on the structure of max--stable processes schlather & de haan (2009)stationary max{stable elds...

Preliminaries Minimality Continuous/Discrete Stationarity and Flows Co–spectrum

On the Structure of Max–Stable Processes

Stilian Stoev (joint work with Yizao Wang)University of Michigan, Ann Arbor

Graybill VIII: Extreme Value AnalysisFort Collins, Colorado, June 24, 2009

Some references:

• The talk is based on the work:Wang & S. (2009) On the structure and representations of max–stableprocesses, Preprint.

• Closely related works:de Haan & Pickands (1986) Stationary min-stable stochastic processes.Probab. Theory Relat. Fields, 72:477–492, 1986.Kabluchko, Schlather & de Haan (2009) Stationary max–stable fieldsassociated to negative definite functions, Preprint.Kabluchko (2009) Spectral representations of sum– and max–stableprocesses, Preprint.Wang & S. (2009) On the Association of Sum– and Max– StableProcesses, Preprint.

There is a beautiful parallel world out there.

Sum–stable processes:Hardin (1982) On the spectral representation of symmetric stableprocesses. Journal of Multivariate Analysis, 12:385–401, 1982.Rosinski (1995) On the structure of stationary stable processes. Ann.Probab., 23(3):1163–1187, 1995.Rosinski & Samorodnitsky (1996) Classes of mixing stable processes.Bernoulli, 2(4):365–377, 1996.Pipiras & Taqqu (2004) Stable stationary processes related to cyclicflows. Ann. Probab., 32(3A):2222–2260, 2004.Samorodnitsky (2005) Null flows, positive flows and the structure ofstationary symmetric stable processes. Ann. Probab., 33:1782–1803,2005.Our goal:

To catch up! Provide tools to achieve similar structural resultsfor max–stable processes.

There is a beautiful parallel world out there.

Sum–stable processes:Hardin (1982) On the spectral representation of symmetric stableprocesses. Journal of Multivariate Analysis, 12:385–401, 1982.Rosinski (1995) On the structure of stationary stable processes. Ann.Probab., 23(3):1163–1187, 1995.Rosinski & Samorodnitsky (1996) Classes of mixing stable processes.Bernoulli, 2(4):365–377, 1996.Pipiras & Taqqu (2004) Stable stationary processes related to cyclicflows. Ann. Probab., 32(3A):2222–2260, 2004.Samorodnitsky (2005) Null flows, positive flows and the structure ofstationary symmetric stable processes. Ann. Probab., 33:1782–1803,2005.Our goal: To catch up! Provide tools to achieve similar structural resultsfor max–stable processes.

1 Preliminaries

2 Minimal representations

3 Continuous–Discrete Spectral Decomposition

4 Stationary Max–stable Processes and Flows

5 Classification via Co–spectral functions

Preliminaries

Motivation: Consider heavy–tailed i.i.d. processes Y (i)t t∈T . If for some

an ∼ n1/α`(n) (α > 0), we have 1

∨1≤i≤n

f .d.d.−→ Xtt∈T , (n→∞),

then X = Xtt∈T is max–stable (α−Frechet).

Def The process X = Xtt∈T is max–stable (α−Frechet) if:

X (1)t ∨ · · · ∨ X

(n)t t∈T

d= n1/αXtt∈T ,

for all n ∈ N, where X (i) = X (i)t t∈T are independent copies of X and

where α > 0. The margins of X are then α−Frechet (α > 0), namely:

PXt ≤ x = exp−σαt x−α, x > 0,

with σαt > 0.Fine print: For simplicity, we focus on max–stable processes with α−Frechet marginals. By transforming the

margins, our theory applies to the most general definition of max–stable processes where the margins can be

extreme value distributions of different types.

Preliminaries

an ∼ n1/α`(n) (α > 0), we have 1

∨1≤i≤n

f .d.d.−→ Xtt∈T , (n→∞),

then X = Xtt∈T is max–stable (α−Frechet).Def The process X = Xtt∈T is max–stable (α−Frechet) if:

X (1)t ∨ · · · ∨ X

(n)t t∈T

d= n1/αXtt∈T ,

where α > 0.

