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 Bul. Acad. Ştiinţe Repub. Mold. Mat., 2012, Number 1, Pages 90–107 (Mi basm304)

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.

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Document Type: Article
MSC: 60K35, 60J60, 60J65, 82C41, 82C70
Language: English

Citation: Alexander D. Kolesnik, “Moment analysis of the telegraph random process”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2012, no. 1, 90–107

Citation in format AMSBIB
\Bibitem{Kol12} \by Alexander D.~Kolesnik \paper Moment analysis of the telegraph random process \jour Bul. Acad. \c Stiin\c te Repub. Mold. Mat. \yr 2012 \issue 1 \pages 90--107 \mathnet{http://mi.mathnet.ru/basm304} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2987330} \zmath{https://zbmath.org/?q=an:06100376} 

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• http://mi.mathnet.ru/eng/basm/y2012/i1/p90

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Kolesnik A.D., “Probability Distribution Function For the Euclidean Distance Between Two Telegraph Processes”, Adv. Appl. Probab., 46:4 (2014), 1172–1193
2. Lopez O., Ratanov N., “on the Asymmetric Telegraph Processes”, J. Appl. Probab., 51:2 (2014), 569–589
3. Kolesnik A.D., “the Explicit Probability Distribution of the Sum of Two Telegraph Processes”, Stoch. Dyn., 15:2 (2015), 1550013
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