Moment analysis of the telegraph random process

Alexander D. Kolesnik

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2012), pp. 90-107

Résumé

We consider the Goldstein–Kac telegraph process $X(t)$, $t>0$, on the real line $\mathbb R^1$ 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.