The margins of X are then α−Frechet (α > 0), namely:

Preliminaries

an ∼ n1/α`(n) (α > 0), we have 1

∨1≤i≤n

f .d.d.−→ Xtt∈T , (n→∞),

then X = Xtt∈T is max–stable (α−Frechet).Def The process X = Xtt∈T is max–stable (α−Frechet) if:

X (1)t ∨ · · · ∨ X

(n)t t∈T

d= n1/αXtt∈T ,

where α > 0. The margins of X are then α−Frechet (α > 0), namely:

Spectral Representations

For an α−Frechet process (α > 0) X = Xtt∈T , we have:

(a) de Haan’s spectral representation:

Xt =∞∨

ft(Ui )/Γ1/αi , (t ∈ T )

for a Poisson point process (Γi ,Ui ) on (0,∞)× U with intensitydx × µ(du).(b) Extremal integral representation:

ft(u)Mα(du), (t ∈ T )

for an α−Frechet random sup–measure Mα(du) on (U, µ).• The deterministic functions ft(u) ≥ 0 are called spectral functions of Xand satisfy: ∫

ft(u)αµ(du) <∞, (t ∈ T ).

Fine print: The measure space (U, µ) can be chosen ([0, 1], dx) if the process Xtt∈T is separable in

probability, in particular, continuous in probability when T is a separable metric space.

For an α−Frechet process (α > 0) X = Xtt∈T , we have:(a) de Haan’s spectral representation:

Xt =∞∨

ft(Ui )/Γ1/αi , (t ∈ T )

for a Poisson point process (Γi ,Ui ) on (0,∞)× U with intensitydx × µ(du).

(b) Extremal integral representation:

ft(u)αµ(du) <∞, (t ∈ T ).

Xt =∞∨

ft(Ui )/Γ1/αi , (t ∈ T )

for an α−Frechet random sup–measure Mα(du) on (U, µ).

• The deterministic functions ft(u) ≥ 0 are called spectral functions of Xand satisfy: ∫

ft(u)αµ(du) <∞, (t ∈ T ).

Xt =∞∨

ft(Ui )/Γ1/αi , (t ∈ T )

ft(u)αµ(du) <∞, (t ∈ T ).

Extremal Integrals

Let Mα be a random α−Frechet sup–measure on (U, µ). For simple functions f (u) =

∑ni=1 ai 1Ai (u), f (u) ≥ 0:∫e

f (u)Mα(du) :=∨

1≤i≤n

ai Mα(Ai ).

The def of∫e

UfdMα extends to all f ∈ Lα+(µ) and

P ∫e

fdMα ≤ x

= exp−‖f ‖αLα(µ)x−α, x > 0.

For f , g ∈ Lα+(µ):∫eU

(af ∨ bg)dMα = a

fdMα ∨ b

gdMα (max–linearity)

UfdMα and

gdMα are independent if and only if fg = 0, (mod µ).

Benefits: For any ft ∈ Lα+(µ), t ∈ T , we get a max–stable process:

ftdMα

For the finite–dimensional distributions, we have:

PXti ≤ xi , 1 ≤ i ≤ d = P∫e

(∨1≤i≤d x−1i fti )dMα ≤ 1

= exp−∫

(∨f αti/xαi )dµ.

Examples:• (moving maxima)

f (t − x)Mα(dx), t ∈ R,

with (U, µ) ≡ (R, dx) and f ∈ Lα+(dx). Smith’s storm processes are moving maxima.

Examples (cont’d)

• (mixed moving maxima) With (U, µ) = (R× V , dx × dν):

∫eR×V

f (t − x , v)Mα(dx , dv), f (x , v) ∈ Lα+(dx , dν).

A continuous–time version of the M3 processes.

• (doubly stochastic)Let (U, µ) be a probability space and ξt(u)≥ 0, t ∈ R a stochasticprocess over (U, µ). If Eµξαt <∞, then

ξt(u)Mα(du), t ∈ R,

is an α−Frechet process on the probability space (Ω,F ,P). Schlater’s processes are doubly stochastic with particular ξt ’s.• (Brown–Resnick) With (U, µ) a probability space and wt(u)t∈R astandard Brownian motion on (U, µ):

ewt (u)−α|t|/2Mα(du), t ∈ R.

Examples (cont’d)

∫eR×V

A continuous–time version of the M3 processes. • (doubly stochastic)Let (U, µ) be a probability space and ξt(u)≥ 0, t ∈ R a stochasticprocess over (U, µ). If Eµξαt <∞, then

is an α−Frechet process on the probability space (Ω,F ,P). Schlater’s processes are doubly stochastic with particular ξt ’s.

