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 $k\to \mathrm{\infty }$, 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.

Bibliographie

[1] Bartlett M., “Some problems associated with random velocity”, Publ. Inst. Stat. Univ. Paris, 6 (1957), 261–270 | MR | Zbl

[2] Bartlett M., “A note on random walks at constant speed”, Adv. Appl. Prob., 10 (1978), 704–707 | DOI | Zbl

[3] Bateman H., Erdelyi A., Tables of Integral Transforms, v. 1, McGraw-Hill, NY, 1954 | Zbl

[4] Bogachev L., Ratanov N., “Occupation time distributions for the telegraph process”, Stoch. Process. Appl., 121 (2011), 1816–1844 | DOI | MR | Zbl

[5] Cane V., “Random walks and physical processes”, Bull. Intern. Statist. Inst., 42 (1967), 622–640

[6] Cane V., “Diffusion models with relativity effects”, Perspectives in Probability and Statistics, Applied Probability Trust, Sheffield, 1975, 263–273 | MR | Zbl

[7] Di Crescenzo A., “On random motion with velocities alternating at Erlang-distributed random times”, Adv. Appl. Probab., 33 (2001), 690–701 | DOI | MR | Zbl

[8] Di Crescenzo A., Martinucci B., “A damped telegraph random process with logistic stationary distributions”, J. Appl. Probab., 47 (2010), 84–96 | DOI | MR | Zbl

[9] Di Crescenzo A., Martinucci B., “On the effect of random alternating perturbations on hazard rates”, Sci. Math. Japon., 64 (2006), 381–394 | MR | Zbl

[10] Foong S. K., “First-passage time, maximum displacement and Kac's solution of the telegrapher's equation”, Phys. Rev. A, 46 (1992), 707–710 | DOI | MR

[11] Foong S. K., Kanno S., “Properties of the telegrapher's random process with or without a trap”, Stoch. Process. Appl., 53 (2002), 147–173 | DOI | MR

[12] Goldstein S., “On diffusion by discontinuous movements and on the telegraph equation”, Quart. J. Mech. Appl. Math., 4 (1951), 129–156 | DOI | MR | Zbl

[13] Gradshteyn I. S., Ryzhik I. M., Tables of Integrals, Series and Products, Academic Press, NY, 1980 | Zbl

[14] Iacus S. M., Yoshida N., “Estimation for the discretely observed telegraph process”, Theory Probab. Math. Stat., 78 (2009), 37–47 | DOI | MR

[15] Kabanov Yu. M., “Probabilistic representation of a solution of the telegraph equation”, Theory Probab. Appl., 37 (1992), 379–380 | DOI | MR | Zbl

[16] Kac M., “A stochastic model related to the telegrapher's equation”, Rocky Mountain J. Math., 4 (1974), 497–509 | DOI | MR | Zbl

[17] Kaplan S., “Differential equations in which the Poisson process plays a role”, Bull. Amer. Math. Soc., 70 (1964), 264–267 | DOI | MR

[18] Kisynski J., “On M. Kac's probabilistic formula for the solution of the telegraphist's equation”, Ann. Polon. Math., 29 (1974), 259–272 | MR | Zbl

[19] Kolesnik A. D., “Moments of the Markovian random evolutions in two and four dimensions”, Bull. Acad. Sci. Moldova Ser. Math., 2008, no. 2(57), 68–80 | MR | Zbl

[20] Kolesnik A. D., “A four-dimensional random motion at finite speed”, J. Appl. Probab., 43 (2006), 1107–1118 | DOI | MR | Zbl

[21] Kolesnik A. D., “The equations of Markovian random evolution on the line”, J. Appl. Probab., 35 (1998), 27–35 | DOI | MR | Zbl

[22] Kolesnik A. D., Orsingher E., “A planar random motion with an infinite number of directions controlled by the damped wave equation”, J. Appl. Probab., 42 (2005), 1168–1182 | DOI | MR | Zbl

[23] Kolesnik A. D., Turbin A. F., “The equation of symmetric Markovian random evolution in a plane”, Stoch. Process. Appl., 75 (1998), 67–87 | DOI | MR | Zbl

[24] Lachal A., “Cyclic random motions in $\mathbb R^d$-space with $n$ directions”, ESAIM: Probab. Stat., 10 (2006), 277–316 | DOI | MR | Zbl

[25] Masoliver J., Weiss G. H., “First-passage times for a generalized telegrapher's equation”, Physica A, 183 (1992), 537–548 | DOI | MR

[26] Masoliver J., Weiss G. H., “On the maximum displacement of a one-dimensional diffusion process described by the telegrapher's equation”, Physica A, 195 (1993), 93–100 | DOI | Zbl

[27] Orsingher E., “Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws”, Stoch. Process. Appl., 34 (1990), 49–66 | DOI | MR | Zbl

[28] Orsingher E., “Motions with reflecting and absorbing barriers driven by the telegraph equation”, Random Operat. Stoch. Equat., 3 (1995), 9–21 | DOI | MR | Zbl

[29] Pinsky M. A., Lectures on Random Evolution, World Sci., River Edge, NJ, 1991 | MR | Zbl

[30] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integrals and Series. Special Functions, Nauka, Moscow, 1983 (in Russian) | MR | Zbl

[31] Samoilenko I. V., “Moments of Markov random evolutions”, Ukrain. Math. J., 53 (2001), 1002–1008 | DOI | MR

[32] Stadje W., Zacks S., “Telegraph processes with random velocities”, J. Appl. Probab., 41 (2004), 665–678 | DOI | MR | Zbl

[33] Turbin A. F., Samoilenko I. V., “A probabilistic method for solving the telegraph equation with real-analytic initial conditions”, Ukrain. Math. J., 52 (2000), 1292–1299 | DOI | MR | Zbl