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.
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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).
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Translated from Sibirskii Matematicheskii Zhurnal, vol. 60, no. 1, pp. 37–54, January–February, 2019; DOI: 10.17377/smzh.2019.60.104.
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Borovkov, A.A. Functional Limit Theorems for Compound Renewal Processes. Sib Math J 60, 27–40 (2019). https://doi.org/10.1134/S003744661901004X
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DOI: https://doi.org/10.1134/S003744661901004X