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

This article is cited in 3 scientific papers (total in 3 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.

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Bibliographic databases:

Document Type: Article
MSC: 60K35, 60J60, 60J65, 82C41, 82C70
Received: 14.11.2011
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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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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  crossref  mathscinet  zmath  isi
    2. Lopez O., Ratanov N., “on the Asymmetric Telegraph Processes”, J. Appl. Probab., 51:2 (2014), 569–589  crossref  mathscinet  zmath  isi
    3. Kolesnik A.D., “the Explicit Probability Distribution of the Sum of Two Telegraph Processes”, Stoch. Dyn., 15:2 (2015), 1550013  crossref  mathscinet  zmath  isi  scopus
  • Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
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