Functional Limit Theorems for Compound Renewal Processes
- 5 Downloads
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.
KeywordsAnscombe’s theorem functional limit theorems compound renewal processes invariance principle convergence to a stable process
Unable to display preview. Download preview PDF.
- 11.Billingsley P. P., Convergence of Probability Measures [Russian translation], Nauka, Moscow (1977).Google Scholar
- 18.Gnedenko B. V. and Kolmogorov A. N., Limit Distributions for Sums of Independent Random Variables [Russian], Gostekhizdat, Moscow (1949).Google Scholar
- 20.Borovkov A. A., Asymptotic Analysis of Random Walks: Rapidly Decreasing Jumps [Russian], Fizmatlit, Moscow (2013).Google Scholar