• (Brown–Resnick) With (U, µ) a probability space and wt(u)t∈R astandard Brownian motion on (U, µ):

Examples (cont’d)

∫eR×V

A continuous–time version of the M3 processes. • (doubly stochastic)Let (U, µ) be a probability space and ξt(u)≥ 0, t ∈ R a stochasticprocess over (U, µ). If Eµξαt <∞, then

is an α−Frechet process on the probability space (Ω,F ,P). Schlater’s processes are doubly stochastic with particular ξt ’s.• (Brown–Resnick) With (U, µ) a probability space and wt(u)t∈R astandard Brownian motion on (U, µ):

Max-linear Isometries

Consider a max–stable process:

ftdMα, (t ∈ T ).

Some Natural Questions:• How does the structure of the ft(u)’s determine the structure ofX = Xtt∈T and vice versa?

• Given another representation gt ⊂ Lα+(V , ν)

Xtt∈Td= ∫e

gtdMα

what is the relationship between ftt∈T and gtt∈T ?Some Answers: For all ai ≥ 0, ti ∈ T , we have:

‖∨

ai fti‖αLα(µ) = ‖∨

ai gti‖αLα(ν).

• Thus, there exists a max–linear isometry I : Lα+(U, µ)→ Lα+(V , ν), such

that I(ft) = gt , for all t ∈ T .

ftdMα, (t ∈ T ).

Some Natural Questions:• How does the structure of the ft(u)’s determine the structure ofX = Xtt∈T and vice versa?• Given another representation gt ⊂ Lα+(V , ν)

Xtt∈Td= ∫e

gtdMα

what is the relationship between ftt∈T and gtt∈T ?

Some Answers: For all ai ≥ 0, ti ∈ T , we have:

‖∨

ai gti‖αLα(ν).

ftdMα, (t ∈ T ).

Xtt∈Td= ∫e

gtdMα

‖∨

ai gti‖αLα(ν).

ftdMα, (t ∈ T ).

Xtt∈Td= ∫e

gtdMα

‖∨

ai gti‖αLα(ν).

Def I : Lα+(U, µ)→ Lα+(V , ν) is a max–linear isometry, if:

I(af ∨ bg) = aI(f ) ∨ bI(g), ∀f , g ∈ Lα+(U, µ), a, b ≥ 0.

‖I(f )‖αLα(ν) =

I(f )αdν =

f αdµ = ‖f ‖Lα(µ).

Conversely, any max–linear isometry I : Lα+(U, µ)→ Lα+(V , ν) yields anequivalent spectral represenation gt := I(ft) of the process X over (V , ν).

It is important to clarify the structure of max–linear isometries!

• In Wang & S., 2009, we extend results of Hardin, 1981/82 on thestructure of linear isometries to the max–linear case.

• The importance of these results for max–stable processes is bestunderstood via the notion of minimal representation.

I(f )αdν =

Minimal spectral representations

Def The spectral representation ftt∈T ⊂ Lα+(U, µ) of X is minimal if:(i) (full support) suppft(u), t ∈ T = U (mod µ)(ii) (non-redundancy) For any measurable A ⊂ U, there exists

B ∈ ρft , t ∈ T ≡ σft/fs , t, s ∈ T,

such that µ(A∆B) = 0.

This def is identical to the one of Rosinski (1995) in the sum–stablecase, similar to Hardin (1982), and to the proper pistons of de Haan andPickands (1986).

• Why are minimal reps called ’minimal’?

Minimal spectral representations

Def The spectral representation ftt∈T ⊂ Lα+(U, µ) of X is minimal if:(i) (full support) suppft(u), t ∈ T = U (mod µ)(ii) (non-redundancy) For any measurable A ⊂ U, there exists

B ∈ ρft , t ∈ T ≡ σft/fs , t, s ∈ T,

such that µ(A∆B) = 0.

This def is identical to the one of Rosinski (1995) in the sum–stablecase, similar to Hardin (1982), and to the proper pistons of de Haan andPickands (1986).

• Why are minimal reps called ’minimal’?

Why ’minimal’?

The full–support condition is natural.

Consider the process

∫e[0,1]

t2 sin2(u)Mα(du) = t2Z ,

where Z =∫e

[0,1]sin2(u)Mα(du).

