On functional records and champions
8th International Conference of the ERCIM WG onComputational and Methodological Statistics (ERCIM 2015),
London, UK
Clément Dombry, Michael Falk and Maximilian Zott
Université de Franche-Comté, Besançon, France,University of Wuerzburg, Germany
December 12, 2015
1 / 21
Concurrence, champions and records
Let X1, . . . ,Xn be iid random vectors in Rd , resp. stochasticprocesses with sample paths inC(S) = {f : S → R : f continuous}, S being a compact metricspace.
Dombry, Ribatet and Stoev (2015): Extremes are sampleconcurrent, if maxk=1,...,n Xk = Xi for some i ∈ {1, . . . ,n}.
In the sample concurrence case, we call Xi a champion amongX1, . . . ,Xn.
Xn is a complete record, if Xn > max (X1, . . . ,Xn−1).
Πn(X ) := P (Xn is a complete record)
Xn is a simple record, if Xn 6≤ max (X1, . . . ,Xn−1).
πn(X ) := P (Xn is a simple record)
All operations such as > or max are meant componentwise.
2 / 21
Concurrence, champions and records
Let X1, . . . ,Xn be iid random vectors in Rd , resp. stochasticprocesses with sample paths inC(S) = {f : S → R : f continuous}, S being a compact metricspace.
Dombry, Ribatet and Stoev (2015): Extremes are sampleconcurrent, if maxk=1,...,n Xk = Xi for some i ∈ {1, . . . ,n}.
In the sample concurrence case, we call Xi a champion amongX1, . . . ,Xn.
Xn is a complete record, if Xn > max (X1, . . . ,Xn−1).
Πn(X ) := P (Xn is a complete record)
Xn is a simple record, if Xn 6≤ max (X1, . . . ,Xn−1).
πn(X ) := P (Xn is a simple record)
All operations such as > or max are meant componentwise.
2 / 21
Concurrence, champions and records
Let X1, . . . ,Xn be iid random vectors in Rd , resp. stochasticprocesses with sample paths inC(S) = {f : S → R : f continuous}, S being a compact metricspace.
Dombry, Ribatet and Stoev (2015): Extremes are sampleconcurrent, if maxk=1,...,n Xk = Xi for some i ∈ {1, . . . ,n}.
In the sample concurrence case, we call Xi a champion amongX1, . . . ,Xn.
Xn is a complete record, if Xn > max (X1, . . . ,Xn−1).
Πn(X ) := P (Xn is a complete record)
Xn is a simple record, if Xn 6≤ max (X1, . . . ,Xn−1).
πn(X ) := P (Xn is a simple record)
All operations such as > or max are meant componentwise.
2 / 21
Concurrence, champions and records
Let X1, . . . ,Xn be iid random vectors in Rd , resp. stochasticprocesses with sample paths inC(S) = {f : S → R : f continuous}, S being a compact metricspace.
Dombry, Ribatet and Stoev (2015): Extremes are sampleconcurrent, if maxk=1,...,n Xk = Xi for some i ∈ {1, . . . ,n}.
In the sample concurrence case, we call Xi a champion amongX1, . . . ,Xn.
Xn is a complete record, if Xn > max (X1, . . . ,Xn−1).
Πn(X ) := P (Xn is a complete record)
Xn is a simple record, if Xn 6≤ max (X1, . . . ,Xn−1).
πn(X ) := P (Xn is a simple record)
All operations such as > or max are meant componentwise.
2 / 21
Concurrence, champions and records
Let X1, . . . ,Xn be iid random vectors in Rd , resp. stochasticprocesses with sample paths inC(S) = {f : S → R : f continuous}, S being a compact metricspace.
Dombry, Ribatet and Stoev (2015): Extremes are sampleconcurrent, if maxk=1,...,n Xk = Xi for some i ∈ {1, . . . ,n}.
In the sample concurrence case, we call Xi a champion amongX1, . . . ,Xn.
Xn is a complete record, if Xn > max (X1, . . . ,Xn−1).
Πn(X ) := P (Xn is a complete record)
Xn is a simple record, if Xn 6≤ max (X1, . . . ,Xn−1).
πn(X ) := P (Xn is a simple record)
All operations such as > or max are meant componentwise.
