Max stable processes: representations, ergodic properties and some statistical applications

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1 Max stable processes: representations, ergodic properties and some statistical applications Stilian Stoev University of Michigan, Ann Arbor Oberwolfach March 21, 2008

2 The focus Let X = {X t } t R be a (strictly) stationary max stable process. Questions: When is X ergodic or mixing? How can one check ergodicity/mixing for X? Answers: Must be in terms of representations for max stable processes.

3 1 The focus 2 Representations of max stable processes 3 The problems of ergodicity and mixing 4 Results 5 xamples 6 Statistical applications 7 References

4 Max stable processes The process X = {X t } t R is a max stable process if its finite dimensional distributions (fdd) are multivariate max stable: Definition A vector X = {X j } 1 j d is max stable if for all k N, there exist a k R d +, b k R d, such that 1 a k k X (i) d b k = X, i=1 where X (i), 1 i k are independent copies of X. The marginals of X are either Fréchet, Gumbel or negative Fréchet.

5 Fréchet processes ξ is an α Fréchet variable (α > 0) with scale σ > 0, if P{ξ x} = exp{ (x/σ) α } = exp{ σ α x α }, x (0, ). P{ξ > x} σ α x α, as x. Definition {X t } t R is an α Fréchet process if for all a i > 0 and t i, 1 i d, the max linear combinations are α Fréchet. max{a i X ti, 1 i d} = d a i X ti, i=1

6 xamples (moving maxima) For iid α Fréchet Z i s, X t := c i Z t i, t Z i=0 is well defined stationary α Fréchet process if c i 0 are such that ci α <. i=0 Observe that, x > 0 : P{X t x} = exp{ (x/c i ) α } = exp{ ( ci α )x α }. i=0 i=0 Similarly, one gets that a 1 X t1 a d X td is α Fréchet, for all a i 0, t i Z.

7 Fréchet and max stable processes The α Fréchet processes are max stable. How rich is the class of α Fréchet processes? Theorem (de Haan (1978)) A process X with α Fréchet marginals is max stable if and only if it is an α Fréchet process. The class of Fréchet processes is sufficient for the study of ergodicity. We focus on max-stable processes with Fréchet marginals or, equivalently, Fréchet processes.

8 The de Haan s spectral representation Let X = {X t } t R be a continuous in probability α Fréchet process. There exists a measure ρ on (0, 1), and f t (u) 0, 1 0 f t α (u)ρ(du) <, such that { fdd {X t } t R = k=1 f t (U k ) } ɛ 1/α, t R k where {(U k, ɛ k )} is a Poisson point process on (0, 1) (0, ) with intensity ρ(du) dx. Note: Similar to the LePage Woodroofe Zinn series representation for α stable processes (e.g. for 0 < α < 1): {Y t } t R d = { k=1 f t (U k ) } ɛ 1/α t R k

9 Fréchet random sup measures quivalently, one can use extremal integrals: { e } fdd {X t } t R = f t (u)m α (du) (0,1) t R where M α is an α Fréchet sup measure on (0, 1) with control measure ρ(du). Definition M α is α Fréchet sup measure on (, ) with control measure µ, if: (i) independently scattered (ii) P{M α (A) x} = exp{ µ(a)x α }, x > 0 (iii) σ sup additive: M α ( n=1a n ) = M α (A n ), n=1 almost surely. Analogous to the α stable measures in the sum stable case (see Samorodnitsky and Taqqu (1994)).

10 xtremal integrals Intuition: Let A B = and a, b > 0. By independence, for x > 0 P{aM α (A) bm α (B) x} = e µ(a)(x/a) α e µ(b)(x/b) α = e (aα µ(a)+b α µ(b))x α. For a simple function f (u) = n k=1 a k1 Ak (u), a k 0, define I (f ) := e f (u)m α (du) := n a k M α (A k ), k=1 to be the extremal integral of f w.r.t. M α.

11 xtremal integrals: properties 1 For simple f (u) 0, P{I (f ) x} = e R f α dµ x α, x > 0. 2 For constants a, b > 0, and simple f (u), g(u) 0: I (af bg) = ai (f ) bi (g). 3 I (f i ), 1 i d are independent if and only if f i, 1 i d have disjoint supports (µ a.e.). e Note: fdm α can be defined for all measurable f (u) 0, with f α dµ <.

