin (27)

This completes the proof of the lemma

We are now ready for the proofs of the two main results

Proof of Theorem 1 We firstly consider the tail probability of119885 = 119883 + 119884 In case (i) since 119883 and 119884 are QAI and 119865 isin C119866 isin C by usingTheorem 31 of Chen and Yuen [23] we have

119875 (119885 gt 119909) sim 119865 (119909) + 119866 (119909) (28)

implying 119865119885 isin C In case (ii) by (119883 119884) isin BRVminus120572(] 119867) wehave

119875 (119885 gt 119909) = 119875((119883 119884)119909 isin 119860) sim ] (119860)119867 (119909) (29)

implying 119865119885 isin Rminus120572 sub C In case (iii) applying Lemma 4gives 119875 (119885 gt 119909) sim 119865 (119909) (30)

also implying 119865119885 isin C The above three relations togetherwith (21) lead to

120595lowast (119909) sim 120582119888 (1 minus 120588)intinfin

119909119865119885 (119906) 119889119906

sim

120582119888 (1 minus 120588)intinfin

119909(119865 (119906) + 119866 (119906)) 119889119906 in case (i)120582] (119860)119888 (1 minus 120588)intinfin

119909119867(119906) 119889119906 in case (ii)120582119888 (1 minus 120588)intinfin

119909119865 (119906) 119889119906 in case (iii)

(31)

By the first equivalence of (31) we have120595lowast(119909+119910) sim 120595lowast(119909) forany fixed 119910 gt 0 because of 119865119885 isin C sub L Then we can followthe same line of Li [17] to verify120595 (119909) sim 120595lowast (119909) (32)

Therefore the desired relations (15)ndash(17) hold from (31) and(32)

Proof of Theorem 2 Under the conditions of Theorem 2 by(28)ndash(30) we have 119865119885 isin Rminus120574 with 120574 gt 1 specified inTheorem 2 By Lemma 6 we have that for every 119905 gt 0

120595lowast (119909 119909119905) sim 120582119888 (1 minus 120588)int119909(1+119888(1minus120588)119905)

119909119865119885 (119906) 119889119906 (33)

by this and (31) according to Karamatarsquos theorem we derive

lim119909997888rarrinfin

120595lowast (119909 119909119905)120595lowast (119909) = lim119909997888rarrinfin

int119909(1+119888(1minus120588)119905)119909

119865119885 (119906) 119889119906intinfin119909119865119885 (119906) 119889119906

= 1 minus (1 + 119888 (1 minus 120588) 119905)minus120574+1 (34)

Similarly to (32) Li [17] proved120595 (119909 119909119905) sim 120595lowast (119909 119909119905) (35)

Thus the desired relation (18) follows from (32) (34) and(35) This completes the proof of the theorem

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by National Natural Science Foun-dation of China (NSFC 71671166)

6 Complexity

References

[1] K C Yuen J Guo andKWNg ldquoOnultimate ruin in a delayed-claims risk modelrdquo Journal of Applied Probability vol 42 no 1pp 163ndash174 2005

[2] K C Yuen and J Y Guo ldquoRuin probabilities for time-correlatedclaims in the compound binomial modelrdquo Insurance Mathe-matics and Economics vol 29 no 1 pp 47ndash57 2001

[3] Y Xiao and J Guo ldquoThe compound binomial risk model withtime-correlated claimsrdquo InsuranceMathematics and Economicsvol 41 no 1 pp 124ndash133 2007

[4] L Wu and S Li ldquoOn a discrete time risk model with time-delayed claims and a constant dividend barrierrdquo InsuranceMarkets and Companies Analyses and Actuarial Computationsvol 3 pp 50ndash57 2012

[5] Q Tang ldquoAsymptotics for the finite time ruin probability in therenewalmodel with consistent variationrdquo StochasticModels vol20 no 3 pp 281ndash297 2004

[6] R Leipus and J Siaulys ldquoAsymptotic behaviour of the finite-time ruin probability under subexponential claim sizesrdquo Insur-ance Mathematics and Economics vol 40 no 3 pp 498ndash5082007