2 / 21
Concurrence, champions and records
Let X1, . . . ,Xn be iid random vectors in Rd , resp. stochasticprocesses with sample paths inC(S) = {f : S → R : f continuous}, S being a compact metricspace.
Dombry, Ribatet and Stoev (2015): Extremes are sampleconcurrent, if maxk=1,...,n Xk = Xi for some i ∈ {1, . . . ,n}.
In the sample concurrence case, we call Xi a champion amongX1, . . . ,Xn.
Xn is a complete record, if Xn > max (X1, . . . ,Xn−1).
Πn(X ) := P (Xn is a complete record)
Xn is a simple record, if Xn 6≤ max (X1, . . . ,Xn−1).
πn(X ) := P (Xn is a simple record)
All operations such as > or max are meant componentwise.2 / 21
Some comments
X1 is a record by definition.Each champion defines a record.Possibly more than one record among X1, . . . ,Xn, but notmore than one champion.
Let X1, . . . ,Xn iid copies of X , then the sample concurrenceprobability is given by
pn(X ) := P (sample concurrence in X1, . . . ,Xn)
= nP (Xn is a complete record) = nΠn(X ).
Different to that, nπn(X ) can not be interpreted as a probability.
3 / 21
Some comments
X1 is a record by definition.Each champion defines a record.Possibly more than one record among X1, . . . ,Xn, but notmore than one champion.
Let X1, . . . ,Xn iid copies of X , then the sample concurrenceprobability is given by
pn(X ) := P (sample concurrence in X1, . . . ,Xn)
= nP (Xn is a complete record) = nΠn(X ).
Different to that, nπn(X ) can not be interpreted as a probability.
3 / 21
Some comments
X1 is a record by definition.Each champion defines a record.Possibly more than one record among X1, . . . ,Xn, but notmore than one champion.
Let X1, . . . ,Xn iid copies of X , then the sample concurrenceprobability is given by
pn(X ) := P (sample concurrence in X1, . . . ,Xn)
= nP (Xn is a complete record) = nΠn(X ).
Different to that, nπn(X ) can not be interpreted as a probability.
3 / 21
Example: Simple and complete records, n = 5
5 observations X1, . . . ,X5:
4 / 21
Example: Simple and complete records, n = 5
5 observations X1, . . . ,X5:
5 / 21
Example: Simple and complete records, n = 5
5 observations X1, . . . ,X5:
6 / 21
Example: Simple and complete records, n = 5
5 observations X1, . . . ,X5:
7 / 21
Example: Simple and complete records, n = 5
5 observations X1, . . . ,X5:
8 / 21
Univariate champions and records
Univariate records have been studied extensively, cf. Chandler(1952), Rényi (1962) and Resnick (1987). If X ,X1, . . . ,Xn areiid rv on the real line with continuous df, thenΠn(X ) = πn(X ) = 1/n.
In the univariate case, pn(X ) = 1, and the champion coincideswith the largest order statistic max(X1, . . . ,Xn).
Multivariate records have been studied by e. g. Goldie andResnick (1989, 1995), Gnedin (1993,1998) or Hashorva andHüsler (2005). −→ Much more difficult to handle!
9 / 21
Univariate champions and records
Univariate records have been studied extensively, cf. Chandler(1952), Rényi (1962) and Resnick (1987). If X ,X1, . . . ,Xn areiid rv on the real line with continuous df, thenΠn(X ) = πn(X ) = 1/n.
In the univariate case, pn(X ) = 1, and the champion coincideswith the largest order statistic max(X1, . . . ,Xn).
Multivariate records have been studied by e. g. Goldie andResnick (1989, 1995), Gnedin (1993,1998) or Hashorva andHüsler (2005). −→ Much more difficult to handle!
9 / 21
Univariate champions and records
Univariate records have been studied extensively, cf. Chandler(1952), Rényi (1962) and Resnick (1987). If X ,X1, . . . ,Xn areiid rv on the real line with continuous df, thenΠn(X ) = πn(X ) = 1/n.
In the univariate case, pn(X ) = 1, and the champion coincideswith the largest order statistic max(X1, . . . ,Xn).
