Cyclic random motions in d -space with n directions
ESAIM: Probability and Statistics, Volume 10 (2006), pp. 277-316.

We study the probability distribution of the location of a particle performing a cyclic random motion in d . The particle can take n possible directions with different velocities and the changes of direction occur at random times. The speed-vectors as well as the support of the distribution form a polyhedron (the first one having constant sides and the other expanding with time t). The distribution of the location of the particle is made up of two components: a singular component (corresponding to the beginning of the travel of the particle) and an absolutely continuous component. We completely describe the singular component and exhibit an integral representation for the absolutely continuous one. The distribution is obtained by using a suitable expression of the location of the particle as well as some probability calculus together with some linear algebra. The particular case of the minimal cyclic motion (n=d+1) with Erlangian switching times is also investigated and the related distribution can be expressed in terms of hyper-Bessel functions with several arguments.

DOI: 10.1051/ps:2006012
Classification: 33E99, 60K99, 62G30
Keywords: cyclic random motions, linear image of a random vector, singular and absolutely continuous measures, convexity, hyper-Bessel functions with several arguments
     author = {Lachal, Aim\'e},
     title = {Cyclic random motions in $\mathbb {R}^d$-space with $n$ directions},
     journal = {ESAIM: Probability and Statistics},
     pages = {277--316},
     publisher = {EDP-Sciences},
     volume = {10},
     year = {2006},
     doi = {10.1051/ps:2006012},
     zbl = {1183.33028},
     mrnumber = {2247923},
     language = {en},
     url = {}
AU  - Lachal, Aimé
TI  - Cyclic random motions in $\mathbb {R}^d$-space with $n$ directions
JO  - ESAIM: Probability and Statistics
PY  - 2006
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PB  - EDP-Sciences
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DO  - 10.1051/ps:2006012
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ID  - PS_2006__10__277_0
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%0 Journal Article
%A Lachal, Aimé
%T Cyclic random motions in $\mathbb {R}^d$-space with $n$ directions
%J ESAIM: Probability and Statistics
%D 2006
%P 277-316
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%I EDP-Sciences
%R 10.1051/ps:2006012
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Lachal, Aimé. Cyclic random motions in $\mathbb {R}^d$-space with $n$ directions. ESAIM: Probability and Statistics, Volume 10 (2006), pp. 277-316. doi : 10.1051/ps:2006012.

[1] R.B. Cooper, S.C. Niu and M.M. Srinivasan, Setups in polling models: does it make sense to set up if no work is waiting? J. Appl. Prob. 36 (1999) 585-592. | Zbl

[2] A. Di Crescenzo, On random motions with velocities alternating at Erlang-distributed random times. Adv. Appl. Prob. 33 (2001) 690-701. | Zbl

[3] A. Di Crescenzo, Exact transient analysis of a planar random motion with three directions. Stoch. Stoch. Rep. 72 (2002) 175-189. | Zbl

[4] V.A. Fok, Works of the State Optical Institute, 4, Leningrad Opt. Inst. 34 (1926) (in Russian).

[5] S. Goldstein, On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. Appl. Math. 4 (1951) 129-156. | Zbl

[6] R. Griego and R. Hersh, Theory of random evolutions with applications to partial differential equations. Trans. Amer. Math. Soc. 156 (1971) 405-418. | Zbl

[7] M. Kac, A stochastic model related to the telegrapher's equation. Rocky Mountain J. Math. 4 (1974) 497-509. | Zbl

[8] A.D. Kolesnik and E. Orsingher, Analysis of a finite-velocity planar random motion with reflection. Theory Prob. Appl. 46 (2002) 132-140. | Zbl

[9] A. Lachal, S. Leorato and E. Orsingher, Random motions in n -space with (n+1) directions, to appear in Ann. Inst. Henri Poincaré Sect. B.

[10] S. Leorato and E. Orsingher, Bose-Einstein-type statistics, order statistics and planar random motions with three directions. Adv. Appl. Probab. 36(3) (2004) 937-970. | Zbl

[11] S. Leorato, E. Orsingher and M. Scavino, An alternating motion with stops and the related planar, cyclic motion with four directions. Adv. Appl. Probab. 35(4) (2003) 1153-1168. | Zbl

[12] E. Orsingher, Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws. Stoch. Proc. Appl. 34 (1990) 49-66. | Zbl

[13] E. Orsingher, Exact joint distribution in a model of planar random motion. Stoch. Stoch. Rep. 69 (2000) 1-10. | Zbl

[14] E. Orsingher, Bessel functions of third order and the distribution of cyclic planar motions with three directions. Stoch. Stoch. Rep. 74 (2002) 617-631. | Zbl

[15] E. Orsingher and A.D. Kolesnik, Exact distribution for a planar random motion model, controlled by a fourth-order hyperbolic equation. Theory Prob. Appl. 41 (1996) 379-387. | Zbl

[16] E. Orsingher and N. Ratanov, Planar random motions with drift. J. Appl. Math. Stochastic Anal. 15 (2002) 205-221. | Zbl

[17] E. Orsingher and N. Ratanov, Exact distributions of random motions in inhomogeneous media, submitted.

[18] E. Orsingher and A. San Martini, Planar random evolution with three directions, in Exploring stochastic laws, A.V. Skorokhod and Yu.V. Borovskikh, Eds., VSP, Utrecht (1995) 357-366. | Zbl

[19] E. Orsingher and A.M. Sommella, A cyclic random motion in 3 with four directions and finite velocity. Stoch. Stoch. Rep. 76(2) (2004) 113-133. | Zbl

[20] M.A. Pinsky, Lectures on random evolution. World Scientific, River Edge (1991). | MR | Zbl

[21] I.V. Samoilenko, Markovian random evolutions in n . Random Oper. Stochastic Equ. 9 (2001) 139-160. | Zbl

[22] I.V. Samoilenko, Analytical theory of Markov random evolutions in n . Doctoral thesis, University of Kiev (in Russian) (2001).

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