[7] R Leipus and J Siaulys ldquoAsymptotic behaviour of the finite-time ruin probability in renewal riskmodelsrdquoApplied StochasticModels in Business and Industry vol 25 no 3 pp 309ndash321 2009

[8] Y Yang R Leipus J Siaulys and Y Cang ldquoUniform estimatesfor the finite-time ruin probability in the dependent renewalrisk modelrdquo Journal of Mathematical Analysis and Applicationsvol 383 no 1 pp 215ndash225 2011

[9] Y Wang Z Cui K Wang and X Ma ldquoUniform asymptoticsof the finite-time ruin probability for all timesrdquo Journal ofMathematical Analysis and Applications vol 390 no 1 pp 208ndash223 2012

[10] X Liu Q Gao and YWang ldquoA note on a dependent risk modelwith constant interest raterdquo Statistics amp Probability Letters vol82 no 4 pp 707ndash712 2012

[11] Y Yang and K C Yuen ldquoFinite-time and infinite-time ruinprobabilities in a two-dimensional delayed renewal risk modelwith Sarmanov dependent claimsrdquo Journal of MathematicalAnalysis and Applications vol 442 no 2 pp 600ndash626 2016

[12] Y Yang K C Yuen and J-F Liu ldquoAsymptotics for ruinprobabilities in Levy-driven risk models with heavy-tailedclaimsrdquo Journal of Industrial and Management Optimizationvol 14 no 1 pp 231ndash247 2018

[13] Y Yang K Wang J Liu and Z Zhang ldquoAsymptotics fora bidimensional risk model with two geometric Levy priceprocessesrdquo Journal of Industrial andManagement Optimizationvol 15 no 2 pp 481ndash505 2019

[14] Y Chen Y Yang and T Jiang ldquoUniform asymptotics for finite-time ruin probability of a bidimensional risk modelrdquo Journal ofMathematical Analysis and Applications vol 469 no 2 pp 525ndash536 2019

[15] J Li ldquoOn pairwise quasi-asymptotically independent randomvariables and their applicationsrdquo Statistics amp Probability Lettersvol 83 no 9 pp 2081ndash2087 2013

[16] K Fu and H Li ldquoAsymptotic ruin probability of a renewalrisk model with dependent by-claims and stochastic returnsrdquoJournal of Computational andAppliedMathematics vol 306 pp154ndash165 2016

[17] J Li ldquoA note on a by-claim risk model asymptotic resultsrdquoCommunications in StatisticsmdashTheory and Methods vol 46 no22 pp 11289ndash11295 2017

[18] P Embrechts C Kluppelberg and T Mikosch Extremal Eventsin Finance and Insurance Springer New York NY USA 1997

[19] S FossDKorshunov and S ZacharyAn Introduction toHeavy-Tailed and Subexponential Distributions Springer New YorkNY USA 2011

[20] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Concepts Techniques and Tools Princeton Seriesin Finance Princeton University Press Princeton NJ USA2015

[21] Q Tang Z Tang and Y Yang ldquoSharp asymptotics for largeportfolio losses under extreme risksrdquo European Journal ofOperational Research vol 276 no 2 pp 710ndash722 2019

[22] Q Tang and Y Yang ldquoInterplay of insurance and financial risksin a stochastic environmentrdquo Scandinavian Actuarial Journalvol 2019 no 5 pp 432ndash451 2019

[23] Y Chen andK C Yuen ldquoSums of pairwise quasi-asymptoticallyindependent random variables with consistent variationrdquoStochastic Models vol 25 no 1 pp 76ndash89 2009

[24] K Wang Y Wang and Q Gao ldquoUniform asymptotics for thefinite-time ruin probability of a dependent risk model with aconstant interest raterdquo Methodology and Computing in AppliedProbability vol 15 no 1 pp 109ndash124 2013

[25] N Ebrahimi and M Ghosh ldquoMultivariate negative depen-dencerdquoCommunications in StatisticsmdashTheory andMethods vol10 no 4 pp 307ndash337 1981

[26] K Alam and K M L Saxena ldquoPositive dependence in multi-variate distributionsrdquo Communications in Statistics-Theory andMethods vol 10 pp 1183ndash1196 1981

