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Sibirsk. Mat. Zh., 2019, Volume 60, Number 1, Pages 37–54 (Mi smj3057)  

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

Functional limit theorems for compound renewal processes

A. A. Borovkovab

a Novosibirsk State University, Novosibirsk, Russia
b Sobolev Institute of Mathematics, Novosibirsk, Russia

Abstract: We generalize Anscombe’s Theorem to the case of stochastic processes converging to a continuous random process. As applications, we find a simple proof of an invariance principle for compound renewal processes (CRPs) in the case of finite variance of the elements of the control sequence. We find conditions, close to minimal ones, of the weak convergence of CRPs in the metric space D with metrics of two types to stable processes in the case of infinite variance. They turn out narrower than the conditions for convergence of a distribution in this space.

Keywords: Anscombe's theorem, functional limit theorems, compound renewal processes, invariance principle, convergence to a stable process.

Funding Agency Grant Number
Siberian Branch of Russian Academy of Sciences I.1.3 (проект № 0314-2016-0008)
Russian Foundation for Basic Research 18-01-00101_а
The author was partially supported by the Program of Basic Scientific Research of the Siberian Branch of the Russian Academy of Sciences (Program No. I.1.3, Project 0314-2016-0008) and the Russian Foundation for Basic Research (Grant 18-01-00101a).


DOI: https://doi.org/10.33048/smzh.2019.60.104

Full text: PDF file (340 kB)
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English version:
Siberian Mathematical Journal, 2019, 60:1, 27–40

Bibliographic databases:

UDC: 519.21
MSC: 35R30
Received: 19.05.2018
Revised: 19.05.2018
Accepted:23.05.2018

Citation: A. A. Borovkov, “Functional limit theorems for compound renewal processes”, Sibirsk. Mat. Zh., 60:1 (2019), 37–54; Siberian Math. J., 60:1 (2019), 27–40

Citation in format AMSBIB
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\pages 27--40
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    This publication is cited in the following articles:
    1. A. A. Borovkov, “Rasprostranenie printsipa invariantnosti dlya obobschennykh protsessov vosstanovleniya na oblasti umerenno bolshikh i malykh uklonenii”, Teoriya veroyatn. i ee primen., 65:4 (2020), 651–670  mathnet  crossref
  • Сибирский математический журнал Siberian Mathematical Journal
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