Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-23T07:44:18.510Z Has data issue: false hasContentIssue false
  • English
  • Français

A Generalized Telegraph Process with Velocity Driven by Random Trials

Part of: Special processes

Published online by Cambridge University Press:  04 January 2016

Irene Crimaldi ,
Antonio Di Crescenzo ,
Antonella Iuliano  and
Barbara Martinucci
Irene Crimaldi*
Affiliation:
IMT
Antonio Di Crescenzo*
Affiliation:
Università di Salerno
Antonella Iuliano*
Affiliation:
Istituto per le Applicazioni del Calcolo ‘Mauro Picone’
Barbara Martinucci*
Affiliation:
Università di Salerno
*
∗ Postal address: Institute for Advanced Studies, IMT, Piazza San Ponziano 6, I-55100 Lucca, Italy. Email address: irene.crimaldi@imtlucca.it
∗∗ Postal address: Dipartimento di Matematica, Università di Salerno, Via Ponte don Melillo, Fisciano (SA), I-84084, Italy.
∗∗∗∗ Postal address: Istituto per le Applicazioni del Calcolo ‘Mauro Picone’, CNR, Via Pietro Castellino 111, I-80131 Napoli, Italy. Email address: a.iuliano@na.iac.cnr.it
∗∗ Postal address: Dipartimento di Matematica, Università di Salerno, Via Ponte don Melillo, Fisciano (SA), I-84084, Italy.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Telegraph processrandom intertimerandom velocityBernoulli schemePólya urn modellogistic stationary density

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60K37: Processes in random environments
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 4 , December 2013 , pp. 1111 - 1136
Copyright
© Applied Probability Trust 

References

Abramowitz, M. and Stegun, I. A. (1970). Handbook of Mathematical Functions with Formulas, Graph, and Mathematical Tables. Dover, New York.Google Scholar
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.Google Scholar
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.Google Scholar
Bartlett, M. S. (1957). Some problems associated with random velocity. Publ. Inst. Statist. Univ. Paris. 6, 261–270.Google Scholar
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.Google Scholar
Berti, P., Pratelli, L. and Rigo, P. (2004). Limit theorems for a class of identically distributed random variables. Ann. Prob. 32, 2029–2052.Google Scholar
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.Google Scholar
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.Google Scholar
Cohen, A. and Sackrowitz, H. B. (1993). Evaluating tests for increasing intensity of a Poisson process. Technometrics 35, 446–448.Google Scholar
Crimaldi, I. (2009). An almost sure conditional convergence result and an application to a generalized Pólya urn. Internat. Math. Forum 4, 1139–1156.Google Scholar
De Gregorio, A. (2010). Stochastic velocity motions and processes with random time. Adv. Appl. Prob. 42, 1028–1056.Google Scholar
De Gregorio, A. (2012). On random flights with non-uniformly distributed directions. J. Statist. Phys. 147, 382–411.Google Scholar
De Gregorio, A. and Orsingher, E. (2012). Flying randomly in R d with Dirichlet displacements. Stoch. Process. Appl. 122, 676–713.Google Scholar
Di Crescenzo, A. (2001). On random motions with velocities alternating at Erlang-distributed random times. Adv. Appl. Prob. 33, 690–701.CrossRefGoogle Scholar
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 Moreno-Dı´az, R. et al., Springer, Berlin, pp. 163–170.Google Scholar
Di Crescenzo, A. and Martinucci, B. (2009). On a first-passage-time problem for the compound power-law process. Stoch. Models 25, 420–435.Google Scholar
Di Crescenzo, A. and Martinucci, B. (2010). A damped telegraph random process with logistic stationary distribution. J. Appl. Prob. 47, 84–96.Google Scholar
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 Perna, C. and Sibillo, M., Springer, Dordrecht, pp. 175–182.Google Scholar
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.Google Scholar
Goldstein, S. (1951). On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. Appl. Math. 4, 129–156.Google Scholar
Gradshteyn, I. S. and Ryzhik, I. M. (2007). Tables of Integrals, Series, and Products, 7th edn. Academic Press, Amsterdam.Google Scholar
Iacus, S. M. (2001). Statistical analysis of the inhomogeneous telegrapher's process. Statist. Prob. Lett. 55, 83–88.Google Scholar
Kac, M. (1974). A stochastic model related to the telegrapher's equation. Rocky Mount. J. Math. 4, 497–509.Google Scholar
Mahmoud, H. M. (2008). Pólya Urn Models. Chapman & Hall/CRC.CrossRefGoogle Scholar
Moschopoulos, P. G. (1985). The distribution of the sum of independent gamma random variables. Ann. Inst. Statist. Math. 37, 541–544.Google Scholar
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.Google Scholar
Orsingher, E. (1995). Motions with reflecting and absorbing barriers driven by the telegraph equation. Random Operators Stoch. Equat. 1, 9–21.Google Scholar
Pinsky, M. A. (1991). Lectures on Random Evolutions. World Scientific, River Edge, NJ.Google Scholar
Pólya, G. (1931). Sur quelques points de la théorie des probabilités. Ann. Inst. H. Poincaré 1, 117–161.Google Scholar
Serino, C. A. and Redner, S. (2010). The Pearson walk with shrinking steps in two dimensions. J. Statist. Mech. Theory Exp. 2010, P01006, 12 pp.Google Scholar
Stadje, W. and Zacks, S. (2004). Telegraph processes with random velocities. J. Appl. Prob. 41, 665–678.Google Scholar
Wolff, R. W. (1989). Stochastic Modeling and the Theory of Queues. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
Zacks, S. (2004). Generalized integrated telegraph processes and the distribution of related stopping times. J. Appl. Prob. 41, 497–507.Google Scholar