Advances in Applied Probability

A generalized telegraph process with velocity driven by random trials

Irene Crimaldi, Antonio Di Crescenzo, Antonella Iuliano, and Barbara Martinucci

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We consider a random trial-based telegraph process, which describes a motion on the real line with two constant velocities along opposite directions. At each epoch of the underlying counting process the new velocity is determined by the outcome of a random trial. Two schemes are taken into account: Bernoulli trials and classical Pólya urn trials. We investigate the probability law of the process and the mean of the velocity of the moving particle. We finally discuss two cases of interest: (i) the case of Bernoulli trials and intertimes having exponential distributions with linear rates (in which, interestingly, the process exhibits a logistic stationary density with nonzero mean), and (ii) the case of Pólya trials and intertimes having first gamma and then exponential distributions with constant rates.

Article information

Adv. in Appl. Probab., Volume 45, Number 4 (2013), 1111-1136.

First available in Project Euclid: 12 December 2013

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Zentralblatt MATH identifier

Primary: 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60K37: Processes in random environments

Telegraph process random intertime random velocity Bernoulli scheme Pólya urn model logistic stationary density


Crimaldi, Irene; Di Crescenzo, Antonio; Iuliano, Antonella; Martinucci, Barbara. A generalized telegraph process with velocity driven by random trials. Adv. in Appl. Probab. 45 (2013), no. 4, 1111--1136. doi:10.1239/aap/1386857860.

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