Surfaces aléatoires : approximation du temps local
[Random fields: approximation of local time]
Thèses d'Orsay, no. 241 (1989) , 126 p.

Let X ( t , ω ) t d , ω Ω , d 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 , 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 = {http://archive.numdam.org/item/BJHTUP11_1989__0241__A1_0/}
}
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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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