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IMI/Publicaţii/BASM/Ediţii/BASM n.1 (68), 2012/

Moment analysis of the telegraph random process

Authors: Kolesnik Alexander

Abstract

We consider the Goldstein-Kac telegraph process $X(t), \; t>0,$ on the real line $\Bbb R^1$ performed by the random motion at finite speed $c$ and controlled by a homogeneous Poisson process of rate $\lambda>0$. Using a formula for the moment function $\mu_{2k}(t)$ of $X(t)$ we study its asymptotic behaviour, as $c$, $\lambda$ and $t$ vary in different ways. Explicit asymptotic formulas for $\mu_{2k}(t)$, as $k\to\infty$, are derived and numerical comparison of their effectiveness is given. We also prove that the moments $\mu_{2k}(t)$ for arbitrary fixed $t>0$ satisfy the Carleman condition and, therefore, the distribution of the telegraph process is completely determined by its moments. Thus, the moment problem is completely solved for the telegraph process $X(t)$. We obtain an explicit formula for the Laplace transform of $\mu_{2k}(t)$ and give a derivation of the the moment generating function based on direct calculations. A formula for the semi-invariants of $X(t)$ is also presented.

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