TY - JOUR

T1 - Functional Limit Theorems for Compound Renewal Processes

AU - Borovkov, A. A.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Anscombe’s theorem

KW - compound renewal processes

KW - convergence to a stable process

KW - functional limit theorems

KW - invariance principle

UR - http://www.scopus.com/inward/record.url?scp=85065244952&partnerID=8YFLogxK

U2 - 10.1134/S003744661901004X

DO - 10.1134/S003744661901004X

M3 - Article

AN - SCOPUS:85065244952

VL - 60

SP - 27

EP - 40

JO - Siberian Mathematical Journal

JF - Siberian Mathematical Journal

SN - 0037-4466

IS - 1

ER -