Discrete approximation of stable white noise - Application to spatial linear filtering

Motivated by the simulation of stable random fields, we consider the issue of discrete approximations of independently scattered stable noise. Two approaches are proposed: grid approximations available when the underlying space is \bbR^d and shot noise approximations available on more general spaces. Limit theorems stating the convergence of discrete random noises to stable white noise are proved. These results are then applied to study moving average spatial random fields with heavy-tailed innovations and related limit theorems. A second application deals with discrete approximation for Brownian L\'evy motion on the sphere or on the euclidean space.Comment: 24

Ergodic decompositions of stationary max-stable processes in terms of their spectral functions

We revisit conservative/dissipative and positive/null decompositions of stationary max-stable processes. Originally, both decompositions were defined in an abstract way based on the underlying non-singular flow representation. We provide simple criteria which allow to tell whether a given spectral function belongs to the conservative/dissipative or positive/null part of the de Haan spectral representation. Specifically, we prove that a spectral function is null-recurrent iff it converges to $0$ in the Ces\`{a}ro sense. For processes with locally bounded sample paths we show that a spectral function is dissipative iff it converges to $0$. Surprisingly, for such processes a spectral function is integrable a.s. iff it converges to $0$ a.s. Based on these results, we provide new criteria for ergodicity, mixing, and existence of a mixed moving maximum representation of a stationary max-stable process in terms of its spectral functions. In particular, we study a decomposition of max-stable processes which characterizes the mixing property.Comment: 21 pages, no figure

Strong mixing properties of max-infinitely divisible random fields

Let $\eta=(\eta(t))_{t\in T}$ be a sample continuous max-infinitely random field on a locally compact metric space $T$. For a closed subset $S\in T$, we note $\eta_{S}$ the restriction of $\eta$ to $S$. We consider $\beta(S_1,S_2)$ the absolute regularity coefficient between $\eta_{S_1}$ and $\eta_{S_2}$, where $S_1,S_2$ are two disjoint closed subsets of $T$. Our main result is a simple upper bound for $\beta(S_1,S_2)$ involving the exponent measure $\mu$ of $\eta$: we prove that $\beta(S_1,S_2)\leq 2\int \bbP[\eta\not<_{S_1} f,\ \eta\not <_{S_2} f]\,\mu(df)$, where$f\not<_{S} g$means that there exists$s\in S$such that$f(s)\geq g(s)$. If$\eta$is a simple max-stable random field, the upper bound is related to the so-called extremal coefficients: for countable disjoint sets$S_1$and$S_2$, we obtain$\beta(S_1,S_2)\leq 4\sum_{(s_1,s_2)\in S_1\times S_2}(2-\theta(s_1,s_2))$, where$\theta(s_1,s_2)$is the pair extremal coefficient. As an application, we show that these new estimates entail a central limit theorem for stationary max-infinitely divisible random fields on$\bbZ^d$. In the stationary max-stable case, we derive the asymptotic normality of three simple estimators of the pair extremal coefficient.Comment: 27