**Arsen L. Yakymiv**

Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow 119991, Russia; arsen@mi-ras.ru

Received: 30 September 2020; Accepted: 13 November 2020; Published: 19 November 2020

**Abstract:** We study the behavior of multiple power series distributions at the boundary points of their existence. In previous papers, the necessary and sufficient conditions for the integral limit theorem were obtained. Here, the necessary and sufficient conditions for the corresponding local limit theorem are established. This article is dedicated to the memory of my teacher, professor V.M. Zolotarev.

**Keywords:** multiple power series distribution; integral limit theorem; local limit theorem; Tauberian lemma; *R*-weakly one-sided oscillation of the multiple sequence at infinity along the given multiple sequence
