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Siberian Mathematical Journal

, Volume 60, Issue 1, pp 27–40 | Cite as

Functional Limit Theorems for Compound Renewal Processes

  • A. A. BorovkovEmail author
Article
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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 

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Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia

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