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Lois gaussiennes inverses (généralisées), lois de Kummer et méthode de Stein

Abstract : In this thesis, we make a contribution to the study of the properties of the generalized inverse Gaussian and Kummer distributions in the context of Stein's method. This is to contribute to establishing the mathematical tools necessary for the application of Stein's method to the case where the target law is one of the two aforementioned laws, on the one hand, and to applying Stein's method to give a bound of the rate of convergence in some limit theorems involving these two laws, on the other hand. We retrieve the Stein operator of each of these two laws, solve the corresponding differential equation and bound the solution obtained as well as its successive derivatives (for the derivatives, the bound is not always explicit but is established by an iterative technique). The techniques used to obtain the bounds of the solution and of its derivatives are essentially based on the fact that these two laws belong to the family of probability laws whose density g satisfies the differential equation (s(x)g(x))^'=τ(x)g(x) where s and τ are polynomial functions satisfying certain conditions. By Stein's method, we establish a bound for the rate of convergence of the generalized inverse Gaussian and Kummer distributions to the gamma distribution, of the generalized hyperbolic distribution to the generalized inverse Gaussian distribution and a sequence of random variables to the reciprocal inverse Gaussian distribution. Our approach in estimating these rate of convergence by Stein's method is essentially based on the fact that the law of the sequence of random variables considered and the limit law are involved in a convolution relation.
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Submitted on : Monday, February 1, 2021 - 4:33:45 PM
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Essomanda Konzou. Lois gaussiennes inverses (généralisées), lois de Kummer et méthode de Stein. Mathématiques [math]. Université de Lorraine; Université de Lomé (Togo), 2020. Français. ⟨NNT : 2020LORR0147⟩. ⟨tel-03127767⟩



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