• This representation is clearly redundant! Note that

ρ(ft , t ∈ T ) = ∅, [0, 1] 6' B[0,1].

We have a natural, simpler representation:

ft(u)Mα(du), with ft(u) = t2,

and trivial U = 0, and µ(du) = cδ0(du), c :=∫

[0,1]sin2α(x)dx .

• The ratio σ−algebra ρ(ft , t ∈ T ) captures best the minimal

information needed to represent the process.

Why ’minimal’?

The full–support condition is natural. Consider the process

∫e[0,1]

where Z =∫e

ρ(ft , t ∈ T ) = ∅, [0, 1] 6' B[0,1].

[0,1]sin2α(x)dx .

Why ’minimal’?

∫e[0,1]

where Z =∫e

ρ(ft , t ∈ T ) = ∅, [0, 1] 6' B[0,1].

[0,1]sin2α(x)dx .

Why ’minimal’?

∫e[0,1]

where Z =∫e

ρ(ft , t ∈ T ) = ∅, [0, 1] 6' B[0,1].

[0,1]sin2α(x)dx .

What are the benefits of minimal representations?

Thm 1. (Wang & S.(2009)) Let ftt∈T ⊂ Lα+(µ) be a minimalmeasurable rep of X . If gtt∈T ⊂ Lα+(V , ν) is another measurable rep ofX , then:

gt(v) = h(v)ft(φ(v)), ν − a.e.

for some measurable h ≥ 0 and φ : V → U. The map φ is unique (modν).If gt is also minimal, then φ is bi–measurable, ν ∼ µ φ and

dµ φdν

(v) = hα(v) > 0.

Note: h and φ are independent of ’time’ t ∈ T point mappings.

Minimal representations with standardized support

Consider the setsSI,N = (0, I) ∪ 1, 2, · · · ,N,

where I ∈ 0, 1 and 0 ≤ N ≤ ∞: For example:

S1,3 = (0, 1) ∪ 1, 2, 3, S0,∞ = 1, 2, · · · , and S1,0 = (0, 1).

• Equip SI,N with the measure

λI,N (x) = dx +N∑

δi(dx).

Fine print: Every standard Lebesgue space is isomorphic to some (SI,N , λI,N ).

Def A minimal representation ftt∈T ⊂ Lα+(U, µ) is said to havestandardized support if, for some I,N: (U, µ) ≡ (SI,N , λI,N ).

Thm 2. (Wang & S., 2009) Every separable in probability α−Frechetprocess X has a minimal representation with standardized support:

Xtt∈Td= ∫e

ftdMα

Continuous–Discrete Decomposition

Consider an α−Frechet process X with the minimal rep of standardizedsupport:

Xtt∈Td= ∫e

ftdMα

By setting

X It :=

∫eSI,N∩(0,1)

ftdMα and X Nt :=

∫eSI,N∩N

ftdMα,

we obtain the continuous–discrete decomposition:

Xtt∈Td= X I

t ∨ X Nt t∈T .

The components X It and X N

t are independent.

Intuition: Suppose I = 1 and N > 0. Then,

X It =

∫e(0,1)

ftdMα and X Nt =

N∨i=1

ft(i)Zi ,

where Zi = Mαi are i.i.d. standard α−Frechet, independent of X It .

Continuous–Discrete Decomposition

Consider an α−Frechet process X with the minimal rep of standardizedsupport:

Xtt∈Td= ∫e

ftdMα

By setting

X It :=

∫eSI,N∩(0,1)

ftdMα and X Nt :=

∫eSI,N∩N

ftdMα,

we obtain the continuous–discrete decomposition:

Xtt∈Td= X I

t ∨ X Nt t∈T .

The components X It and X N

t are independent.Intuition: Suppose I = 1 and N > 0. Then,

X It =

∫e(0,1)

ftdMα and X Nt =

N∨i=1

ft(i)Zi ,

where Zi = Mαi are i.i.d. standard α−Frechet, independent of X It .

X It and X N

t t∈T are called the continuous and discrete spectralcomponents of X .Fine print: One of them may be zero.

• The continous–discerete decomposition does not depend on therepresentation.Thm (Wang & S., 2009) Let gtt∈T ⊂ Lα+(SI ′,N′ , λI ′,N′) be anotherminimal rep of X with standardized support, then (I ,N) ≡ (I ′,N ′),

d= X I ′

t t∈T and X Nt

d= X N′

t t∈T ,

where X I ′

t :=∫e

SI,N∩(0,1)gtdMα and X N′

t =∫e

SI,N∩N gtdMα.

