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Moment analysis of the telegraph random process
Alexander D. Kolesnik Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, Kishinev, Moldova
Аннотация:
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.
Ключевые слова и фразы:
random evolution, random flight, persistent random walk, telegraph process, moments, Carleman condition, moment problem, asymptotic behaviour, semi-invariants.
MSC: 60K35,
60J60,
60J65,
82C41,
82C70 Поступила в редакцию: 14.11.2011
Язык публикации: английский
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