Local limit theorems for complex valued sequences: Old & New

Coulombel, Jean-François; Faye, Grégory

Journées équations aux dérivées partielles (2024), Exposé no. 5, 15 p.

In probability theory, local limit theorems provide an asymptotic expansion of the convolution powers of a probability distribution supported on $\mathbb{Z}$ with uniform bounds on the remainders. In this review, we present some recent results for the iterated convolution of complex valued integrable sequences in one space dimension. In the so-called parabolic case, we give a complete expansion, at any accuracy order, for these convolution powers and we provide sharp, pointwise, generalized Gaussian bounds for the remainders. We also present an extension of our main result to the semi-discrete setting (time-continuous convolution problems), and discuss several natural perspectives.

En théorie des probabilités, les théorèmes de la limite locale donnent un développement asymptotique des puissances itérées d’une loi de probabilité supportée sur $\mathbb{Z}$ avec des estimations uniformes sur les termes de reste. Dans cette revue, nous présentons des résultats récents pour les convolutions itérées de suites complexes intégrables en une dimension d’espace. Dans le cas parabolique, nous donnons un développement complet, à tous les ordres, et nous obtenons une estimation ponctuelle précise des termes de reste sous forme de Gaussiennes généralisées. Nous présentons également une extension de notre résultat principal dans le cas semi-discret (problèmes continus en temps de convolution), et nous discutons plusieurs perspectives naturelles à ce travail.

Classification : 42A85, 35K25, 60F99, 65M12
Keywords: Convolution, asymptotic expansion, stability, local limit theorem
Mots-clés : Convolution, développement asymptotique, stabilité, théorème de la limite locale

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