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

Abstract : 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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Anne Estrade, Julie Fournier. FINITE VARIANCE OF THE NUMBER OF STATIONARY POINTS OF A GAUSSIAN RANDOM FIELD. MAP5 2015-26. 2015. 〈hal-01192805〉

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