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 Mat. Sb., 2016, Volume 207, Number 2, Pages 143–172 (Mi msb8507)

A Tauberian theorem for multiple power series

A. L. Yakymiv

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: 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
$$\{m=m(k)=(m_1(k),…,m_n(k)), m_j(k)>0 \forall j=1,…,n, k\in \mathbb N\}$$
such that $m_j(k)\to\infty$ as $k\to\infty$ for $j=1,…,n$ is introduced. The asymptotic behaviour of the sequence $a(x_1m_1,…, x_nm_n)$ (for fixed positive numbers $x_1,…,x_n$) 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 $s=(e^{-\lambda_1/m_1},…,e^{-\lambda_n/m_n})$ (where $\lambda_1,…,\lambda_n$ 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.

Keywords: $\sigma$-finite measures, weak convergence of monotone functions and $\sigma$-finite measures, multiple power series, weakly one-sided oscillating multiple sequences and functions, Tauberian theorem.

 Funding Agency Grant Number Russian Foundation for Basic Research 14-01-00318-à This research was carried out with the support of the Russian Foundation for Basic Research (grant no. 14-01-00318-a).

DOI: https://doi.org/10.4213/sm8507

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English version:
Sbornik: Mathematics, 2016, 207:2, 286–313

Bibliographic databases:

UDC: 517.521.75+517.521.5
MSC: Primary 40B05, 40E05; Secondary 44A10

Citation: A. L. Yakymiv, “A Tauberian theorem for multiple power series”, Mat. Sb., 207:2 (2016), 143–172; Sb. Math., 207:2 (2016), 286–313

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Yu. N. Drozhzhinov, “Multidimensional Tauberian theorems for generalized functions”, Russian Math. Surveys, 71:6 (2016), 1081–1134
2. D. V. Khlopin, “A uniform Tauberian theorem in dynamic games”, Sb. Math., 209:1 (2018), 122–144
3. A. L. Yakymiv, “On the distribution of multiple power series regularly varying at the boundary point”, Discrete Math. Appl., 29:6 (2019), 409–421
4. A. L. Yakymiv, “On the order of random permutation with cycle weights”, Theory Probab. Appl., 63:2 (2018), 209–226
5. “Abstracts of talks given at the 3rd International Conference on Stochastic Methods”, Theory Probab. Appl., 64:1 (2019), 124–169
6. A. L. Yakymiv, “Abelian theorem for the regularly varying measure and its density in orthant”, Theory Probab. Appl., 64:3 (2019), 385–400
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