Multivariate records have been studied by e. g. Goldie andResnick (1989, 1995), Gnedin (1993,1998) or Hashorva andHüsler (2005). −→ Much more difficult to handle!
9 / 21
Standard max-stable processes
Let η = (ηs)s∈S be a standard max-stable process (SMSP) withsample paths in C−(S) := {f ∈ C(S) : f ≤ 0}, i. e.
η =D n maxi=1,...,n
ηi ,
where η1, . . . ,ηn are iid copies of η.
Theorem (Giné et al. (1990), de Haan and Ferreira (2006))
η SMSP iff there is a process Z = (Zs)s∈S inC+(S) := {f ∈ C(S) : f ≥ 0} with E(Zs) = 1, s ∈ S, andE (sups∈S Zs) <∞ such that
P(η ≤ f ) = exp(−‖f‖D) := exp(−E
(sups∈S
(|f (s)|Zs)
)),
for all f ∈ E−(S):= set of bounded functions f : S → (−∞,0]with only finitely many discontinuities.
Univariate margins of η: P(ηs ≤ x) = exp(x), x ≤ 0, s ∈ S.
10 / 21
Standard max-stable processes
Let η = (ηs)s∈S be a standard max-stable process (SMSP) withsample paths in C−(S) := {f ∈ C(S) : f ≤ 0}, i. e.
η =D n maxi=1,...,n
ηi ,
where η1, . . . ,ηn are iid copies of η.
Theorem (Giné et al. (1990), de Haan and Ferreira (2006))
η SMSP iff there is a process Z = (Zs)s∈S inC+(S) := {f ∈ C(S) : f ≥ 0} with E(Zs) = 1, s ∈ S, andE (sups∈S Zs) <∞ such that
P(η ≤ f ) = exp(−‖f‖D) := exp(−E
(sups∈S
(|f (s)|Zs)
)),
for all f ∈ E−(S):= set of bounded functions f : S → (−∞,0]with only finitely many discontinuities.
Univariate margins of η: P(ηs ≤ x) = exp(x), x ≤ 0, s ∈ S.
10 / 21
Standard max-stable processes
Let η = (ηs)s∈S be a standard max-stable process (SMSP) withsample paths in C−(S) := {f ∈ C(S) : f ≤ 0}, i. e.
η =D n maxi=1,...,n
ηi ,
where η1, . . . ,ηn are iid copies of η.
Theorem (Giné et al. (1990), de Haan and Ferreira (2006))
η SMSP iff there is a process Z = (Zs)s∈S inC+(S) := {f ∈ C(S) : f ≥ 0} with E(Zs) = 1, s ∈ S, andE (sups∈S Zs) <∞ such that
P(η ≤ f ) = exp(−‖f‖D) := exp(−E
(sups∈S
(|f (s)|Zs)
)),
for all f ∈ E−(S):= set of bounded functions f : S → (−∞,0]with only finitely many discontinuities.
Univariate margins of η: P(ηs ≤ x) = exp(x), x ≤ 0, s ∈ S.10 / 21
Copulas in the domain of attraction of an SMSP
Suppose a copula process U = (Us)s∈S (continuous samplepaths, uniform univariate margins) is in the max-domain ofattraction of an SMSP η (U ∈ D(η)) with D-norm ‖·‖Dgenerated by Z = (Zs)s∈S, i. e.
n(
maxi=1,...,n
Ui − 1)→D η, (1)
where U1,U2, . . . are iid copies of U.
Eq. (1) implies forf ∈ E−(S)
nP (n(U − 1) 6≤ f )→n→∞ E(
sups∈S|f (s)|Zs
)= ‖f‖D , (2)
see Aulbach et al. (2013). Also, if (1) holds,
nP (n(U − 1) > f )→n→∞ E(
infs∈S|f (s)|Zs
)=: oo f ooD, (3)
for f ∈ E−(S), see Dombry, Falk and Z. (2015).