[27] K Joag-Dev and F Proschan ldquoNegative Association of RandomVariables with ApplicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983

[28] Y Wang and D Cheng ldquoBasic renewal theorems for randomwalkswithwidely dependent incrementsrdquo Journal ofMathemat-ical Analysis and Applications vol 384 no 2 pp 597ndash606 2011

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 5

For the upper bound by the right-hand side of (24) we havethat for sufficiently large 119909 and all 119905 isin [119891(119909)infin)

120595lowast (119909 119905) le 1120583int119909+120583120582119905

119909119865119885 (119906) 119889119906

sdot (1 + int119909+(1+120598)120583120582119905119909+120583120582119905

119865119885 (119906) 119889119906int119909+120583120582119905119909

119865119885 (119906) 119889119906 )le 1 + 120598120583 int119909+120583120582119905

119909119865119885 (119906) 119889119906

(25)

which concludes the upper bound by the arbitrariness of 120598 gt0 We next consider the lower bound Similarly by the left-hand side of (24) we have that for sufficiently large 119909 and all119905 isin [119891(119909)infin)

120595lowast (119909 119905) ge 1120583int119909+120583120582119905

119909119865119885 (119906) 119889119906

sdot (1 minus int119909+120583120582119905119909+(1minus120598)120583120582119905

119865119885 (119906) 119889119906int119909+120583120582119905119909

119865119885 (119906) 119889119906 )ge 1120583int119909+120583120582119905

119909119865119885 (119906) 119889119906

sdot (1 minus 120598 sdot 119865119885 (119909 + (1 minus 120598) 120583120582119905)119865119885 (119909 + 120583120582119905) )ge 1 minus 1198880120598120583 int119909+120583120582119905

119909119865119885 (119906) 119889119906

(26)

for some 1198880 gt 0 where the last step holds because by 0 lt 120598 lt12 and 119865119885 isin C

lim sup119909997888rarrinfin

sup119905ge119891(119909)

119865119885 (119909 + (1 minus 120598) 120583120582119905)119865119885 (119909 + 120583120582119905)le lim sup

119909997888rarrinfinsup119905ge119891(119909)

119865119885 ((12) (119909 + 120583120582119905))119865119885 (119909 + 120583120582119905) lt infin (27)

This completes the proof of the lemma

We are now ready for the proofs of the two main results

Proof of Theorem 1 We firstly consider the tail probability of119885 = 119883 + 119884 In case (i) since 119883 and 119884 are QAI and 119865 isin C119866 isin C by usingTheorem 31 of Chen and Yuen [23] we have

119875 (119885 gt 119909) sim 119865 (119909) + 119866 (119909) (28)

implying 119865119885 isin C In case (ii) by (119883 119884) isin BRVminus120572(] 119867) wehave

119875 (119885 gt 119909) = 119875((119883 119884)119909 isin 119860) sim ] (119860)119867 (119909) (29)

implying 119865119885 isin Rminus120572 sub C In case (iii) applying Lemma 4gives 119875 (119885 gt 119909) sim 119865 (119909) (30)

also implying 119865119885 isin C The above three relations togetherwith (21) lead to

120595lowast (119909) sim 120582119888 (1 minus 120588)intinfin

119909119865119885 (119906) 119889119906

sim

120582119888 (1 minus 120588)intinfin

119909(119865 (119906) + 119866 (119906)) 119889119906 in case (i)120582] (119860)119888 (1 minus 120588)intinfin

119909119867(119906) 119889119906 in case (ii)120582119888 (1 minus 120588)intinfin

119909119865 (119906) 119889119906 in case (iii)

(31)

By the first equivalence of (31) we have120595lowast(119909+119910) sim 120595lowast(119909) forany fixed 119910 gt 0 because of 119865119885 isin C sub L Then we can followthe same line of Li [17] to verify120595 (119909) sim 120595lowast (119909) (32)

Therefore the desired relations (15)ndash(17) hold from (31) and(32)

Proof of Theorem 2 Under the conditions of Theorem 2 by(28)ndash(30) we have 119865119885 isin Rminus120574 with 120574 gt 1 specified inTheorem 2 By Lemma 6 we have that for every 119905 gt 0

