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Some results on the telegraph process driven by gamma components

Part of: Special processes

Published online by Cambridge University Press:  14 June 2022

Barbara Martinucci Open the ORCID record for Barbara Martinucci [Opens in a new window] ,
Alessandra Meoli  and
Shelemyahu Zacks
Barbara Martinucci*
Affiliation:
University of Salerno
Alessandra Meoli*
Affiliation:
University of Salerno
Shelemyahu Zacks*
Affiliation:
Binghamton University
*
*Postal address: Fisciano (SA), I-84084, Italy.
*Postal address: Fisciano (SA), I-84084, Italy.
**Postal address: Binghamton, NY 13902-6000, USA. Email address: shelly@math.binghamton.edu
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Abstract

We study the integrated telegraph process $X_t$ under the assumption of general distribution for the random times between consecutive reversals of direction. Specifically, $X_t$ represents the position, at time t, of a particle moving U time units upwards with velocity c and D time units downwards with velocity $-c$ . The latter motions are repeated cyclically, according to independent alternating renewals. Explicit expressions for the probability law of $X_t$ are given in the following cases: (i) (U, D) gamma-distributed; (ii) U exponentially distributed and D gamma-distributed. For certain values of the parameters involved, the probability law of $X_t$ is provided in a closed form. Some expressions for the moment generating function of $X_t$ and its Laplace transform are also obtained. The latter allows us to prove the existence of a Kac-type condition under which the probability density function of the integrated telegraph process, with identically distributed gamma intertimes, converges to that of the standard Brownian motion.

Finally, we consider the square of $X_t$ and disclose its distribution function, specifying the expression for some choices of the distribution of (U, D).

Keywords

Random motions with finite velocitytelegraph processgamma distributionmoment generating functionKac conditions

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60K99: None of the above, but in this section
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 3 , September 2022 , pp. 808 - 848
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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