# Cyclic random motions in ${ℝ}^{d}$-space with n directions

• Volume: 10, page 277-316
• ISSN: 1292-8100

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## Abstract

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

## How to cite

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Lachal, Aimé. "Cyclic random motions in $\mathbb{R}^d$-space with n directions." ESAIM: Probability and Statistics 10 (2006): 277-316. <http://eudml.org/doc/249743>.

@article{Lachal2006,
abstract = { 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. },
author = {Lachal, Aimé},
journal = {ESAIM: Probability and Statistics},
keywords = {Cyclic random motions; linear image of a random vector; singular and absolutely continuous measures; convexity; hyper-Bessel functions with several arguments.; linear image of a random vector; convexity; hyper-Bessel functions with several arguments},
language = {eng},
month = {9},
pages = {277-316},
publisher = {EDP Sciences},
title = {Cyclic random motions in $\mathbb\{R\}^d$-space with n directions},
url = {http://eudml.org/doc/249743},
volume = {10},
year = {2006},
}

TY - JOUR
AU - Lachal, Aimé
TI - Cyclic random motions in $\mathbb{R}^d$-space with n directions
JO - ESAIM: Probability and Statistics
DA - 2006/9//
PB - EDP Sciences
VL - 10
SP - 277
EP - 316
AB - 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.
LA - eng
KW - Cyclic random motions; linear image of a random vector; singular and absolutely continuous measures; convexity; hyper-Bessel functions with several arguments.; linear image of a random vector; convexity; hyper-Bessel functions with several arguments
UR - http://eudml.org/doc/249743
ER -

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