A Tauberian theorem for multiple power series

Multiple sequences $\{a(i)\geqslant 0,\ i\in Z_+^n\}$ are considered. The notion of weak one-sided oscillation of such a sequence along a sequence
$$ \bigl\{m=m(k)=(m_1(k),\dots,m_n(k)),\ m_j(k)>0 \ \forall\,j=1,\dots,n,\ k\in \mathbb N\bigr\} $$
such that $m_j(k)\to\infty$ as $k\to\infty$ for $j=1,\dots,n$ is introduced. The asymptotic behaviour of the sequence $a(x_1m_1,\dots, x_nm_n)$ (for fixed positive numbers $x_1,\dots,x_n$) is deduced from the asymptotic behaviour as ${k\to\infty}$ of the generating function $A(s)$, $s\in[0,1)^n$, of the multiple sequence under consideration for $s=(e^{-\lambda_1/m_1},\dots,e^{-\lambda_n/m_n})$ (where $\lambda_1,\dots,\lambda_n$ are positive and fixed). The Tauberian theorem thus established generalizes several Tauberian theorems due to the author, which were established while investigating certain classes of random substitutions and random maps of a finite set to itself. Karamata's well-known Tauberian theorem for the generating functions of sequences was the starting point for research in this direction.
Bibliography: 36 titles.