A generalized telegraph process with velocity driven by random trials

Abstract

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

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

Dates
First available in Project Euclid: 12 December 2013

https://projecteuclid.org/euclid.aap/1386857860

Digital Object Identifier
doi:10.1239/aap/1386857860

Mathematical Reviews number (MathSciNet)
MR3161299

Zentralblatt MATH identifier
1291.60182

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

Citation

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. https://projecteuclid.org/euclid.aap/1386857860

References

• Abramowitz, M. and Stegun, I. A. (1970). Handbook of Mathematical Functions with Formulas, Graph, and Mathematical Tables. Dover, New York.
• Aldous, D. J. (1985). Exchangeability and related topics. In École d'Été de Probabilités de Saint-Flour, XIII-1983 (Lecture Notes Math. 1117), Springer, Berlin, pp. 1–198.
• Aletti, G., May, C. and Secchi, P. (2009). A central limit theorem, and related results, for a two-color randomly reinforced urn. Adv. Appl. Prob. 41, 829–844.
• Bartlett, M. S. (1957). Some problems associated with random velocity. Publ. Inst. Statist. Univ. Paris. 6, 261–270.
• Beghin, L., Nieddu, L. and Orsingher, E. (2001). Probabilistic analysis of the telegrapher's process with drift by means of relativistic transformations. J. Appl. Math. Stoch. Anal. 14, 11–25.
• Berti, P., Pratelli, L. and Rigo, P. (2004). Limit theorems for a class of identically distributed random variables. Ann. Prob. 32, 2029–2052.
• Berti, P., Crimaldi, I., Pratelli, L. and Rigo, P. (2011). A central limit theorem and its applications to multicolor randomly reinforced urns. J. Appl. Prob. 48, 527–546.
• Brown, M. (2011). Detection of changes of multiple Poisson processes monitored at discrete time points where the arrival rates are unknown. Sequent. Anal. 30, 280–296.
• Cohen, A. and Sackrowitz, H. B. (1993). Evaluating tests for increasing intensity of a Poisson process. Technometrics 35, 446–448.
• Crimaldi, I. (2009). An almost sure conditional convergence result and an application to a generalized Pólya urn. Internat. Math. Forum 4, 1139–1156.
• De Gregorio, A. (2010). Stochastic velocity motions and processes with random time. Adv. Appl. Prob. 42, 1028–1056.
• De Gregorio, A. (2012). On random flights with non-uniformly distributed directions. J. Statist. Phys. 147, 382–411.
• De Gregorio, A. and Orsingher, E. (2012). Flying randomly in $\mathbb {R}^d$ with Dirichlet displacements. Stoch. Process. Appl. 122, 676–713.
• Di Crescenzo, A. (2001). On random motions with velocities alternating at Erlang-distributed random times. Adv. Appl. Prob. 33, 690–701.
• Di Crescenzo, A. and Martinucci, B. (2007). Random motion with gamma-distributed alternating velocities in biological modeling. In Computer Aided Systems Theory, EUROCAST 2007 (Lecture Notes Comput. Sci. 4739), eds R. Moreno-Díaz et al., Springer, Berlin, pp. 163–170.
• Di Crescenzo, A. and Martinucci, B. (2009). On a first-passage-time problem for the compound power-law process. Stoch. Models 25, 420–435.
• Di Crescenzo, A. and Martinucci, B. (2010). A damped telegraph random process with logistic stationary distribution. J. Appl. Prob. 47, 84–96.
• Di Crescenzo, A., Martinucci, B. and Zacks, S. (2011). On the damped geometric telegrapher's process. In Mathematical and Statistical Methods for Actuarial Sciences and Finance, eds C. Perna and M. Sibillo, Springer, Dordrecht, pp. 175–182.
• Foong, S. K. and Kanno, S. (1994). Properties of the telegrapher's random process with or without a trap. Stoch. Process. Appl. 53, 147–173.
• Goldstein, S. (1951). On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. Appl. Math. 4, 129–156.
• Gradshteyn, I. S. and Ryzhik, I. M. (2007). Tables of Integrals, Series, and Products, 7th edn. Academic Press, Amsterdam.
• Iacus, S. M. (2001). Statistical analysis of the inhomogeneous telegrapher's process. Statist. Prob. Lett. 55, 83–88.
• Kac, M. (1974). A stochastic model related to the telegrapher's equation. Rocky Mount. J. Math. 4, 497–509.
• Mahmoud, H. M. (2008). Pólya Urn Models. Chapman & Hall/CRC.
• Moschopoulos, P. G. (1985). The distribution of the sum of independent gamma random variables. Ann. Inst. Statist. Math. 37, 541–544.
• Orsingher, E. (1990). Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws. Stoch. Process. Appl. 34, 49–66.
• Orsingher, E. (1995). Motions with reflecting and absorbing barriers driven by the telegraph equation. Random Operators Stoch. Equat. 1, 9–21.
• Pinsky, M. A. (1991). Lectures on Random Evolutions. World Scientific, River Edge, NJ.
• Pólya, G. (1931). Sur quelques points de la théorie des probabilités. Ann. Inst. H. Poincaré 1, 117–161.
• Serino, C. A. and Redner, S. (2010). The Pearson walk with shrinking steps in two dimensions. J. Statist. Mech. Theory Exp. 2010, P01006, 12pp.
• Stadje, W. and Zacks, S. (2004). Telegraph processes with random velocities. J. Appl. Prob. 41, 665–678.
• Wolff, R. W. (1989). Stochastic Modeling and the Theory of Queues. Prentice Hall, Englewood Cliffs, NJ.
• Zacks, S. (2004). Generalized integrated telegraph processes and the distribution of related stopping times. J. Appl. Prob. 41, 497–507.