• Moreover, for the discrete component, we have that:

X Nt t∈T

d= N∨

φt(i)Zi

for some unique set of functions φt(i), 1 ≤ i ≤ N.

X It and X N

• The continous–discerete decomposition does not depend on therepresentation.

Thm (Wang & S., 2009) Let gtt∈T ⊂ Lα+(SI ′,N′ , λI ′,N′) be anotherminimal rep of X with standardized support, then (I ,N) ≡ (I ′,N ′),

d= X I ′

t t∈T and X Nt

d= X N′

t t∈T ,

where X I ′

t :=∫e

t =∫e

SI,N∩N gtdMα.

X Nt t∈T

d= N∨

φt(i)Zi

X It and X N

• The continous–discerete decomposition does not depend on therepresentation.Thm (Wang & S., 2009) Let gtt∈T ⊂ Lα+(SI ′,N′ , λI ′,N′) be anotherminimal rep of X with standardized support, then (I ,N) ≡ (I ′,N ′),

d= X I ′

t t∈T and X Nt

d= X N′

t t∈T ,

where X I ′

t :=∫e

t =∫e

SI,N∩N gtdMα.

X Nt t∈T

d= N∨

φt(i)Zi

Discrete Principal Components

Consider the spectrally discrete component of the process Xtt∈T :

X Nt =

N∨i=1

φt(i)Zi , (t ∈ T ),

for i.i.d. standard α−Frechet Zi ’s.• The functions t 7→ φt(i), 1 ≤ i ≤ N are unique up to permutation ofthe indices. The φt(i)’s are the discrete principal components of X .

• Not all sequences of positive functions can be discrete principalcomponents.Fine print: Prop: (Wang & S., 2009) A countable set of functions φ := φt (i) ≥ 0, 1 ≤ i ≤ N can be

discrete principal components of an α−Frechet process if and only if, φ is a minimal representation. Namely, if (i)PNi=1 φ

αt (i) <∞ and (ii) ρ(φt (·), t ∈ T ) = 21,··· ,N.

Discrete Principal Components

Consider the spectrally discrete component of the process Xtt∈T :

X Nt =

N∨i=1

φt(i)Zi , (t ∈ T ),

for i.i.d. standard α−Frechet Zi ’s.• The functions t 7→ φt(i), 1 ≤ i ≤ N are unique up to permutation ofthe indices. The φt(i)’s are the discrete principal components of X .• Not all sequences of positive functions can be discrete principalcomponents.Fine print: Prop: (Wang & S., 2009) A countable set of functions φ := φt (i) ≥ 0, 1 ≤ i ≤ N can be

discrete principal components of an α−Frechet process if and only if, φ is a minimal representation. Namely, if (i)PNi=1 φ

αt (i) <∞ and (ii) ρ(φt (·), t ∈ T ) = 21,··· ,N.

Discrete Principal Components: Applications and Implications

Applications: Given a spectrally discrete statistical model, estimate:

The order N, if finite. The (unique) principal component functions t 7→ φt(i), for 1 ≤ i ≤ N.

Thm (Wang & S., 2009) Let Xtt∈R be a measurable and stationaryα−Frechet process. Then, the spectrally discrete component of X iseither zero or trivial, i.e.

X Nt t∈R

d= Zt∈R,

for some random variable Z .• Certainly, with i.i.d. α−Frechet Zi ’s:

Xt :=∨i∈Z

f (t − i)Zi , (t ∈ Z),

is a non–trivial stationary and spectrally discrete process.

Applications: Given a spectrally discrete statistical model, estimate: The order N, if finite.

The (unique) principal component functions t 7→ φt(i), for 1 ≤ i ≤ N.

X Nt t∈R

d= Zt∈R,

Xt :=∨i∈Z

f (t − i)Zi , (t ∈ Z),

Applications: Given a spectrally discrete statistical model, estimate: The order N, if finite. The (unique) principal component functions t 7→ φt(i), for 1 ≤ i ≤ N.

X Nt t∈R

d= Zt∈R,

Xt :=∨i∈Z

f (t − i)Zi , (t ∈ Z),

X Nt t∈R

d= Zt∈R,

for some random variable Z .