11 / 21
Copulas in the domain of attraction of an SMSP
Suppose a copula process U = (Us)s∈S (continuous samplepaths, uniform univariate margins) is in the max-domain ofattraction of an SMSP η (U ∈ D(η)) with D-norm ‖·‖Dgenerated by Z = (Zs)s∈S, i. e.
n(
maxi=1,...,n
Ui − 1)→D η, (1)
where U1,U2, . . . are iid copies of U. Eq. (1) implies forf ∈ E−(S)
nP (n(U − 1) 6≤ f )→n→∞ E(
sups∈S|f (s)|Zs
)= ‖f‖D , (2)
see Aulbach et al. (2013).
Also, if (1) holds,
nP (n(U − 1) > f )→n→∞ E(
infs∈S|f (s)|Zs
)=: oo f ooD, (3)
for f ∈ E−(S), see Dombry, Falk and Z. (2015).
11 / 21
Copulas in the domain of attraction of an SMSP
Suppose a copula process U = (Us)s∈S (continuous samplepaths, uniform univariate margins) is in the max-domain ofattraction of an SMSP η (U ∈ D(η)) with D-norm ‖·‖Dgenerated by Z = (Zs)s∈S, i. e.
n(
maxi=1,...,n
Ui − 1)→D η, (1)
where U1,U2, . . . are iid copies of U. Eq. (1) implies forf ∈ E−(S)
nP (n(U − 1) 6≤ f )→n→∞ E(
sups∈S|f (s)|Zs
)= ‖f‖D , (2)
see Aulbach et al. (2013). Also, if (1) holds,
nP (n(U − 1) > f )→n→∞ E(
infs∈S|f (s)|Zs
)=: oo f ooD, (3)
for f ∈ E−(S), see Dombry, Falk and Z. (2015).11 / 21
Dual D-norm functions
For a given D-norm ‖f‖D = E (sups∈S |f (s)|Zs), the mapping
oo · ooD : E(S)→ R, f 7→ oo f ooD := E(
infs∈S|f (s)|Zs
),
is called dual D-norm function.
The mapping ‖·‖D → oo · ooD is well-defined, but notone-to-one.oo · ooD ≡ 0 if at least two components of the SMSP η in (1)are independent.Finitedimensional setup: dual D-norm function is the tailcopula, cf. Schmidt and Stadtmüller (2006).
12 / 21
Dual D-norm functions
For a given D-norm ‖f‖D = E (sups∈S |f (s)|Zs), the mapping
oo · ooD : E(S)→ R, f 7→ oo f ooD := E(
infs∈S|f (s)|Zs
),
is called dual D-norm function.
The mapping ‖·‖D → oo · ooD is well-defined, but notone-to-one.
oo · ooD ≡ 0 if at least two components of the SMSP η in (1)are independent.Finitedimensional setup: dual D-norm function is the tailcopula, cf. Schmidt and Stadtmüller (2006).
12 / 21
Dual D-norm functions
For a given D-norm ‖f‖D = E (sups∈S |f (s)|Zs), the mapping
oo · ooD : E(S)→ R, f 7→ oo f ooD := E(
infs∈S|f (s)|Zs
),
is called dual D-norm function.
The mapping ‖·‖D → oo · ooD is well-defined, but notone-to-one.oo · ooD ≡ 0 if at least two components of the SMSP η in (1)are independent.
Finitedimensional setup: dual D-norm function is the tailcopula, cf. Schmidt and Stadtmüller (2006).
12 / 21
Dual D-norm functions
For a given D-norm ‖f‖D = E (sups∈S |f (s)|Zs), the mapping
oo · ooD : E(S)→ R, f 7→ oo f ooD := E(
infs∈S|f (s)|Zs
),
is called dual D-norm function.
The mapping ‖·‖D → oo · ooD is well-defined, but notone-to-one.oo · ooD ≡ 0 if at least two components of the SMSP η in (1)are independent.Finitedimensional setup: dual D-norm function is the tailcopula, cf. Schmidt and Stadtmüller (2006).
12 / 21
Complete Records: Extremal concurrence probabilityof U
Remember
Πn(U) = P (Un is a complete record) = P(
Un > maxi=1,...,n−1
Ui
).
Functional version of the Dombry-Ribatet-Stoev theorem, seealso Hashorva and Hüsler (2005):
Theorem
Let U,U1,U2, . . . be iid copula processes with U ∈ D(η), whereη is an SMSP with D-norm ‖·‖D. The sample concurrenceprobability fulfills
pn(U) = nΠn(U)→n→∞ E (oo η ooD) .
