Nonsingular Group Actions and Stationary S alpha S Random Fields

This paper deals with measurable stationary symmetric stable random fields indexed by R^d and their relationship with the ergodic theory of nonsingular R^d-actions. Based on the phenomenal work of Rosinski(2000), we establish extensions of some structure results of stationary S alpha S processes to S\alpha S fields. Depending on the ergodic theoretical nature of the underlying action, we observe different behaviors of the extremes of the field.


Introduction
X := {X t } t∈R d is called a symmetric α-stable (SαS) random field if for all c 1 , c 2 , . . . , c k ∈ R and t 1 , t 2 , . . . , t k ∈ R d , k j=1 c j X tj follows a symmetric α-stable distribution. See Samorodnitsky and Taqqu (1994) for more information on SαS distributions and processes. In this paper we will further assume that {X t } t∈R d is measurable and stationary with α ∈ (0, 2).
The Hopf decomposition of nonsingular flows (see Aaronson (1997)) gives rise to a useful decomposition of stationary SαS processes into two independent components; see Rosiński (1995). For a general d > 1, Rosiński (2000) established a similar decomposition of SαS random fields. We show the connection between this work and the conservative-dissipative decomposition of nonsingular R d -actions. This connection with ergodic theory enables us to study the rate of growth of the partial maxima {M τ } τ >0 of the random field X t as t runs over a d-dimensional hypercube with an edge length τ increasing to infinity. This is a straightforward extension of the one-dimensional version of this result available in Samorodnitsky (2004b). See Samorodnitsky (2004a) and Roy and Samorodnitsky (2008) for the discrete parameter case. This paper is organized as follows. In Section 2, we develop the theory of nonsingular R d -actions based on Aaronson (1997) and Kolodyński and Rosiński (2003). We extend some of the structure results of stationary SαS processes available in Rosiński (1995) to the d > 1 case in Section 3 and use these results in Section 4 to compute the rate of growth of the partial maxima M τ of the field as τ increases to infinity.

Nonsingular R d -actions
In this section we present the theory of nonsingular R d -actions in parallel to the corresponding discrete-parameter results discussed in Section 2 in Roy and Samorodnitsky (2008). Most of the notions discussed in this section can be found in Aaronson (1997) and Krengel (1985).
Let {φ t } t∈R d be a nonsingular R d -action on a σ-finite standard measure space (S, S, µ). This means that {φ t } t∈R d is a collection of measurable transformations The following result is a partial extension of Corollary 1.6.5 in Aaronson (1997) to nonsingular R d -actions.
Proof. Let C n be the conservative part of {φ t } t∈Γn for all n ≥ 0 and λ be the Lebesgue measure on R d . By Theorem A.1 in Kolodyński and Rosiński (2003), there exists a strictly positive measurable function (t, s) for µ-almost all s ∈ S, and for all t, h ∈ R d and for all s ∈ S (2.1) w t+h (s) = w h (s)w t (φ h (s)).
Motivated by Proposition 2.1, we define the conservative (resp. dissipative) part of {φ t } t∈R d to be C 0 (resp. D 0 := S \ C 0 ). Then from the proof of Proposition 2.1, we get the following continuous parameter analogue of Corollary 2.4 in Roy and Samorodnitsky (2008).
where w t (s) is as above.
Remark 2.3. Note that Theorem A.1 in Kolodyński and Rosiński (2003) takes care of the measurability issues regarding the Radon Nikodym derivatives very nicely.
As in the discrete case, the action {φ t } is called conservative if S = C and dissipative if S = D. Recall that nonsingular group actions {φ t } t∈R d and {ψ t } t∈R d , defined on standard measure spaces (S, S, µ) and (T, T , ν) resp., are equivalent if there is a Borel isomorphism Φ : In light of Corollary 2.2, we can rephrase Theorem 2.2 in Rosiński (2000) to obtain Krengel's structure theorem (see Krengel (1969)) for dissipative nonsingular R d -actions.