120595lowast (119909 119909119905) sim 120582119888 (1 minus 120588)int119909(1+119888(1minus120588)119905)

119909119865119885 (119906) 119889119906 (33)

by this and (31) according to Karamatarsquos theorem we derive

lim119909997888rarrinfin

120595lowast (119909 119909119905)120595lowast (119909) = lim119909997888rarrinfin

int119909(1+119888(1minus120588)119905)119909

119865119885 (119906) 119889119906intinfin119909119865119885 (119906) 119889119906

= 1 minus (1 + 119888 (1 minus 120588) 119905)minus120574+1 (34)

Similarly to (32) Li [17] proved120595 (119909 119909119905) sim 120595lowast (119909 119909119905) (35)

Thus the desired relation (18) follows from (32) (34) and(35) This completes the proof of the theorem

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by National Natural Science Foun-dation of China (NSFC 71671166)

6 Complexity

References

[1] K C Yuen J Guo andKWNg ldquoOnultimate ruin in a delayed-claims risk modelrdquo Journal of Applied Probability vol 42 no 1pp 163ndash174 2005

[2] K C Yuen and J Y Guo ldquoRuin probabilities for time-correlatedclaims in the compound binomial modelrdquo Insurance Mathe-matics and Economics vol 29 no 1 pp 47ndash57 2001

[3] Y Xiao and J Guo ldquoThe compound binomial risk model withtime-correlated claimsrdquo InsuranceMathematics and Economicsvol 41 no 1 pp 124ndash133 2007

[4] L Wu and S Li ldquoOn a discrete time risk model with time-delayed claims and a constant dividend barrierrdquo InsuranceMarkets and Companies Analyses and Actuarial Computationsvol 3 pp 50ndash57 2012

[5] Q Tang ldquoAsymptotics for the finite time ruin probability in therenewalmodel with consistent variationrdquo StochasticModels vol20 no 3 pp 281ndash297 2004

[6] R Leipus and J Siaulys ldquoAsymptotic behaviour of the finite-time ruin probability under subexponential claim sizesrdquo Insur-ance Mathematics and Economics vol 40 no 3 pp 498ndash5082007

[7] R Leipus and J Siaulys ldquoAsymptotic behaviour of the finite-time ruin probability in renewal riskmodelsrdquoApplied StochasticModels in Business and Industry vol 25 no 3 pp 309ndash321 2009

[8] Y Yang R Leipus J Siaulys and Y Cang ldquoUniform estimatesfor the finite-time ruin probability in the dependent renewalrisk modelrdquo Journal of Mathematical Analysis and Applicationsvol 383 no 1 pp 215ndash225 2011

[9] Y Wang Z Cui K Wang and X Ma ldquoUniform asymptoticsof the finite-time ruin probability for all timesrdquo Journal ofMathematical Analysis and Applications vol 390 no 1 pp 208ndash223 2012

[10] X Liu Q Gao and YWang ldquoA note on a dependent risk modelwith constant interest raterdquo Statistics amp Probability Letters vol82 no 4 pp 707ndash712 2012

[11] Y Yang and K C Yuen ldquoFinite-time and infinite-time ruinprobabilities in a two-dimensional delayed renewal risk modelwith Sarmanov dependent claimsrdquo Journal of MathematicalAnalysis and Applications vol 442 no 2 pp 600ndash626 2016

[12] Y Yang K C Yuen and J-F Liu ldquoAsymptotics for ruinprobabilities in Levy-driven risk models with heavy-tailedclaimsrdquo Journal of Industrial and Management Optimizationvol 14 no 1 pp 231ndash247 2018

[13] Y Yang K Wang J Liu and Z Zhang ldquoAsymptotics fora bidimensional risk model with two geometric Levy priceprocessesrdquo Journal of Industrial andManagement Optimizationvol 15 no 2 pp 481ndash505 2019

[14] Y Chen Y Yang and T Jiang ldquoUniform asymptotics for finite-time ruin probability of a bidimensional risk modelrdquo Journal ofMathematical Analysis and Applications vol 469 no 2 pp 525ndash536 2019

[15] J Li ldquoOn pairwise quasi-asymptotically independent randomvariables and their applicationsrdquo Statistics amp Probability Lettersvol 83 no 9 pp 2081ndash2087 2013

