### 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.

Original language | English |
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Pages (from-to) | 27-40 |

Number of pages | 14 |

Journal | Siberian Mathematical Journal |

Volume | 60 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

### Keywords

- Anscombe’s theorem
- compound renewal processes
- convergence to a stable process
- functional limit theorems
- invariance principle

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## Cite this

Borovkov, A. A. (2019). Functional Limit Theorems for Compound Renewal Processes.

*Siberian Mathematical Journal*,*60*(1), 27-40. https://doi.org/10.1134/S003744661901004X