• Certainly, with i.i.d. α−Frechet Zi ’s:

Xt :=∨i∈Z

f (t − i)Zi , (t ∈ Z),

X Nt t∈R

d= Zt∈R,

Xt :=∨i∈Z

f (t − i)Zi , (t ∈ Z),

Stationary Max–Stable Processes and Flows

Let now Xt =∫e

Uft(u)Mα(du), (t ∈ R) be stationary.

As in the sum–stable case, if ftt∈T is minimal:

ft(u) =(dµ φt

dµ(u))1/α

f0(φt(u)),

where φt : U → U is a non–singular flow:

(i) φt+s = φt φs , ∀t, s ∈ R, (ii) φ0 = id, and (iii) µ φt ∼ µ, ∀t ∈ R.

Intuition: by stationarity and Thm 1:

ft+s(u) = ht+s(u)f0(φt+s(u)) = ht(u)fs(φt(u)) = ht(u)hs(u)f0(φt(φs(u))).

• The flow φtt∈R associated with X is unique (does not depend on theminimal rep), up to flow–equivalence. As in the sum–stable case, the structure of φtt∈R’s motivatesclassifications of the X ’s.

Let now Xt =∫e

ft(u) =(dµ φt

dµ(u))1/α

f0(φt(u)),

where φt : U → U is a non–singular flow:(i) φt+s = φt φs , ∀t, s ∈ R, (ii) φ0 = id, and (iii) µ φt ∼ µ, ∀t ∈ R.

Let now Xt =∫e

ft(u) =(dµ φt

dµ(u))1/α

f0(φt(u)),

Let now Xt =∫e

ft(u) =(dµ φt

dµ(u))1/α

f0(φt(u)),

• The flow φtt∈R associated with X is unique (does not depend on theminimal rep), up to flow–equivalence.

As in the sum–stable case, the structure of φtt∈R’s motivatesclassifications of the X ’s.

Let now Xt =∫e

ft(u) =(dµ φt

dµ(u))1/α

f0(φt(u)),

Hopf Decomposition

Let φ : U → U be non–singular bijection.• B ⊂ U is a wandering set for φ if φk (B) ∩ φj (B) = ∅, (∀k 6= j ∈ Z).

Hopf decomposition: We have U = C ∪ D, where:(i) C ∩ D = ∅ and C and D are φ−invariant.(ii) C has no wandering sub–set of positive measure (for φ).(iii) D = ∪k∈Zφ

k (B) for some wandering set B ⊂ D.

For a measurable flow φtt∈R, we have U = Ct ∪ Dt , where

Ct = C and Dt = D , ( mod µ).

Characterization: for any strictly positive f ∈ Lα+(U, µ),

ft(u)αdt =∞

and D =

ft(u)αdt <∞

with ft(u) =(

dµφt

dµ (u))1/α

f (φt(u)).

Hopf Decomposition

Let φ : U → U be non–singular bijection.• B ⊂ U is a wandering set for φ if φk (B) ∩ φj (B) = ∅, (∀k 6= j ∈ Z).Hopf decomposition: We have U = C ∪ D, where:(i) C ∩ D = ∅ and C and D are φ−invariant.(ii) C has no wandering sub–set of positive measure (for φ).(iii) D = ∪k∈Zφ

ft(u)αdt =∞

and D =

ft(u)αdt <∞

with ft(u) =(

dµφt

dµ (u))1/α

f (φt(u)).

Hopf Decomposition

ft(u)αdt =∞

and D =

ft(u)αdt <∞

with ft(u) =(

dµφt

dµ (u))1/α

f (φt(u)).

Hopf Decomposition

ft(u)αdt =∞

and D =

ft(u)αdt <∞

with ft(u) =(

dµφt

dµ (u))1/α

f (φt(u)).

Conservative–Dissipative Decomposition

Classification of stationary α–Frechet processes via the underlying flowstructure:

Thm (Wang & S., 2009) Let Xtt∈R be a stationary α–Frechet processwith spectral representation ftt∈R ∈ Lα+(U, du). Then,

Xtt∈Rd=

X Ct ∨ X D

with X Ct =

ftdMα and X Dt =

ftdMα. This decomposition is uniquein distribution.

We say X Ct t∈R is generated by a conservative flow and X D

t t∈R is

generated by a dissipative flow.