13 / 21
Complete Records: Extremal concurrence probabilityof U
Remember
Πn(U) = P (Un is a complete record) = P(
Un > maxi=1,...,n−1
Ui
).
Functional version of the Dombry-Ribatet-Stoev theorem, seealso Hashorva and Hüsler (2005):
Theorem
Let U,U1,U2, . . . be iid copula processes with U ∈ D(η), whereη is an SMSP with D-norm ‖·‖D. The sample concurrenceprobability fulfills
pn(U) = nΠn(U)→n→∞ E (oo η ooD) .
13 / 21
Complete Records: Comments and examples
The number E (oo η ooD) ∈ [0,1] is called extremalconcurrence probability.
If ‖·‖D = ‖·‖∞ (complete dependence case), thenE (oo η ooD) = 1.If ηs, ηt are independent for some s, t ∈ S, thenE (oo η ooD) = 0.More examples in Dombry, Ribatet and Stoev (2015).(logistic model, max-linear model, extremal process etc.)
14 / 21
Complete Records: Comments and examples
The number E (oo η ooD) ∈ [0,1] is called extremalconcurrence probability.If ‖·‖D = ‖·‖∞ (complete dependence case), thenE (oo η ooD) = 1.
If ηs, ηt are independent for some s, t ∈ S, thenE (oo η ooD) = 0.More examples in Dombry, Ribatet and Stoev (2015).(logistic model, max-linear model, extremal process etc.)
14 / 21
Complete Records: Comments and examples
The number E (oo η ooD) ∈ [0,1] is called extremalconcurrence probability.If ‖·‖D = ‖·‖∞ (complete dependence case), thenE (oo η ooD) = 1.If ηs, ηt are independent for some s, t ∈ S, thenE (oo η ooD) = 0.
More examples in Dombry, Ribatet and Stoev (2015).(logistic model, max-linear model, extremal process etc.)
14 / 21
Complete Records: Comments and examples
The number E (oo η ooD) ∈ [0,1] is called extremalconcurrence probability.If ‖·‖D = ‖·‖∞ (complete dependence case), thenE (oo η ooD) = 1.If ηs, ηt are independent for some s, t ∈ S, thenE (oo η ooD) = 0.More examples in Dombry, Ribatet and Stoev (2015).(logistic model, max-linear model, extremal process etc.)
14 / 21
Simple records: Limit probability
Now focus on the multivariate case, i. e. U,U1,U2, . . . are iidobservations following a copula on Rd . Remember
πn(U) = P (Un is a simple record) = P(
Un 6≤ maxi=1,...,n−1
Ui
).
Theorem (Dombry, Falk and Z. (2015))
Let U,U1,U2, . . . be iid random vectors in Rd with U ∈ D(η),where η is standard max-stable with D-norm ‖·‖D, i. e.P(η ≤ x) = exp (−‖x‖D), x ≤ 0 ∈ Rd . Then
nπn(U)→n→∞ E (‖η‖D) .
15 / 21
Simple records: Limit probability
Now focus on the multivariate case, i. e. U,U1,U2, . . . are iidobservations following a copula on Rd . Remember
πn(U) = P (Un is a simple record) = P(
Un 6≤ maxi=1,...,n−1
Ui
).
Theorem (Dombry, Falk and Z. (2015))
Let U,U1,U2, . . . be iid random vectors in Rd with U ∈ D(η),where η is standard max-stable with D-norm ‖·‖D, i. e.P(η ≤ x) = exp (−‖x‖D), x ≤ 0 ∈ Rd . Then
nπn(U)→n→∞ E (‖η‖D) .
15 / 21
Simple record times
Take an iid sequence X ,X1,X2, . . . following a continuous df Fon Rd .
Let N(n) be the (random) index at which the n-th simple recordoccurs, N(1) := 1. Furthermore, ∆n := N(n)− N(n − 1), n ≥ 2,∆1 := 1 (inter record waiting times). For every n ∈ N
P(XN(n+1) ≤ x
∣∣XN(n) = y)
= P(X ≤ x
∣∣X 6≤ y),
∆n+1∣∣XN(n) = x ∼ Geom(1− F (x)).
