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Regenerative processes in the infinite mean cycle case

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

K. V. Mitov  and
N. M. Yanev
K. V. Mitov*
Affiliation:
Bulgarian Academy of Sciences
N. M. Yanev*
Affiliation:
Bulgarian Academy of Sciences
*
∗ Postal address: Department of Probability and Statistics, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 Acad. G. Bonchev Str., 1113 Sofia, Bulgaria.
∗ Postal address: Department of Probability and Statistics, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 Acad. G. Bonchev Str., 1113 Sofia, Bulgaria.
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Abstract

A class of non-negative alternating regenerative processes is considered, where the process stays at zero random time (waiting period), then it jumps to a random positive level and hits zero after some random period (life period), depending on the evolution of the process. It is assumed that the waiting time and the lifetime belong to the domain of attraction of stable laws with parameters in the interval (½,1]. An integral representation for the distribution functions of the regenerative process is obtained, using the spent time distributions of the corresponding alternating renewal process. Given the asymptotic behaviour of the process in the regenerative cycle, different types of limiting distributions are proved, applying some new results for the corresponding renewal process and two limit theorems for the convergence in distribution.

Keywords

alternating regenerative processesalternating renewal processesinfinite mean cycleslimiting distributions

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 165 - 179
Copyright
Copyright © by the Applied Probability Trust 2001 

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