Motions with finite velocity analyzed with order statistics and differential equations
Authors:
A. de Gregorio, E. Orsingher and L. Sakhno
Journal:
Theor. Probability and Math. Statist. 71 (2005), 63-79
MSC (2000):
Primary 60K99; Secondary 62G30, 35L25
DOI:
https://doi.org/10.1090/S0094-9000-05-00648-4
Published electronically:
December 28, 2005
MathSciNet review:
2144321
Full-text PDF Free Access
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Additional Information
Abstract: The aim of this paper is to derive the explicit distribution of the position of randomly moving particles on the line and in the plane (with different velocities taken cyclically) by means of order statistics and by studying suitable problems of differential equations. The two approaches are compared when both are applicable (case of the telegraph process). In some specific cases (alternating motions with skipping) it is possible to use the order statistics approach also to solve the equations governing the distribution. Finally, the approach based on order statistics is also applied in order to obtain the distribution of the position in the case of planar motion with three velocities conditioned on the number of changes of directions recorded.
References
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- S. Leorato, E. Orsingher, and M. Scavino, An alternating motion with stops and the related planar, cyclic motion with four directions, Adv. in Appl. Probab. 35 (2003), no. 4, 1153–1168. MR 2014274, DOI https://doi.org/10.1239/aap/1067436339
- Enzo Orsingher, Bessel functions of third order and the distribution of cyclic planar motions with three directions, Stoch. Stoch. Rep. 74 (2002), no. 3-4, 617–631. MR 1943582, DOI https://doi.org/10.1080/1045112021000060755
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References
- S. K. Foong and S. Kanno, Properties of the telegrapher’s random process with or without a trap, Stochastic Processes and their Applications 53 (1994), 147–173. MR 1290711 (95g:60089)
- S. Goldstein, On diffusion by discontinuous movements and telegraph equation, Quart. J. Mech. Appl. Math. 4 (1951), 129–156. MR 0047963 (13,960b)
- S. Leorato and E. Orsingher, Bose–Einstein-type statistics, order statistics and planar random motions with three directions, Advances in Applied Probability 36 (2004), no. 3, 937–970. MR 2079922 (2005g:60171)
- S. Leorato, E. Orsingher, and M. Scavino, An alternating motion with stops and the related planar, cyclic motion with four directions, Advances in Applied Probability 35 (2003), no. 4, 1153–1168. MR 2014274 (2004g:60132)
- E. Orsingher, Bessel functions of third order and the distribution of cyclic planar motions with three directions, Stochastics and Stochastics Reports 74 (2002), 617–631. MR 1943582 (2003j:60144)
- I. V. Samoilenko, Markovian random evolutions in $R^n$, Random Operators and Stochastic Equations 9 (2001), 139–160. MR 1832161 (2002e:60116)
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Additional Information
A. de Gregorio
Affiliation:
Dipartimento di Scienze Statistiche, University of Padua, via Cesare Battisti 241, 35121, Padua, Italy
E. Orsingher
Affiliation:
Dipartimento di Statistica, Probabilità e Statistiche Applicate, University of Rome “La Sapienza”, p. le Aldo Moro 5, 00185, Rome, Italy
Email:
enzo.orsingher@uniroma1.it
L. Sakhno
Affiliation:
Department of Probability Theory and Mathematical Statistics, Kyiv National Taras Shevchenko University, Volodymyrska 64, 01033, Kyiv, Ukraine
Keywords:
Order statistics,
Bessel functions of higher order,
random motions
Received by editor(s):
June 17, 2003
Published electronically:
December 28, 2005
Additional Notes:
This work was partially supported by the NATO grant PST.CLG.976361.
Article copyright:
© Copyright 2005
American Mathematical Society
