Hostname: page-component-84b7d79bbc-g78kv Total loading time: 0 Render date: 2024-07-25T23:27:32.583Z Has data issue: false hasContentIssue false
  • English
  • Français

An alternating motion with stops and the related planar, cyclic motion with four directions

Part of: Special processes Nonparametric inference Hyperbolic equations and systems

Published online by Cambridge University Press:  01 July 2016

S. Leorato ,
E. Orsingher  and
M. Scavino
S. Leorato*
Affiliation:
University of Rome ‘La Sapienza’
E. Orsingher*
Affiliation:
University of Rome ‘La Sapienza’
M. Scavino*
Affiliation:
Universidad de la República, Montevideo
*
∗ Postal address: Dipartimento di Statistica, Probabilità e Statistiche Applicate, University of Rome ‘La Sapienza’, Piazzale Aldo Moro, 5, 00185 Rome, Italy.
∗ Postal address: Dipartimento di Statistica, Probabilità e Statistiche Applicate, University of Rome ‘La Sapienza’, Piazzale Aldo Moro, 5, 00185 Rome, Italy.
∗∗∗ Postal address: Universidad de la República, Facultad de Ciencias Económicas y de Administración, Instituto de Estadística, Eduardo Acevedo 1139, 11200 Montevideo, Uruguay.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study a planar random motion (X(t), Y(t)), t>0, with orthogonal directions taken cyclically at Poisson paced times. The process is split into one-dimensional motions with alternating displacements interrupted by exponentially distributed stops. The distributions of X = X(t) (conditional and nonconditional) are obtained by means of order statistics and the connection with the telegrapher's process is derived and discussed. We are able to prove that the distributions involved in our analysis are solutions of a certain differential system and of the related fourth-order hyperbolic equation.

Keywords

Order statisticshypergeometric functiontelegrapher's processrandom motionexchangeabilityhyperbolic equation

MSC classification

Primary: 60K99: None of the above, but in this section
Secondary: 62G30: Order statistics; empirical distribution functions 35L25: Higher-order hyperbolic equations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 4 , December 2003 , pp. 1153 - 1168
Copyright
Copyright © Applied Probability Trust 2003 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arfken, G. B. (1985). Mathematical Methods for Physicists. Academic Press, Orlando, FL.Google Scholar
Di Crescenzo, A. (2002). Exact transient analysis of a planar motion with three directions. Stoch. Stoch. Reports 72, 175–189.Google Scholar
Di Matteo, I. and Orsingher, E. (1997). Detailed probabilistic analysis of the integrated three-valued telegraph signal. J. Appl. Prob. 34, 671–684.Google Scholar
Kolesnik, A. D. (1991). Markov equations of random evolutions. Doctoral Thesis, University of Kiev (in Russian).Google Scholar
Kolesnik, A. D. and Orsingher, E. (2002). Analysis of a finite-velocity planar random motion with reflection. Theory Prob. Appl. 46, 132–140.Google Scholar
Orsingher, E. (2000). Exact joint distribution in a model of planar random motion. Stoch. Stoch. Reports 69, 1–10.CrossRefGoogle Scholar
Orsingher, E. (2002). Bessel functions of third order and the distribution of cyclic planar motions with three directions. Stoch. Stoch. Reports 74, 617–631.Google Scholar
Orsingher, E. and Ratanov, N. (2002). Exact distributions of random motions in inhomogeneous media. Submitted.Google Scholar
Samoilenko, I. V. (2001). Markovian random evolutions in Rn . Random Operators Stoch. Equat. 9, 139–160.Google Scholar