Regenerative processes in the infinite mean cycle case

Citation
Kv. Mitov et Nm. Yanev, Regenerative processes in the infinite mean cycle case, J APPL PROB, 38(1), 2001, pp. 165-179
Citations number
23
Categorie Soggetti
Mathematics
Journal title
JOURNAL OF APPLIED PROBABILITY
ISSN journal
00219002 → ACNP
Volume
38
Issue
1
Year of publication
2001
Pages
165 - 179
Database
ISI
SICI code
0021-9002(200103)38:1<165:RPITIM>2.0.ZU;2-H
Abstract
A class of non-negative alternating regenerative processes is considered, w here 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 per iod), depending on the evolution of the process. It is assumed that the wai ting time and the lifetime belong to the domain of attraction of stable law s with parameters in the interval (1/2, 1]. An integral representation for the distribution functions of the regenerative process is obtained, using t he spent time distributions of the corresponding alternating renewal proces s. Given the asymptotic behaviour of the process in the regenerative cycle, different types of limiting distributions are proved, applying some new re sults for the corresponding renewal process and two limit theorems for the convergence in distribution.