Random walks: the problem of attaining a set

Authors: Krysyaev R.Yu.
Published in issue: #4(9)/2017
DOI: 10.18698/2541-8009-2017-4-74
Category: Mathematics | Chapter: Computational Mathematics
Keywords: markov processes, random walks, walk trajectories, stationary distri-bution, limiting characteristics, functional equation, Student’s t-distribution, linear operator
Published: 10.04.2017

The work deals with the problem of a particle performing a random walk attaining a certain set. We consider the definition of a random walk as a stochastic Markov process and characteristics of random walk functional trajectories. We present a stationary distribution and limiting characteristics for random walks. We formulate the problem of attaining a set. We solve the problem of attaining a horizontal line. We found an expression to search for the probability of landing in a set. We describe the limit behaviour of the landing coordinate, the distribution density of which approaches the density of Student's t-distribution. We supply the results concerning the distribution of the moment when the particle first lands in the set. We derive an expression for the Markov operator norm. We estimate the first landing moment time for a limit case.

References

[1] Rozanov Yu.A. Sluchaynye protsessy (kratkiy kurs) [Stochastic processes: short course]. Moscow, Nauka Publ., 1971. 288 p. (in Russ.).

[2] Kolmogorov A.N., Fomin S.V. Elementy teorii funktsiy i funktsional’nogo analiza [Elements of function theory and functional analysis]. Moscow, Nauka Publ., 1972. 496 p. (in Russ.).

[3] Ito K. Veroyatnostnye protsessy. Vypusk I [Random processes. Iss. 1]. Moscow, IL Publ., 1960. 135 p. (in Russ.).

[4] Spicer F., Axler S., Gehring F.W., Ribet K.A., eds. Principles of random walk. 2nd ed. New York, Springer-Verlag, 1976. 422 p.

[5] Zorich V.A. Matematicheskiy analiz. Chast’ II [Mathematical analysis. Part II]. Moscow, MTsNMO Publ., 2002. 794 p. (in Russ.).

[6] Polyanin A.D., Manzhirov A.V. Spravochnik po integral’nym uravneniyam: Tochnye resh-eniya [Handbook on integral equations: exact solutions]. Moscow, Faktorial Publ., 1998. 432 p. (in Russ.).

[7] Kanatnikov A.N., Krishchenko A.P. Analiticheskaya geometriya [Analytic geometry]. Mos-cow, Bauman MSTU Publ., 2002. 388 p. (in Russ.).