12 xtremal integrals: benefits If X t = i=0 c e iz t i, t Z, then X t = R f (t u)m α(du), with M α (i 1, i] = Z i, i Z, where f (u) := i=0 c i1 [i,i+1) (u), 0 u <. General α Fréchet processes: X t = e f t (u)m α (du), Finite dimensional distributions: t R, f t (u) L α +(,, µ(du)). P{X tj x j, 1 j d} = P{ d j=1x 1 j X tj 1} e { ( d P{ ( d j=1x 1 f tj ) αdµ } j f tj )dµ 1} = exp x j One can explicitly handle the fdd s via the spectral functions f t! j=1

13 Fréchet spaces and max linear isometries Consider M = {I (f ), f L α +(µ)}. The class M is closed w.r.t: max linear combinations convergence in probability which is metrized by ρ M (I (f ), I (g)) := ρ α (f, g) := f α g α dµ (see Davis and Resnick (1993) context α = 1). Definition U : L α +(µ) L α +(µ) is a max linear isometry if: (i) U(af bg) = au(f ) bu(g) (ii) f α dµ = U(f ) α dµ.

14 Problem formulation The process: Let X = {X t } t R, X t := e U t (f )dm α, < t <, where f L α +(µ) and {U t } is a group of max linear isometries on L α +(µ). Then X is a stationary, α Fréchet process. Intuition: For f (u) L α +(du), think U t (f )(u) = f (t + u)! Goals: we look for necessary and sufficient conditions (on {U t } and f ) for ergodicity/mixing of X. Notes: The max linear isometires {U t } are essentially the pistons of de Haan and Pickands (see S. (2007)). X = {X t } t R has a measurable modification if and only if it is continuous in probability (see S. (2007)).

15 General results: ergodicity Theorem (S. (2007)) X = {X t } t R is ergodic, if and only if, for some (any) p > 0, 1 T U τ g g p T L α (µ) dτ 0, as T, 0 for all g F := span{u t f, t R}. Here F L α +(µ) contains all positive max linear combinations of U t f s and is closed in the metric ρ α. Namely, all limits in ρ α of the max linear combinations a 1 U t1 (f ) a n U tn (f ), a i 0, t i R.

16 General results: mixing Theorem (S. (2007)) X = {X t } t R is mixing, if and only if, U τ h g L α + (µ) 0, as τ, for all g, h F = span{u t f, t R}. The proofs borrow ideas from the α (sum)stable case Cambanis, Hardin and Weron (1987). Key Idea: f g L α + (µ) measures the dependence b/w e ξ = fdm e α and η = gdm α.

17 Outline of the proof (mixing) It is enough to focus on events of the type: A = {X si x i, 1 i d} and B = {X ti y i, 1 i d}, with s i, t i R and x i, y i 0. Then, B τ = {X τ+ti y i, 1 i d} and also: P(A) = P{ d i=1x 1 i X si 1} = exp{ P(B) = exp{ h α dµ}, where g(u) = d i=1x 1 i g α dµ} f si (u) and h(u) = d i=1y 1 f ti (u). Note that P(B τ ) = P{ d i=1y 1 i X τ+ti 1} = exp{ i U τ (h) α dµ}.

18 Proof... and, similarly, by the max linearity of the extremal integrals: P(A B τ ) = exp{ (g α U τ (h) α )dµ}. Thus P(A B τ )/P(A)P(B) 1 as τ if and only if exp{ g α dµ + h α dµ g α U τ (h) α dµ} = exp{ g α U τ (h) α dµ} 1 (τ )

19 A natural measure of dependence Definition For jointly α Fréchet ξ and η, d α (ξ, η) := ξ α α + η α α ξ η α α, is a measure of the dependence between ξ and η. Here ξ α is the scale of ξ: P{ξ x} = exp{ ξ α αx α }, (x > 0). ξ and η are independent if and only if d α (ξ, η) = 0. e If ξ = fdm e α and η = gdm α then d α (ξ, η) = f α dµ + g α dµ f α g α dµ = f α g α dµ

20 Mixing: a simpler criterion Theorem (S. (2007)) A continuous in probability stationary α Fréchet process X = {X t } t R is mixing if and only if d α (X t, X 0 ) 0, as t. The condition d α (X τ, X 0 ) 0, τ 0 is easy to check. One and the same process X = {X t } may have different representations (f, {U t }, M α ), but has only one dependence function d α (τ) = d α (X τ, X 0 ) = X τ α α + X 0 α α X τ X 0 α α.