Xtt∈Rd=

X Ct ∨ X D

with X Ct =

ftdMα and X Dt =

t t∈R is

Xtt∈Rd=

X Ct ∨ X D

with X Ct =

ftdMα and X Dt =

t t∈R is

Mixed Moving Maxima Characterization

As in the sum–stable case, the purely dissipative processes are preciselythe mixed moving maxima:

Thm (Wang & S., 2009) X is generated by a dissipative flow, iff

Xtt∈Rd= ∫e

g(t + x , v)Mα(dx , dv)

t∈R,

for some g(t, v) ∈ Lα+(R× V , dx × ν(dv)).• Samorodnitsky’s positive–null decomposition also translates to themax–stable case.Fine print: Recently Kabluchko 2009, has independently obtaineddecompositions of max–stable processes by association with thesum–stable setting.

Moreover, he has shown that a max–stable X is ergodic if and only if it is

generated by a null flow. An exact parallel to Samorodnitsky 2005.

As in the sum–stable case, the purely dissipative processes are preciselythe mixed moving maxima:Thm (Wang & S., 2009) X is generated by a dissipative flow, iff

Xtt∈Rd= ∫e

t∈R,

for some g(t, v) ∈ Lα+(R× V , dx × ν(dv)).

• Samorodnitsky’s positive–null decomposition also translates to themax–stable case.Fine print: Recently Kabluchko 2009, has independently obtaineddecompositions of max–stable processes by association with thesum–stable setting.

Xtt∈Rd= ∫e

t∈R,

for some g(t, v) ∈ Lα+(R× V , dx × ν(dv)).• Samorodnitsky’s positive–null decomposition also translates to themax–stable case.

Fine print: Recently Kabluchko 2009, has independently obtaineddecompositions of max–stable processes by association with thesum–stable setting.

Xtt∈Rd= ∫e

t∈R,

Xtt∈Rd= ∫e

t∈R,

Generalized Brown–Resnick Processes

Thm (Kabluchko, Schlather and de Haan, 2009) Let wt(u)t∈R be zeromean, continuous path, Gaussian process with stationary increments onthe prob space (U, µ). Then the max–stable process

ewt (u)−ασ2t /2Mα(du), t ∈ R,

is stationary.

• When is X = Xtt∈R dissipative? By our results: X is dissipative, if and only if∫

Reαwt (u)−α2σ2

t /2dt <∞ (µ− a.e.). (1)

• If wt is the fractional Brownian motion, then X is dissipative andhence a mixed moving maxima.Fine print: (1) follows from the law of the iterated logarithm of Oodaira, 1972.

is stationary.• When is X = Xtt∈R dissipative?

By our results: X is dissipative, if and only if∫R

eαwt (u)−α2σ2t /2dt <∞ (µ− a.e.). (1)

is stationary.• When is X = Xtt∈R dissipative? By our results: X is dissipative, if and only if∫

Reαwt (u)−α2σ2

t /2dt <∞ (µ− a.e.). (1)

is stationary.• When is X = Xtt∈R dissipative? By our results: X is dissipative, if and only if∫

Reαwt (u)−α2σ2

t /2dt <∞ (µ− a.e.). (1)

A bonus and an open problem

• From Kabluchko, Schlather & de Haan, 2009, we have that ageneralized Brown–Resnick process is dissipative if:

lim|t|→∞

(wt − σ2t /2) = −∞. (2)

By combining with our NSC, we get the bonus:Thm Consider a Gaussian wt process with zero mean, continuouspaths, and stationary increments. The condition (2) implies

ewt (u)−σ2t /2dt <∞

Open question: (To me!) Is there a zero–one law:

= 1 or 0.

for the wt in the above Thm.

A bonus and an open problem

• From Kabluchko, Schlather & de Haan, 2009, we have that ageneralized Brown–Resnick process is dissipative if:

lim|t|→∞

(wt − σ2t /2) = −∞. (2)

By combining with our NSC, we get the bonus:Thm Consider a Gaussian wt process with zero mean, continuouspaths, and stationary increments. The condition (2) implies

Open question: (To me!) Is there a zero–one law:

= 1 or 0.

for the wt in the above Thm.

Co–spectral Functions

Let now T be a separable metric space, equipped with a Borel measure λ.Consider the α−Frechet process X = Xtt∈T :

f (t, u)Mα(du), (t ∈ T ),

where (t, u) 7→ f (t, u) is measurable.

