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On random motions with velocities alternating at Erlang-distributed random times

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Antonio Di Crescenzo
Antonio Di Crescenzo*
Affiliation:
Università della Basilicata
*
∗ Postal address: Dipartimento di Matematica, Università degli Studi della Basilicata, Loc. Macchia Romana, I-85100 Potenza, Italy. Email address: [email protected]
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Abstract

We analyse a non-Markovian generalization of the telegrapher's random process. It consists of a stochastic process describing a motion on the real line characterized by two alternating velocities with opposite directions, where the random times separating consecutive reversals of direction perform an alternating renewal process. In the case of Erlang-distributed interrenewal times, explicit expressions of the transition densities are obtained in terms of a suitable two-index pseudo-Bessel function. Some results on the distribution of the maximum of the process are also disclosed.

Keywords

Telegrapher's random processalternating renewal processpseudo-Bessel functionsdistribution of the maximum

MSC classification

Primary: 60K99: None of the above, but in this section
Secondary: 60K15: Markov renewal processes, semi-Markov processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 3 , September 2001 , pp. 690 - 701
Copyright
Copyright © Applied Probability Trust 2001 

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