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 Teor. Veroyatnost. i Primenen., 2019, Volume 64, Issue 3, Pages 481–501 (Mi tvp5274)

This article is cited in 1 scientific paper (total in 1 paper)

Abelian theorem for the regularly varying measure and its density in orthant

A. L. Yakymiv

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: The paper is concerned with a $\sigma$-finite measure $U$ concentrated in the positive orthant $\mathbf{R}^n_+=[0,\infty)^n$ such that there exists the Laplace transform $\widetilde{U}(\lambda)$ for $\lambda\in\operatorname{int} \mathbf{R}^n_+$. Let functions $R(t)>0$ and $b(t)=(b_1(t),…,b_n(t))\in\operatorname{int}\mathbf{R}^n_+$ for $t\geq0$ be such that $R(t)\to\infty$, $b_i(t)\to\infty$ for any $i=1,…,n$. Under certain assumptions on these functions, the weak convergence of the measures $U(b(t) {\cdot} )/R(t)$ to $\Phi{( \cdot )}$ as $t\to\infty$ is shown to imply the convergence $\widetilde{U}(\lambda/b(t))\to\widetilde{\Phi}(\lambda)<\infty$ for any $\lambda\in\operatorname{int} \mathbf{R}^n_+$ ($t\to\infty$) (the multiplication and division of vectors are defined componentwise). A function $f\colon \mathbf{R}_+^n\to \mathbf{R}_+$ is said to be regularly varying at infinity in $\mathbf{R}_+^n$ along $b(t)$ if $f(b(t)x(t))/f(b(t))\to\varphi(x)\in(0,\infty)$ as $t\to\infty$ for all $x$, $x(t) \in \mathbf{R}_+^n\setminus\{0\}$ such that $x(t)\to x$. Sufficient conditions are given for such functions to give $\widehat{f}(\lambda/b(t))\equiv\widetilde{U}(\lambda/b(t)) \to\widehat{\phi}(\lambda)\equiv\widetilde{\Phi}(\lambda)<\infty$ for any $\lambda\in\operatorname{int} \mathbf{R}^n_+$\enskip ($t\to\infty$) for $U(dx)=f(x) dx$, $\Phi(dx)=\varphi(x) dx$. The Abelian theorem obtained here is applied at the end of the paper to investigate the limit behavior of multiple power series distributions.

Keywords: weak convergence of sequence of measures, Abelian theorem for a measure and its density, regularly varying functions and measures at infinity in an orthant, integral representation theorem, multiple power series distributions.

DOI: https://doi.org/10.4213/tvp5274

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English version:
Theory of Probability and its Applications, 2019, 64:3, 385–400

Bibliographic databases:

Received: 19.11.2018
Revised: 06.02.2019
Accepted:12.02.2019

Citation: A. L. Yakymiv, “Abelian theorem for the regularly varying measure and its density in orthant”, Teor. Veroyatnost. i Primenen., 64:3 (2019), 481–501; Theory Probab. Appl., 64:3 (2019), 385–400

Citation in format AMSBIB
\Bibitem{Yak19} \by A.~L.~Yakymiv \paper Abelian theorem for the regularly varying measure and its density in orthant \jour Teor. Veroyatnost. i Primenen. \yr 2019 \vol 64 \issue 3 \pages 481--501 \mathnet{http://mi.mathnet.ru/tvp5274} \crossref{https://doi.org/10.4213/tvp5274} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3988270} \zmath{https://zbmath.org/?q=an:1426.28007} \elib{https://elibrary.ru/item.asp?id=38590354} \transl \jour Theory Probab. Appl. \yr 2019 \vol 64 \issue 3 \pages 385--400 \crossref{https://doi.org/10.1137/S0040585X97T989568} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000492370500004} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85074326184} 

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This publication is cited in the following articles:
1. L. Yakymiv, A., “Local limit theorem for the multiple power series distributions”, Mathematics, 8:11 (2020), 2067
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