Local Limit Theorem for the Multiple Power Series Distributions

Arsen L Yakymiv

doi:10.3390/math8112067

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.

Highlights

Let ( a(i ) ≥ 0, i = 0, 1, 2, . . . ) be a sequence with ∞ B( x ) = ∑ a (i ) x i < ∞i =0 for x ∈ (0, 1)
It is said that a random variable ξ x has a power series distribution iff a (i ) x i
Power series distributions were introduced in the fundamental paper of Noack [1] (1950)

Summary

Introduction

Note that power series distributions are widely useful in a generalized allocation scheme (in the one-dimensional case). The multidimensional integral limit theorem was obtained in [20] It is supposed in [20] that the corresponding multiple power series regularly varies at the boundary point of its convergence (see Definition 2). We prove the corresponding local limit theorem The notion of R-weakly one-sided oscillatory sequences at infinity along some sequence (see Definition 3) This concept allows us to give adequate conditions for the validity of both the local limit theorem and the corresponding statement of Tauberian type (Lemma 2).

Some Notations

Tauberian Lemma

Proof of Theorem 1

On Some Previous Results