Indeed,(XN(n)
)n∈N and
((XN(n),∆n
))n∈N are homogenous
Markov chains!
16 / 21
Simple record times
Take an iid sequence X ,X1,X2, . . . following a continuous df Fon Rd .
Let N(n) be the (random) index at which the n-th simple recordoccurs, N(1) := 1. Furthermore, ∆n := N(n)− N(n − 1), n ≥ 2,∆1 := 1 (inter record waiting times). For every n ∈ N
P(XN(n+1) ≤ x
∣∣XN(n) = y)
= P(X ≤ x
∣∣X 6≤ y),
∆n+1∣∣XN(n) = x ∼ Geom(1− F (x)).
Indeed,(XN(n)
)n∈N and
((XN(n),∆n
))n∈N are homogenous
Markov chains!
16 / 21
Simple record times
Take an iid sequence X ,X1,X2, . . . following a continuous df Fon Rd .
Let N(n) be the (random) index at which the n-th simple recordoccurs, N(1) := 1. Furthermore, ∆n := N(n)− N(n − 1), n ≥ 2,∆1 := 1 (inter record waiting times). For every n ∈ N
P(XN(n+1) ≤ x
∣∣XN(n) = y)
= P(X ≤ x
∣∣X 6≤ y),
∆n+1∣∣XN(n) = x ∼ Geom(1− F (x)).
Indeed,(XN(n)
)n∈N and
((XN(n),∆n
))n∈N are homogenous
Markov chains!
16 / 21
Simple record times
Take an iid sequence X ,X1,X2, . . . following a continuous df Fon Rd .
Let N(n) be the (random) index at which the n-th simple recordoccurs, N(1) := 1. Furthermore, ∆n := N(n)− N(n − 1), n ≥ 2,∆1 := 1 (inter record waiting times). For every n ∈ N
P(XN(n+1) ≤ x
∣∣XN(n) = y)
= P(X ≤ x
∣∣X 6≤ y),
∆n+1∣∣XN(n) = x ∼ Geom(1− F (x)).
Indeed,(XN(n)
)n∈N and
((XN(n),∆n
))n∈N are homogenous
Markov chains!
16 / 21
Simple record times: Integrability of N(2)
Take an iid sequence U,U1,U2, . . . following a copula C on[0,1]d .
Now focus on N(2). Easy to see:
E(N(2)) =
∫[0,1]d
11− C(u)
C(du) + 1.
d = 1 :
E(N(2)) =
∫ 1
0
11− u
du + 1 =∞.
d = 2, independent margins: In that case C(u1,u2) = u1u2, and
E(N(2)) =
∫ 1
0
∫ 1
0
11− u1u2
du1du2 + 1 =π2
6+ 1 <∞.
17 / 21
Simple record times: Integrability of N(2)
Take an iid sequence U,U1,U2, . . . following a copula C on[0,1]d .
Now focus on N(2). Easy to see:
E(N(2)) =
∫[0,1]d
11− C(u)
C(du) + 1.
d = 1 :
E(N(2)) =
∫ 1
0
11− u
du + 1 =∞.
d = 2, independent margins: In that case C(u1,u2) = u1u2, and
E(N(2)) =
∫ 1
0
∫ 1
0
11− u1u2
du1du2 + 1 =π2
6+ 1 <∞.
17 / 21
Simple record times: Integrability of N(2)
Take an iid sequence U,U1,U2, . . . following a copula C on[0,1]d .
Now focus on N(2). Easy to see:
E(N(2)) =
∫[0,1]d
11− C(u)
C(du) + 1.
d = 1 :
E(N(2)) =
∫ 1
0
11− u
du + 1 =∞.
d = 2, independent margins: In that case C(u1,u2) = u1u2, and
E(N(2)) =
∫ 1
0
∫ 1
0
11− u1u2
du1du2 + 1 =π2
6+ 1 <∞.
17 / 21
Simple record times: Integrability of N(2)
Take an iid sequence U,U1,U2, . . . following a copula C on[0,1]d .
