We study the probability distribution of the location of a particle performing a cyclic random motion in ${\mathbb{R}}^{d}$. 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.

Keywords: cyclic random motions, linear image of a random vector, singular and absolutely continuous measures, convexity, hyper-Bessel functions with several arguments

@article{PS_2006__10__277_0, 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}, mrnumber = {2247923}, zbl = {1183.33028}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps:2006012/} }

TY - JOUR AU - Lachal, Aimé TI - Cyclic random motions in $\mathbb {R}^d$-space with $n$ directions JO - ESAIM: Probability and Statistics PY - 2006 SP - 277 EP - 316 VL - 10 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2006012/ DO - 10.1051/ps:2006012 LA - en ID - PS_2006__10__277_0 ER -

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. http://www.numdam.org/articles/10.1051/ps:2006012/

[1] Setups in polling models: does it make sense to set up if no work is waiting? J. Appl. Prob. 36 (1999) 585-592. | Zbl

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

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

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

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

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

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

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

and ,[9] Random motions in ${\mathbb{R}}^{n}$-space with $(n+1)$ directions, to appear in Ann. Inst. Henri Poincaré Sect. B.

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

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

, and ,[12] 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] Exact joint distribution in a model of planar random motion. Stoch. Stoch. Rep. 69 (2000) 1-10. | Zbl

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

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

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

and ,[17] Exact distributions of random motions in inhomogeneous media, submitted.

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

and ,[19] A cyclic random motion in ${\mathbb{R}}^{3}$ with four directions and finite velocity. Stoch. Stoch. Rep. 76(2) (2004) 113-133. | Zbl

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

,[21] Markovian random evolutions in ${\mathbb{R}}^{n}$. Random Oper. Stochastic Equ. 9 (2001) 139-160. | Zbl

,[22] Analytical theory of Markov random evolutions in ${\mathbb{R}}^{n}$. Doctoral thesis, University of Kiev (in Russian) (2001).

,*Cited by Sources: *