Permutation Operators and the Central Limit Theorem Associated with Partial Differential Operators / Operateurs de Permutation et Theoreme de la Centrale Associes a des Operateurs aux Derivees Partielles

Abstract

For the differential operators

$$ {\Delta _1}: = {\textstyle{\partial \over {\partial x}}} $$

and

$$ {\Delta _2}: = {\textstyle{{{\partial ^2}} \over {\partial {x^2}}}} + \frac{{2\alpha + 1}}{r}\frac{\partial }{{\partial r}} - {\textstyle{{{\partial ^2}} \over {\partial {x^2}}}} $$

on ℝ and IR ^×₊ ×IR respectively (α∈IR₊) one introduces the corresponding permutation operators ℝ_α and ^tℝ_α commuting with the operators Δ₁ and ${\textstyle{{{\partial ^2}} \over {\partial {x^2}}}}$ respectively. The paper is concerned with problems of harmonic analysis related to the operators Δ₁ and Δ₂ such as generalized Fourier transforms, Plancherel and Paley-Wiener theorems, generalized translation operators, and products of generalized convolution structures. Within this general framework a central limit theorem is proved. More precisely, sufficient conditions in terms of moments up to the fourth order are given for a triangular system of probability measures on IR ^×₊ to converge weakly towards the Gaussian distribution on IR ^×₊ . The main results can be considered as contributions to the analysis and probability theory on two-dimensional hypergroups. The French text follows.