arXiv subjects
Publications and source records attributed to Frédéric Eyi-Minko.
We prove that the class of discrete time stationary max-stable process satisfying the Markov property is equal, up to time reversal, to the class of stationary max-autoregressive processes of order $1$. A similar statement is also proved for continuous time processes.
This paper is devoted to the prediction problem in extreme value theory. Our main result is an explicit expression of the regular conditional distribution of a max-stable (or max-infinitely divisible) process $\{η(t)\}_{t\in T}$ given observations $\{η(t_i)=y_i,\ 1\leq i\leq k\}$. Our starting point is the point process representation of max-infinitely divisible processes by Giné, Hahn and Vatan (1990). We carefully analyze the structure of the underlying point process, introduce the notions of extremal function, sub-extremal function and hitting scenario associated to the constraints and derive the associated distributions. This allows us to explicit the conditional distribution as a mixture over all hitting scenarios compatible with the conditioning constraints. This formula extends a recent related result by Wang and Stoev (2011) dealing with the case of spectrally discrete max-stable random fields. We believe this work offers new tools and perspective for prediction in extreme value theory together with numerous potential applications.
Let $η=(η(t))_{t\in T}$ be a sample continuous max-infinitely random field on a locally compact metric space $T$. For a closed subset $S\in T$, we note $η_{S}$ the restriction of $η$ to $S$. We consider $β(S_1,S_2)$ the absolute regularity coefficient between $η_{S_1}$ and $η_{S_2}$, where $S_1,S_2$ are two disjoint closed subsets of $T$. Our main result is a simple upper bound for $β(S_1,S_2)$ involving the exponent measure $μ$ of $η$: we prove that $β(S_1,S_2)\leq 2\int \bbP[η\not<_{S_1} f,\ η\not <_{S_2} f]\,μ(df)$, where $f\not<_{S} g$ means that there exists $s\in S$ such that $f(s)\geq g(s)$. If $η$ 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 $β(S_1,S_2)\leq 4\sum_{(s_1,s_2)\in S_1\times S_2}(2-θ(s_1,s_2))$, where $θ(s_1,s_2)$ 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.
For each $n\geq 1$, let $ {X_{in}, \quad i \geq 1} $ be independent copies of a nonnegative continuous stochastic process $X_{n}=(X_n(t))_{t\in T}$ indexed by a compact metric space $T$. We are interested in the process of partial maxima [\tilde M_n(u,t) =\max \{X_{in}(t), 1 \leq i\leq [nu]},\quad u\geq 0,\ t\in T.] where the brackets $[\,\cdot\,]$ denote the integer part. Under a regular variation condition on the sequence of processes $X_n$, we prove that the partial maxima process $\tilde M_n$ weakly converges to a superextremal process $\tilde M$ as $n\to\infty$. We use a point process approach based on the convergence of empirical measures. Properties of the limit process are investigated: we characterize its finite-dimensional distributions, prove that it satisfies an homogeneous Markov property, and show in some cases that it is max-stable and self-similar. Convergence of further order statistics is also considered. We illustrate our results on the class of log-normal processes in connection with some recent results on the extremes of Gaussian processes established by Kabluchko.