21 Moving maxima (moving maxima) Let f L α +(dx) and µ(dx) = dx. Then, the process X t := e f (t + x)m α (dx), < t <, R is mixing. Indeed, d α (τ) = f α (τ + x) f α (x)dx 0, as τ. R (mixed moving maxima) (M3 processes of Smith and Weissman) For f (t, v) L α +(dx ν(dv)) and µ(dt, dv) = dt ν(dv), the process is mixing. X t := e R V f (t + x, v)m α (dx, dv), < t <,

22 Non ergodic processes (non ergodicity) Let (,, µ) be ([0, 2π), B, dx) and X t = e sin 2 (t + x)m α (dx), < t <. [0,2π) The process X = {X t } t R is strictly stationary and non ergodic. (a field) The stationary and non isotropic field X (s, t) = e sin 2 (s + u) e t+v M α (du, dv) [0,π) R is non ergodic along s but ergodic along t.

23 An illustration A movie...

24 An example of Brown and Resnick Let (,, µ) be a probability space and w = {w t } t 0 be a standard Brownian motion there. Define the doubly stochastic process X t = e e wt(u) αt/2 M α (du), 0 t <, where M α, α > 0 is defined on a different probability space (Ω, F, P). Brown and Resnick (1977) introduced X = {X t } t 0 and showed that it is stationary. Theorem (S. (2007)) X is exponentially mixing. Namely, d α (τ) e cτ, τ > 0, for some c > 0.

25 stimation of the dependence function Let {X t } t R be stationary and ergodic. Problem: stimate the dependence function d α (τ) := d α (X τ, X 0 ). A consistent estimator of the dependence function is: d α,p,n (τ) := c p,α ( 2 n n k=1 X p k where 0 < p < α and c p,α = Γ(1 p/α) α/p. By ergodicity, for all γ (0, 1) ) α/p cp,α ( n τ 1 (X τ+k X k ) p) α/p, n τ k=1 d α,p,n (τ) a.s. d α (τ) and d α,p,n (τ) d α (τ) γ 0, as n.

26 A numerical example 3 x 10 4 Moving Maxima Time Series (Part): α = 1, N=215 = 32, Dependence Function stimate 80 st. 60 True Spectral Function d α (τ) f(u) τ u

27 Concluding remarks Weintraub (1991) introduced 3 notions of mixing in terms of the de Haan s spectral representation. No connections with the classical notions of ergodicity and mixing was established. We show that Weintraub s 0 mixing is equivalent to mixing. Our work justifies and suggests a range of statistical methods for max stable processes. Some old new tools on modeling and statistics for max stable processes/fields? Further questions on: estimation of the spectral function representations of max stable processes random fields

28 Some references Brown, B. M. & Resnick, S. I. (1977), xtreme values of independent stochastic processes, J. Appl. Probability 14(4), Cambanis, S., Hardin, Jr., C. D. & Weron, A. (1987), rgodic properties of stationary stable processes, Stoch. Proc. Appl. 24, Davis, R. & Resnick, S.I. (1993), Prediction of stationary max stable processes, Ann. Appl. Probab., 3(2), de Haan, L. (1978), A characterization of multidimensional extreme-value distributions, Sankhyā (Statistics). The Indian Journal of Statistics. Series A 40(1), de Haan, L. (1984), A spectral representation for max stable processes, Ann. Probab. 12(4), Stoev, S. & Taqqu, M. S. (2005), xtremal stochastic integrals: a parallel between max stable processes and α stable processes, xtremes 8, Stoev, S. (2007), On the ergodicity and mixing of max stable processes, Stoch. Proc. Appl. To appear. Weintraub, K. S. (1991), Sample and ergodic properties of some min stable processes, Ann. Probab. 19(2),