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Functional limit theorems for compound renewal processes. (English. Russian original) Zbl 1475.60065

Sib. Math. J. 60, No. 1, 27-40 (2019); translation from Sib. Mat. Zh. 60, No. 1, 37-54 (2019).
Summary: 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.

MSC:

60F17 Functional limit theorems; invariance principles
60K05 Renewal theory
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