Structure of Stationary SαS Random Fields
Suppose X = {X t } t∈R d is a stationary measurable SαS random field, 0 < α < 2. Every measurable minimal representation (this exists by Theorem 2.2 in Rosiński (1995)) of X is of the from Rosiński (1995) for the d = 1 case and Rosiński (2000) for a general d.
Conversely, {X t } defined as above is a stationary measurable SαS random field. Without loss of generality we can assume that the family {f t } in (3.1) satisfies the full support assumption and take the Radon-Nikodym derivative in (3.1) to be equal to w t (s) defined in Section 2 by virtue of Theorem A.1 in Kolodyński and Rosiński (2003). We first establish that any measurable stationary random field indexed by R d is continuous in probability. The corresponding one-dimensional result was established by Surgailis et al. (1998) using a result of Cohn (1972).
Proposition 3.1. Suppose X = {X t } t∈R d be a measurable stationary random field. Then X is continuous in probability, i.e., for every t 0 ∈ R d , X t p −→ X t0 whenever t → t 0 .
Proof. Using a truncation argument we can assume without loss of generality that X 0 2 < ∞ where · 2 denotes the L 2 -norm. Define {φ t } t∈R d to be the shift action on the path-space Ω given by φ t (ω)(s) = ω(s + t) for all ω ∈ Ω. By measurability and stationarity of X, {φ t } is an R d -action which preserves the induced probability measure. Using Banach's theorem for Polish groups (see Banach (1932) p. 20) it follows that t → X t is L 2 -continuous (see Section 1.6 in Aaronson (1997)), which implies the result.
As in the discrete parameter case, we say that a measurable stationary SαS random field {X t } t∈R d is generated by a nonsingular R d -action {φ t } on (S, µ) if it has an integral representation of the form (3.1) satisfying (3.2). The following result, which is the continuous parameter analogue of Proposition 3.1 in Roy and Samorodnitsky (2008), yields that the classes of measurable stationary SαS random fields generated by conservative and dissipative actions are disjoint. The corresponding onedimensional result is available in Theorem 4.1 of Rosiński (1995).
Proposition 3.2. Suppose {X t } t∈R d is a measurable stationary SαS random field generated by a nonsingular R d -action {φ t } on (S, µ) and {f t } is given by (3.1). Let C and D be the conservative and dissipative parts of {φ t }. Then we have In particular, if a stationary SαS random field {X t } t∈R d is generated by a conservative (dissipative, resp.) R d -action, then in any other integral representation of {X t } of the form (3.1) satisfying (3.2), the R d -action must be conservative (dissipative, resp.).

Proof. Let
where s ∈ S, a γ > 0 for all γ ∈ Z d and γ∈Z d a γ = 1. Clearly h ∈ L 1 (S, µ) and h > 0 almost surely. By (2.1) and the translation invariance of λ, for all s ∈ S. Hence, by Corollary 2.4 in Roy and Samorodnitsky (2008), we get This completes the proof of the first part. The second part follows by an argument parallel to the one in the proof of Theorem 4.1 in Rosiński (1995).
The following corollary is the continuous parameter analogue of Corollary 3.2 of Roy and Samorodnitsky (2008). The corresponding one-dimensional result is available in Corollary 4.2 of Rosiński (1995) and the same proof works in the ddimensional case.
Corollary 3.3. The measurable stationary SαS random field {X t } t∈R d is generated by a conservative (dissipative, resp.) R d -action if and only if for any (equivalently, some) measurable representation Recall that Surgailis et al. (1993) defined X to be a stable mixed moving average if where f ∈ L α (W × R d , ν ⊗ λ), λ is the Lebesgue measure on R d , ν is a σ-finite measure on a standard Borel space (W, W), and the control measure µ of M equals ν ⊗ λ. The following result gives three equivalent characterizations of stationary SαS random fields generated by dissipative actions.
Theorem 3.4. Suppose {X t } t∈R d is a measurable stationary SαS random field. Then the following are equivalent: (1) {X t } is generated by a dissipative R d -action.
(2) For any measurable representation {f t } of {X t } we have, (3) {X t } is a mixed moving average. (4) {X t } t∈Γn is a mixed moving average for some (all) n ≥ 1.

