On the number of renewals in random time

For the renewal counting process M ( t ) = min { k : S k > t } and the independent of it nonnegative random variable T , we investigate the asymptotic behaviour of P ( M ( t ) < T ) and P ( M ( t ) ≤ K ( t ) x ∣ M ( t ) < T ) in cases when the interarrival times have an infinite mean. These...

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Bibliographic Details
Published in:Statistics & probability letters 2009-11, Vol.79 (21), p.2281-2288
Main Authors: Omey, Edward, Mitov, Georgi K., Mitov, Kosto V.
Format: Article
Language:English
Subjects:
Exact sciences and technology
General topics
Markov processes
Mathematics
Probability and statistics
Probability theory and stochastic processes
Sciences and techniques of general use
Statistics
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Summary:For the renewal counting process M ( t ) = min { k : S k > t } and the independent of it nonnegative random variable T , we investigate the asymptotic behaviour of P ( M ( t ) < T ) and P ( M ( t ) ≤ K ( t ) x ∣ M ( t ) < T ) in cases when the interarrival times have an infinite mean. These quantities appear in a natural way when considering limiting behaviour of random time changed branching processes and shock models.
ISSN:0167-7152
1879-2103
DOI:10.1016/j.spl.2009.08.001