• Focus on the co–spectral functions:

t 7→ f (t, u) ∈ L0+(T , λ), for fixed u ∈ U.

• Can show that the co–spectral functions of X do not depend on therepresentation (up to rescaling)!Fine print: If (U, µ) is standard Lebesgue, then X is measurable, if and only if, (t, u) 7→ f (t, u) has a

measurable modification.

f (t, u)Mα(du), (t ∈ T ),

where (t, u) 7→ f (t, u) is measurable.• Focus on the co–spectral functions:

f (t, u)Mα(du), (t ∈ T ),

• Can show that the co–spectral functions of X do not depend on therepresentation (up to rescaling)!

Fine print: If (U, µ) is standard Lebesgue, then X is measurable, if and only if, (t, u) 7→ f (t, u) has a

f (t, u)Mα(du), (t ∈ T ),

Co–spectral Functions: Classification

Let P be a positive (measurable) cone in L0+(T , λ), i.e. cP ⊂ P.

Consider the partition of U = A ∪ B into disjoint components:

A := u ∈ U : f (·, u) ∈ P and B := U \ A = u ∈ U : f (·, u) 6∈ P.

This yields the decomposition:

Xtt∈Td= X A

t ∨ X Bt t∈T , (3)

X At :=

f (t, u)Mα(du) and X Bt :=

f (t, u)Mα(du)

are two independent processes.

• The decomposition (3) does not depend on the choice of themeasurable rep f (t, u)(t,u)∈T×U .

Idea of proof: WLOG suppose that f (t, u)t∈T is minimal and letg(t, v)t∈T ⊂ Lα+(V , ν) is another measurable rep of X = Xtt∈T .

Then, by Thm 1:

g(t, v) = h(v)f (t, φ(v)), where h(v) ≥ 0.

Since P is a cone,

g(·, v) ∈ P ⇔ f (·, φ(v)) ∈ P,

which shows that the corresponding partition of V is:

V = A ∪ B := φ−1(A) ∪ φ−1(B)

A change of variables, yields: ∫eA

ftdMα

d= ∫e

eA gtdMα

completing the proof.

Applications

Corollary: Let

f (t, u)Mα(du) d=

g(t, v)Mα(dv).

Then, given a cone P ⊂ L0+(T ),

f (·, u) ∈ P, a.e. if and only if g(·, v) ∈ P, a.e.

Thm (Wang & S., 2009) Let (U, µ) ≡ (Rd , dx) and f , g ∈ Lα+(Rd ).Consider the moving maxima random fields

∫eRd

f (t − u)Mα(du) and Yt :=

∫eRd

g(t − u)Mα(du).

Xtt∈Rdd= Ytt∈Rd , iff g(·) = f (·+ τ),

for some τ ∈ Rd .

Proof: Consider the positive cone: P := cf (·+ τ), c > 0, τ ∈ R. We

have g(·+ x) ∈ P, for almost all x!

Applications

Corollary: Let

f (t, u)Mα(du) d=

g(t, v)Mα(dv).

∫eRd

g(t − u)Mα(du).

Applications

Corollary: Let

f (t, u)Mα(du) d=

g(t, v)Mα(dv).

∫eRd

g(t − u)Mα(du).

Thank you!

Some References

Brown, B. M. & Resnick, S. I. (1977), ‘Extreme values of independentstochastic processes’, J. Appl. Probability 14(4), 732–739.Hardin (1982) On the spectral representation of symmetric stableprocesses. Journal of Multivariate Analysis, 12:385–401, 1982.de Haan, L. & Pickands III, J. (1986), ‘Stationary min–stable stochasticprocesses’, Prob. Theo. Rel. Fields 72(4), 477–492.Kabluchko, Schlather & de Haan (2009) Stationary max–stable fieldsassociated to negative definite functions, Preprint.Kabluchko (2009) Spectral representations of sum– and max–stableprocesses, Preprint.Rosinski (1995) On the structure of stationary stable processes. Ann.Probab., 23(3):1163–1187, 1995.Samorodnitsky (2005) Null flows, positive flows and the structure ofstationary symmetric stable processes. Ann. Probab., 33:1782–1803,2005.Wang & S. (2009) On the structure and representations of max–stableprocesses, Preprint.Wang & S. (2009) On the Association of Sum– and Max– StableProcesses, Preprint.

on the structure of max--stable processes schlather & de haan (2009)stationary max{stable elds...

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