Proof.
(1) and (2) are equivalent by Corollary 3.3, (2) and (3) are equivalent by Theorem 2.1 of Rosiński (2000). (1) and (4) are equivalent by Theorem 3.3 in Roy and Samorodnitsky (2008) and Proposition 2.1. Therefore, in order to verify that X is a mixed moving average, it is enough to verify it on a discrete skeleton (e.g., {X t } t∈Z d ) of the random field. Theorem 3.4 allows us to describe the decomposition of a stationary SαS random field given in Theorem 3.7 of Rosiński (2000) in terms of the ergodic-theoretical properties of nonsingular R d -actions generating the field. See Corollary 3.4 in Roy and Samorodnitsky (2008) for the corresponding discrete parameter result.
Corollary 3.5. A stationary SαS random field X has a unique in law decomposition where X C and X D are two independent stationary SαS random fields such that X D is a mixed moving average, and X C is generated by a conservative action.

A Note on the Extreme Values
The extreme values of {X t } are expected to grow at a slower rate if {X t } is generated by a conservative action because of longer memory; see, for example, Samorodnitsky (2004a), Samorodnitsky (2004b) and Roy and Samorodnitsky (2008). This can be formally proved provided X = {X t } t∈R d is assumed to be locally bounded apart from being stationary and measurable. If further X is separable then is a well-defined finite-valued stochastic process. Here u = (u (1) , . . . , u (d) ) ≤ v = (v (1) , . . . , v (d) ) means u (i) ≤ v (i) for all i = 1, 2, . . . , d and 1 := (1, 1, . . . , 1), 0 := (0, 0, . . . , 0). Since X is stationary and measurable, it is continuous in probability by Proposition 3.1. Therefore, as in the one-dimensional case in Samorodnitsky (2004b), taking its separable version the above maxima process can be defined by [u, v] := {s ∈ R d : u ≤ s ≤ v}. This will avoid the usual measurability problems of the uncountable maximum (4.1). The next result is the continuous parameter extension of Theorem 4.3 in Roy and Samorodnitsky (2008). It follows by the exact same argument as in the one-dimensional version of this result (Theorem 2.2 in Samorodnitsky (2004b)) based on Theorem 3.4 and Corollary 3.5.
(i) Suppose that X is not generated by a conservative action (i.e. the component X D in (3.4) generated by the dissipative part is nonzero). Then for any representation of X D in the mixed moving average form (3.3), C α is the stable tail constant (see (1.2.9) in Samorodnitsky and Taqqu (1994)) and Z α is the standard Fréchet-type extreme value random variable with distribution (ii) Suppose that X is generated by a conservative R d -action. Then for all s ∈ S and let U (τ ) j , j = 1, 2 be independent S-valued random variables with common law η τ . Suppose that (4.2) holds and for any ǫ > 0, as τ → ∞. Then 1 b τ M τ ⇒ C 1/α α Z α as τ → ∞. A sufficient condition for (4.3) is lim τ →∞ τ −d/2α b τ = ∞.
Theorem 4.1 gives the exact rate of growth of the maxima only when the underlying group action is not conservative. In the conservative case, the exact rate depends on the group action as well as on the kernel (see the examples in Samorodnitsky (2004a), Samorodnitsky (2004b) and Roy and Samorodnitsky (2008)). For instance, by an obvious extension of Example 6.1 in Roy and Samorodnitsky (2008) to the continous parameter case, it can be observed that the maxima can grow both polynomially as well as logarithmically and it can even converge to a nonextreme value limit after proper normalization.
In the discrete parameter case, depending on the group theoritic properties of the underlying action, a better estimate of this rate is given in Roy and Samorodnitsky (2008); see also Roy (2007b). This connection with abelian group theory is still an open problem in the continuous parameter case and hence needs to be investigated. Two more open problems related to this work are extensions of the results of Samorodnitsky (2005) and Roy (2007a) to the d-dimensional case.
to the anonymous referee for his/her valuable suggestions all of which contributed significantly to this work.