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On functional records and champions · On functional records and champions 8th International...
Transcript of On functional records and champions · On functional records and champions 8th International...
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 <∞.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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Thank you very much for your attention!
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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.
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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.
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