Now focus on N(2). Easy to see:
E(N(2)) =
∫[0,1]d
11− C(u)
C(du) + 1.
d = 1 :
E(N(2)) =
∫ 1
0
11− u
du + 1 =∞.
d = 2, independent margins: In that case C(u1,u2) = u1u2, and
E(N(2)) =
∫ 1
0
∫ 1
0
11− u1u2
du1du2 + 1 =π2
6+ 1 <∞.
17 / 21
Simple record times: Integrability of N(2)
Natural question: When is E(N(2)) finite?
Suppose C ∈ D(G) for some standard max-stable df G withD-norm ‖·‖D, dual D-norm function oo · ooD.
Proposition (Part 1, Dombry, Falk and Z. (2015))
If oo 1 ooD > 0, then E(N(2)) =∞.
Note that oo 1 ooD = 0 if G has at least two independent margins!
Proposition (Sketch of Part 2, Dombry, Falk and Z. (2015))Basically, if G has at least two independent margins, thenE(N(2)) <∞.
Corollary
E(N(2)) <∞ for multivariate normal rv unless all componentsare completely dependent.
18 / 21
Simple record times: Integrability of N(2)
Natural question: When is E(N(2)) finite?
Suppose C ∈ D(G) for some standard max-stable df G withD-norm ‖·‖D, dual D-norm function oo · ooD.
Proposition (Part 1, Dombry, Falk and Z. (2015))
If oo 1 ooD > 0, then E(N(2)) =∞.
Note that oo 1 ooD = 0 if G has at least two independent margins!
Proposition (Sketch of Part 2, Dombry, Falk and Z. (2015))Basically, if G has at least two independent margins, thenE(N(2)) <∞.
Corollary
E(N(2)) <∞ for multivariate normal rv unless all componentsare completely dependent.
18 / 21
Simple record times: Integrability of N(2)
Natural question: When is E(N(2)) finite?
Suppose C ∈ D(G) for some standard max-stable df G withD-norm ‖·‖D, dual D-norm function oo · ooD.
Proposition (Part 1, Dombry, Falk and Z. (2015))
If oo 1 ooD > 0, then E(N(2)) =∞.
Note that oo 1 ooD = 0 if G has at least two independent margins!
Proposition (Sketch of Part 2, Dombry, Falk and Z. (2015))Basically, if G has at least two independent margins, thenE(N(2)) <∞.
Corollary
E(N(2)) <∞ for multivariate normal rv unless all componentsare completely dependent.
18 / 21
Simple record times: Integrability of N(2)
Natural question: When is E(N(2)) finite?
Suppose C ∈ D(G) for some standard max-stable df G withD-norm ‖·‖D, dual D-norm function oo · ooD.
Proposition (Part 1, Dombry, Falk and Z. (2015))
If oo 1 ooD > 0, then E(N(2)) =∞.
Note that oo 1 ooD = 0 if G has at least two independent margins!
Proposition (Sketch of Part 2, Dombry, Falk and Z. (2015))Basically, if G has at least two independent margins, thenE(N(2)) <∞.
Corollary
E(N(2)) <∞ for multivariate normal rv unless all componentsare completely dependent.
18 / 21
Simple record times: Integrability of N(2)
Natural question: When is E(N(2)) finite?
Suppose C ∈ D(G) for some standard max-stable df G withD-norm ‖·‖D, dual D-norm function oo · ooD.
Proposition (Part 1, Dombry, Falk and Z. (2015))
If oo 1 ooD > 0, then E(N(2)) =∞.
Note that oo 1 ooD = 0 if G has at least two independent margins!
Proposition (Sketch of Part 2, Dombry, Falk and Z. (2015))Basically, if G has at least two independent margins, thenE(N(2)) <∞.
Corollary
E(N(2)) <∞ for multivariate normal rv unless all componentsare completely dependent.
18 / 21
Simple record times: Integrability of N(2)
Natural question: When is E(N(2)) finite?
Suppose C ∈ D(G) for some standard max-stable df G withD-norm ‖·‖D, dual D-norm function oo · ooD.
Proposition (Part 1, Dombry, Falk and Z. (2015))
If oo 1 ooD > 0, then E(N(2)) =∞.
Note that oo 1 ooD = 0 if G has at least two independent margins!
Proposition (Sketch of Part 2, Dombry, Falk and Z. (2015))Basically, if G has at least two independent margins, thenE(N(2)) <∞.
