Surfaces aléatoires : approximation du temps local

Berzin, Corinne

Directeur de thèse : Dacunha-Castelle, Didier

Membres du jury : Dacunha-Castelle, Didier ; Bretagnolle, Jean ; Florens-Zmirou, Danielle ; Guyon, Xavier ; Labeyrie, Jacques

Thèses d'Orsay, no. 241 (1989) , 126 p.

Résumé

Let $X (t, ω) t \in ℝ^{d}, ω \in Ω, d \geq 2$ be a real stationary gaussian field, defined on a probability space $(Ω, 𝒜 P)$ . We look at the asymptotic behavior of a particular stochastic integral, with respect to the geometric measure of the $u$ -level sets, $u \in ℝ$ , of the regularized field, obtained by composition of a convolution of $X$ , say $X_{ϵ}$ , with a matrix normalization which contains part of the information contained in the spectral moments matrix of second order of $X_{ϵ}$ .

Under the condition that the covariance function is twice continuously differentiable out of a set of zero Lebesgue's measure, this functional converges in $L^{2} (Ω)$ to the local time of $X$ at the level $u$ . Furthermore, we give a bound for the speed of convergence.

Classification : 60G60, 60G15, 60G10, 60G57, 60J55, 65D10, 60F25, 60D05, 60E07, 60G50
Keywords: Random fields - Gaussian processes - Stationary processes - Random measures - Local time and additive functional - Smoothing, curve fitting -

L^{p}

-limit theorems - Geometric probability, stochastic geometry, random sets - Infinitely divisible distributions, stable distributions - Sums of independent random variables.

@phdthesis{BJHTUP11_1989__0241__A1_0,
     author = {Berzin, Corinne},
     title = {Surfaces al\'eatoires : approximation du temps local},
     series = {Th\`eses d'Orsay},
     publisher = {Universit\'e de Paris-Sud Centre d'Orsay},
     number = {241},
     year = {1989},
     language = {fr},
     url = {https://www.numdam.org/item/BJHTUP11_1989__0241__A1_0/}
}

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%S Thèses d'Orsay
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%N 241
%I Université de Paris-Sud Centre d'Orsay
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Berzin, Corinne. Surfaces aléatoires : approximation du temps local. Thèses d'Orsay, no. 241 (1989), 126 p. http://numdam.org/item/BJHTUP11_1989__0241__A1_0/

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