Tauberian Theorem for Generating Functions of Multiple Series

Abstract

Multiple sequences $a(i)\ge0$, $i\in {\bf Z}_+^n$, are considered. We introduce the concept of one-sided weak oscillation of such sequences along the sequence $(m=m(k)=(m_1(k),\dots,m_n(k)))$, $m_j(k)>0$ $\forall j=1,\dots,n$, $k\in {\bf N}$, such that $m_j(k)\to\infty$ as $k\to\infty$. From asymptotics of the generating function $A(s)$, $s\in[0,1)^n$, of a studied multiple sequence for $s=(e^{-\lambda_1/m_1},\dots,e^{-\lambda_n/m_n})$ as $k\to\infty$ ($\lambda_1,\dots,\lambda_n$ are positive and fixed), we deduce the asymptotics of $a(x_1m_1,\dots,x_nm_n)$ (the numbers $x_1,\dots,x_n$ are also positive and fixed). The Tauberin theorem obtained here generalizes some Tauberian theorems previously proved by the author earlier in research of some classes of random permutations and random mappings of a finite set into it. Thus, the initial result in this direction is the well-known Jovan Karamata's Tauberian theorem for the generating functions of the sequences.