[16] K Fu and H Li ldquoAsymptotic ruin probability of a renewalrisk model with dependent by-claims and stochastic returnsrdquoJournal of Computational andAppliedMathematics vol 306 pp154ndash165 2016

[17] J Li ldquoA note on a by-claim risk model asymptotic resultsrdquoCommunications in StatisticsmdashTheory and Methods vol 46 no22 pp 11289ndash11295 2017

[18] P Embrechts C Kluppelberg and T Mikosch Extremal Eventsin Finance and Insurance Springer New York NY USA 1997

[19] S FossDKorshunov and S ZacharyAn Introduction toHeavy-Tailed and Subexponential Distributions Springer New YorkNY USA 2011

[20] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Concepts Techniques and Tools Princeton Seriesin Finance Princeton University Press Princeton NJ USA2015

[21] Q Tang Z Tang and Y Yang ldquoSharp asymptotics for largeportfolio losses under extreme risksrdquo European Journal ofOperational Research vol 276 no 2 pp 710ndash722 2019

[22] Q Tang and Y Yang ldquoInterplay of insurance and financial risksin a stochastic environmentrdquo Scandinavian Actuarial Journalvol 2019 no 5 pp 432ndash451 2019

[23] Y Chen andK C Yuen ldquoSums of pairwise quasi-asymptoticallyindependent random variables with consistent variationrdquoStochastic Models vol 25 no 1 pp 76ndash89 2009

[24] K Wang Y Wang and Q Gao ldquoUniform asymptotics for thefinite-time ruin probability of a dependent risk model with aconstant interest raterdquo Methodology and Computing in AppliedProbability vol 15 no 1 pp 109ndash124 2013

[25] N Ebrahimi and M Ghosh ldquoMultivariate negative depen-dencerdquoCommunications in StatisticsmdashTheory andMethods vol10 no 4 pp 307ndash337 1981

[26] K Alam and K M L Saxena ldquoPositive dependence in multi-variate distributionsrdquo Communications in Statistics-Theory andMethods vol 10 pp 1183ndash1196 1981

[27] K Joag-Dev and F Proschan ldquoNegative Association of RandomVariables with ApplicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983

[28] Y Wang and D Cheng ldquoBasic renewal theorems for randomwalkswithwidely dependent incrementsrdquo Journal ofMathemat-ical Analysis and Applications vol 384 no 2 pp 597ndash606 2011

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Complexity

References

[1] K C Yuen J Guo andKWNg ldquoOnultimate ruin in a delayed-claims risk modelrdquo Journal of Applied Probability vol 42 no 1pp 163ndash174 2005

[2] K C Yuen and J Y Guo ldquoRuin probabilities for time-correlatedclaims in the compound binomial modelrdquo Insurance Mathe-matics and Economics vol 29 no 1 pp 47ndash57 2001

[3] Y Xiao and J Guo ldquoThe compound binomial risk model withtime-correlated claimsrdquo InsuranceMathematics and Economicsvol 41 no 1 pp 124ndash133 2007

[4] L Wu and S Li ldquoOn a discrete time risk model with time-delayed claims and a constant dividend barrierrdquo InsuranceMarkets and Companies Analyses and Actuarial Computationsvol 3 pp 50ndash57 2012

[5] Q Tang ldquoAsymptotics for the finite time ruin probability in therenewalmodel with consistent variationrdquo StochasticModels vol20 no 3 pp 281ndash297 2004

[6] R Leipus and J Siaulys ldquoAsymptotic behaviour of the finite-time ruin probability under subexponential claim sizesrdquo Insur-ance Mathematics and Economics vol 40 no 3 pp 498ndash5082007

[7] R Leipus and J Siaulys ldquoAsymptotic behaviour of the finite-time ruin probability in renewal riskmodelsrdquoApplied StochasticModels in Business and Industry vol 25 no 3 pp 309ndash321 2009

[8] Y Yang R Leipus J Siaulys and Y Cang ldquoUniform estimatesfor the finite-time ruin probability in the dependent renewalrisk modelrdquo Journal of Mathematical Analysis and Applicationsvol 383 no 1 pp 215ndash225 2011

