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Pré-Publication, Document De Travail Année : 2015

FINITE VARIANCE OF THE NUMBER OF STATIONARY POINTS OF A GAUSSIAN RANDOM FIELD

Résumé

Let X be a real-valued stationary Gaussian random field defined on $R^d$ (d ≥ 1), with almost every realization of class $C^2$. This paper is concerned with the random variable giving the number of points in $T$ (a compact set of $R^d$) where the gradient $X'$ takes a fixed value $v\in R^d$, $N_{X'}(T, v) = \{t \in T : X'(t) = v\}$. More precisely, it deals with the finiteness of the variance of $N_{X'} (T, v)$, under some non-degeneracy hypothesis on $X$. For d = 1, the so-called " Geman condition " has been proved to be a sufficient condition for $N_{X'} (T, v)$ to admit a finite second moment. This condition on the fourth derivative $r^{(4)}$ of the covariance function of $X$ does not depend on $v$ and requires $t \mapsto \frac{ r ^{(4)}(0)−r ^{(4)}(t)}{t} $ to be integrable in a neighbourhood of zero. We prove that for d ≥ 1, a generalization of the Geman condition remains a sufficient condition for $N_{X'} (T, v)$ to admit a second moment. No assumption of isotropy is required.
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Dates et versions

hal-01192805 , version 1 (05-09-2015)

Identifiants

  • HAL Id : hal-01192805 , version 1

Citer

Anne Estrade, Julie Fournier. FINITE VARIANCE OF THE NUMBER OF STATIONARY POINTS OF A GAUSSIAN RANDOM FIELD. 2015. ⟨hal-01192805⟩

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