Corollary
E(N(2)) <∞ for multivariate normal rv unless all componentsare completely dependent.
18 / 21
Simple record times: Integrability of N(n)
Similar to Goldie and Resnick (1989): If F ∈ D(G) for somebivariate max-stable df G, then the number of complete recordsin a sequence X1,X2, . . . following the continuous df F is eitherfinite a. s. or infinite a. s., depending on whether G hasindependent margins or not.
What about E(N(n)), n > 2? Clearly, E(N(n)) =∞ ifE(N(2)) =∞.
Proposition (Dombry, Falk and Z. (2015))
(i) If E(N(2)) =∞, then E(N(n + 1)− N(n)) =∞ for alln ∈ N.
(ii) If E(N(2)) <∞, then E(N(n + 1)− N(n)) <∞ for alln ∈ N.
19 / 21
Simple record times: Integrability of N(n)
Similar to Goldie and Resnick (1989): If F ∈ D(G) for somebivariate max-stable df G, then the number of complete recordsin a sequence X1,X2, . . . following the continuous df F is eitherfinite a. s. or infinite a. s., depending on whether G hasindependent margins or not.
What about E(N(n)), n > 2? Clearly, E(N(n)) =∞ ifE(N(2)) =∞.
Proposition (Dombry, Falk and Z. (2015))
(i) If E(N(2)) =∞, then E(N(n + 1)− N(n)) =∞ for alln ∈ N.
(ii) If E(N(2)) <∞, then E(N(n + 1)− N(n)) <∞ for alln ∈ N.
19 / 21
Simple record times: Integrability of N(n)
Similar to Goldie and Resnick (1989): If F ∈ D(G) for somebivariate max-stable df G, then the number of complete recordsin a sequence X1,X2, . . . following the continuous df F is eitherfinite a. s. or infinite a. s., depending on whether G hasindependent margins or not.
What about E(N(n)), n > 2? Clearly, E(N(n)) =∞ ifE(N(2)) =∞.
Proposition (Dombry, Falk and Z. (2015))
(i) If E(N(2)) =∞, then E(N(n + 1)− N(n)) =∞ for alln ∈ N.
(ii) If E(N(2)) <∞, then E(N(n + 1)− N(n)) <∞ for alln ∈ N.
19 / 21
Thank you very much for your attention!
20 / 21
Some sources
AULBACH, S., FALK, M., HOFMANN, M. (2013). On max-stableprocesses and the functional D-norm. Extremes.
CHANDLER, K. N. (1952). The Distribution and Frequency ofRecord Values. J. R. Statist. Soc. B.
DOMBRY, C., FALK, M., ZOTT, M. (2015). On functional recordsand champions. arXiv:1510.04529
DOMBRY, C., RIBATET, M., STOEV, S. (2015). Probabilities ofconcurrent extremes. arXiv:1503.05748
GINÉ, E., HAHN, M., AND VATAN, P. (1990). Max-infinitelydivisible and max-stable sample continous processes. Probab.Theory Related Fields.
GOLDIE, C. M. AND RESNICK, S. (1989). Records in a partiallyordered set. Ann. Probab.
GOLDIE, C. M. AND RESNICK, S. (1995). Many multivariaterecords. Stochastic Process. Appl.
21 / 21
Some sources
GNEDIN, A. (1993). On Multivariate Extremal Processes. J. Mult.Anal.
GENDIN, A. (1998). Records from a multivariate normal sample.Stat. Probab. Letters
DE HAAN, L., AND FERREIRA, A. (2006). Extreme Value Theory:An Introduction. Springer, New York.
HASHORVA, E. AND HÜSLER, J. (2005). Multiple maxima inmultivariate samples. Stat. Probab. Letters
RÉNYI, A. (1962). Théorie des éléments saillants d’une suited’obervations. Ann. scient. de l’Univ. Clermont-Ferrand
RESNICK, S. (1987). Extreme Values, Regular Variation, andPoint Processes. Springer, New York.
SCHMIDT, R. AND STADTMÜLLER, U. (2006). Non-parametricEstimation of Tail Dependence. Scand. J. Stat.
22 / 21