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Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, 2012, Number 1, Pages 90–107
(Mi basm304)
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This article is cited in 7 scientific papers (total in 7 papers)
Moment analysis of the telegraph random process
Alexander D. Kolesnik Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, Kishinev, Moldova
Abstract:
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.
Keywords and phrases:
random evolution, random flight, persistent random walk, telegraph process, moments, Carleman condition, moment problem, asymptotic behaviour, semi-invariants.
Received: 14.11.2011
Citation:
Alexander D. Kolesnik, “Moment analysis of the telegraph random process”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2012, no. 1, 90–107
Linking options:
https://www.mathnet.ru/eng/basm304 https://www.mathnet.ru/eng/basm/y2012/i1/p90
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