[9] Y Wang Z Cui K Wang and X Ma ldquoUniform asymptoticsof the finite-time ruin probability for all timesrdquo Journal ofMathematical Analysis and Applications vol 390 no 1 pp 208ndash223 2012

[10] X Liu Q Gao and YWang ldquoA note on a dependent risk modelwith constant interest raterdquo Statistics amp Probability Letters vol82 no 4 pp 707ndash712 2012

[11] Y Yang and K C Yuen ldquoFinite-time and infinite-time ruinprobabilities in a two-dimensional delayed renewal risk modelwith Sarmanov dependent claimsrdquo Journal of MathematicalAnalysis and Applications vol 442 no 2 pp 600ndash626 2016

[12] Y Yang K C Yuen and J-F Liu ldquoAsymptotics for ruinprobabilities in Levy-driven risk models with heavy-tailedclaimsrdquo Journal of Industrial and Management Optimizationvol 14 no 1 pp 231ndash247 2018

[13] Y Yang K Wang J Liu and Z Zhang ldquoAsymptotics fora bidimensional risk model with two geometric Levy priceprocessesrdquo Journal of Industrial andManagement Optimizationvol 15 no 2 pp 481ndash505 2019

[14] Y Chen Y Yang and T Jiang ldquoUniform asymptotics for finite-time ruin probability of a bidimensional risk modelrdquo Journal ofMathematical Analysis and Applications vol 469 no 2 pp 525ndash536 2019

[15] J Li ldquoOn pairwise quasi-asymptotically independent randomvariables and their applicationsrdquo Statistics amp Probability Lettersvol 83 no 9 pp 2081ndash2087 2013

[16] K Fu and H Li ldquoAsymptotic ruin probability of a renewalrisk model with dependent by-claims and stochastic returnsrdquoJournal of Computational andAppliedMathematics vol 306 pp154ndash165 2016

[17] J Li ldquoA note on a by-claim risk model asymptotic resultsrdquoCommunications in StatisticsmdashTheory and Methods vol 46 no22 pp 11289ndash11295 2017

[18] P Embrechts C Kluppelberg and T Mikosch Extremal Eventsin Finance and Insurance Springer New York NY USA 1997

[19] S FossDKorshunov and S ZacharyAn Introduction toHeavy-Tailed and Subexponential Distributions Springer New YorkNY USA 2011

[20] A J McNeil R Frey and P Embrechts Quantitative RiskManagement Concepts Techniques and Tools Princeton Seriesin Finance Princeton University Press Princeton NJ USA2015

[21] Q Tang Z Tang and Y Yang ldquoSharp asymptotics for largeportfolio losses under extreme risksrdquo European Journal ofOperational Research vol 276 no 2 pp 710ndash722 2019

[22] Q Tang and Y Yang ldquoInterplay of insurance and financial risksin a stochastic environmentrdquo Scandinavian Actuarial Journalvol 2019 no 5 pp 432ndash451 2019

[23] Y Chen andK C Yuen ldquoSums of pairwise quasi-asymptoticallyindependent random variables with consistent variationrdquoStochastic Models vol 25 no 1 pp 76ndash89 2009

[24] K Wang Y Wang and Q Gao ldquoUniform asymptotics for thefinite-time ruin probability of a dependent risk model with aconstant interest raterdquo Methodology and Computing in AppliedProbability vol 15 no 1 pp 109ndash124 2013

[25] N Ebrahimi and M Ghosh ldquoMultivariate negative depen-dencerdquoCommunications in StatisticsmdashTheory andMethods vol10 no 4 pp 307ndash337 1981

[26] K Alam and K M L Saxena ldquoPositive dependence in multi-variate distributionsrdquo Communications in Statistics-Theory andMethods vol 10 pp 1183ndash1196 1981

[27] K Joag-Dev and F Proschan ldquoNegative Association of RandomVariables with ApplicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983

[28] Y Wang and D Cheng ldquoBasic renewal theorems for randomwalkswithwidely dependent incrementsrdquo Journal ofMathemat-ical Analysis and Applications vol 384 no 2 pp 597ndash606 2011

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom