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Strong Invariance Principle for
Randomly Stopped Stochastic
Processes
Andrii Andrusiv, Nadiia Zinchenko
Introduction
Let be sequence of non-negative i.i.d.r.v. with d.f.
and ch.f. ,
Denote
, , ,
where [a] is entire of a>0.
Let be sequence of non-negative i.i.d.r.v. independent
of with d.f. and ch.f. ,
Denote
, ,
and define the renewal counting process as
n
ii
XnS1
)( 0)0(S ])([)( zSzS
)1:( iXi
F
mEXi
n
ii
ZnZ1
)( 0)0(Z ])([)( aZaZ
})(:0inf{)( txZxtN
)1:( iZi
)1:( iXi 1
F 1 1i
EZ
Main aim
The main aim of this talk is to study the asymptotic behavior
of the random processes S(N(t)) and N(t) when F and F1 are
heavy tailed. This problem has a deep relation with
investigations of risk process U(t) and approximation of ruin
probabilities in Sparre Anderssen collective risk model
)(
1
)(tN
kk
XctutU
)(
1
)(~
1
)(tN
ii
tN
kk
XVutU
Weak invariance principle
Limit theorems for risk process such as (weak) invariance
principle which constitute the weak convergence of U(t) to the
Wiener process W(t) with drift (when , ) or to
α-stable Lévy process (when , ) lead to
useful approximation of the ruin probability as a distribution of
infimum of the Wiener process (Iglehard (1969), Grandell
(1991), Embrechts, Klüppelberg and Mikosch (1997)) or
infimum of the corresponding α-stable process (Furrer, Michna
and Weron (1997), Furrer (1998)).
2
iEZ2
iEX
)(tY 2
iEZ2
iEX
Strong invariance principle
Strong invariance principle (almost sure approximation) is a
general name for the class of limit theorems which ensure the
possibility to construct and Lévy process ,
on the same probability space in such a way that with
probability 1
as
as
were approximation error (rate) is non-random function
depending only on assumption posed on .
)1:( iXi
)(tY 0t
))((|)()(| trotYmttS
))((|)()(| trOtYmttS
t
t
)(r
iX
Strong invariance principle for partial
sums
Based on Skorokhod embeded scheme Strassen (1964) proved
the first variant of the strong invariance principle. In 1970-95
the further investigations were carried out by a number of
authors, so firstly I will summarize their results.
Th.A1. It is possible to construct partial sum process ,
and a standard Wiener process , in such a way that a.s.
,
with:
(i) iff , ,
(ii) iff ,
(iii) can be changed on iff
for some
)(tS 0t
)(tW 0t
))((|)()(| trotWmttS
pttr 1)( p
iXE || 2p
21))log(log()( tttr 2||i
XE
))(( tro )(log))(( tOtrOiuX
Ee
0u
Domain of attraction of stable law
Suppose that ; more precisely we assume that
belongs to the domain of attraction of the stable law .
Here is a d.f. of the stable law with parameters ,
and ch.f.
where .
if for normalized and centered sums
there is a weak convergence
2
iEX )1:( iX
i
,G
21,
G
1|| ))(exp()(,
uKug
))2tan()(sign1(||)()(,
uiuuKuK
)()1:(,
GDNAiXi
*
nS
,
1* ))(( GmnnSnSn
Domain of attraction of stable law
Denote by , , the α-stable Lévy
process with ch.f.
We omit index if it is not essential.
The fact that is not enough to obtain
“good” error term above, thus, certain additional assumptions
are needed. We formulate them in terms of ch.f.
))(exp();();(,,
utKutgutg
)()()(,
tYtYtY 0t
)()1:(,
GDNAiXi
Additional assumption
Assumption (C): there are , and such that
for
where is a ch.f. of .
Put
01
a 02
a
luauguf |||)()(|2,
l
1|| au
)()( ueuf ium )(ii
EXX
]1)}()12(2),1([max{ lA
Strong invariance principle for partial
sums
Th.A2. (Zinchenko) For and under assumption (C)
it is possible to construct α-stable process , such
that a.s.
,
for any
21)(
,tY 0t
)(|)()(|sup 1
,0
TotYmttSTt
))1(41,0(),( Al
Counting renewal processes
Order of magnitude of N(t) is described by following
theorem which includes strong law of large numbers (SLLN),
Marcinkiewich-Zygmund SLLN and law of iterated logarithm
for renewal process.
Counting renewal processes
Th.A3.
(i) If , then a.s.
(ii) if for some then a.s.
(iii) if then
while for the moments we have
,
10i
EZ
ttN )(p
iZE || )2,1(p
0))((1 ttNt p
2)(i
ZVar2321 |)(|)loglog2(suplim ttNtt
t
ttEN ~)( 23~))(( tNVar
α-stable Lévy process
L.A1. If is an α-stable Lévy process with ,
then a.s.
Keeping in mind these and equivalence in weak convergence
for Z(n) and associated N(t) it is natural to ask about a.s.
approximation of N(t).
)(tY 20
)()( 1totY
0
Strong invariance principle for
counting renewal processes
Under assumptions and strong
approximation of the counting process N(t) associated with
partial sum process
was investigated by a number of author. For instance, Csörgő,
Horváth and Steinebach (1986) obtained that for non-negative
r.v. the same error function (see T.A1) provide a.s.
approximation
2EZ 10i
EZ
][
1
)(x
ii
ZxZ
iZ
))(())((|)()(| trOtrotWtNt
)(tr
Strong invariance principle for
counting renewal processes
Let consider the case with and
Th.1. Let satisfy (C) with and
then a.s.
where is any upper function for Lévy process.
)(}{,
GNDAZi
10i
EZ
21
iZ 21 10
iEZ
))((|)()(|,
11 trotYtNt
)(tr
Strong invariance principle
Let recall
Strong invariance principle for D(t) was studied by Csörgő,
Horváth, Steinbach, Deheuvels and other authors.
In the following we focus on the case when
belong to , while can be
attracted to the normal law ( , ) or to the
α2-stable law,
Our approach is close to the methods presented in Csörgő and
Horváth (1993).
))(()( tNStD
2||i
XE )1:( iXi
)(,1
GDNA 211
)1:( iZi
2 2)(i
ZVar
212
Strong invariance principle
Th.2.(Zinchenko) Let satisfy (C) with and . Then a.s.
,
for some
In this case D(t) can be interpreted as total claims until moment t in classic risk model.
Developing such approach we proved rather general result concerning a.s. approximation of the randomly stopped process (not obligatory connected with the partial sum processes).
)1:( iXi
212
iEZ
)(|)()(| 11
,totYtmtD ),0(
01
),(00
l
Strong invariance principle
Let , be two real-valued positive increasing càdlàg
random processes,
– the inverse of is defined by
,
)(* tZ )(* tS
)(* tN )(* tZ
})(:0inf{)( ** txZttN t0
Strong invariance principle
Th.3. Suppose that for some constants m, , and
functions , meet the conditions
, , , as
where is a Wiener process and being α-stable Lévy
process, independent of ,
Then
0a 0
))((|)())((|sup1
*1
0
TrOtWattZTt
)(1
tW
)(tr 0)(
t
trt
))((|)()(|sup *
0
TqOtYmttSTt
)(tY
)(1
tW
)(tq 0)(
t
tq
0
a
tW
a
m
a
tY
a
mttNS
2
** ))((
)))(log()()loglog(( 21)2(1 ttrtqttO
)(tr )(tq
Strong invariance principle
Th.4. Let satisfy (C) with and
satisfy (C) with , . Then a.s.
for some
)()())(( 21
1
1
,totYtmtNS
)1:( iXi
211 )1:( iZ
i
212 21
),(122
l
References1. Alex,M., Steinebach,J. Invariance principles for renewal processes and
some applications. Teor. Imovirnost. ta matem. Statyst., 50,(1994),22-54.
2. Billingsley, P., Convergence of Probability Measures, J.Wiley, New York, (1968).
3. Csörgő,M., Horváth,M., Steinebach,J., Strong approximation for renewal processes, C.R. Math. Rep. Acad. Sci. Canada.8,(1986), 151-154.
4. Csörgő,M., Révész,P., Strong Approximation in Probability and Statistics, J.Wiley, New York (1981).
5. Csörgő,M., Horváth,L., Weighted Approximation in Probability and Statistics, J.Wiley, New York (1993).
6. Embrechts P., Klüppelberg C. and Mikosch T. Modelling extremal events: for insurance and finance. – New York: Springer, 1997, 645p.
7. Giknman I., Skorokhod.,A., The Theory of Stochastic Processes, II, Nauka, Moscow (1973).
8. Gut,A., Stopped Random Walks, Springer, Berlin (1988).
9. Whitt, W., Stochastic-Processes Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues, Springer-Verlag, New York (2002).
10. Zinchenko,N., Strong Invariance Principle for Renewal and Randomly Stopped Processes, Theory of Stochastic Processes Vol. 13 (29), no.4, 2007, pp.233-245.
Thank you for attention!