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}, zbl = {1183.33028}, mrnumber = {2247923}, 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/ UR - https://zbmath.org/?q=an%3A1183.33028 UR - https://www.ams.org/mathscinet-getitem?mr=2247923 UR - https://doi.org/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/

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