Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T20:28:52.609Z Has data issue: false hasContentIssue false
  • English
  • Français

Some explicit results for correlated random walks

Published online by Cambridge University Press:  14 July 2016

Ram Lal  and
U. Narayan Bhat
Ram Lal
Affiliation:
Southern Methodist University
U. Narayan Bhat*
Affiliation:
Southern Methodist University
*
∗ Postal address for both authors: Department of Statistical Science, Southern Methodist University, Dallas, TX 75275, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

In a correlated random walk (CRW) the probabilities of movement in the positive and negative direction are given by the transition probabilities of a Markov chain. The walk can be represented as a Markov chain if we use a bivariate state space, with the location of the particle and the direction of movement as the two variables. In this paper we derive explicit results for the following characteristics of the walk directly from its transition probability matrix: (i) n -step transition probabilities for the unrestricted CRW, (ii) equilibrium distribution for the CRW restricted on one side, and (iii) equilibrium distribution and first-passage characteristics for the CRW restricted on both sides (i.e., with finite state space).

Keywords

MARKOV CHAINEQUILIBRIUM SOLUTIONFIRST-PASSAGE PROBLEM
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 4 , December 1989 , pp. 757 - 766
Copyright
Copyright © Applied Probability Trust 1989 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bender, E. A. and Richmond, L. B. (1984) Correlated random walks. Ann. Prob. 12, 274–278.CrossRefGoogle Scholar
Darroch, J. N. and Whitford, H. J. (1972) Exact fluctuation results for Markov dependent coin tossing. J. Appl. Prob. 9, 158–168.Google Scholar
Gillis, J. (1955) Correlated random walks. Proc. Camb. Phil. Soc. 51, 639–651.CrossRefGoogle Scholar
Gupta, H. C. (1958) Diffusion by discrete movements. Sankhya 20, 295–308.Google Scholar
Henderson, R., Renshaw, E. and Ford, D. (1983) A note on the recurrence of a correlated random walk. J. Appl. Prob. 20, 696–699.Google Scholar
Henderson, R., Renshaw, E. and Ford, D. (1984) A correlated random walk model for two dimensional diffusion. J. Appl. Prob. 31, 233–246.Google Scholar
Jain, G. C. (1971) Some results in a correlated random walk. Canad. Math. Bull. 14, 341–347.Google Scholar
Jain, G. C. (1973) On the expected number of visits of a particle before absorption in a correlated random walk. Canad. Math. Bull. 16, 389–395.Google Scholar
Kemeny, J. G. and Snell, J. L. (1960) Finite Markov Chains. Springer-Verlag, New York.Google Scholar
Lal, R. and Bhat, U. N. (1987) Reduced systems in Markov chains and their applications in queuing theory. Queueing Systems 2, 147–172.Google Scholar
Lal, R. and Bhat, U. N. (1988) Reduced system algorithms for Markov chains. Management Sci. 34, 1202–1220.Google Scholar
Mohan, C. (1955) The gambler's ruin problem with correlation. Biometrika 42, 486–493.Google Scholar
Nain, R. B. and Sen, K. (1979a) Unrestricted correlated random walk with stay. Metron 36, 159–172.Google Scholar
Nain, R. B. and Sen, K. (1979b) Correlated random walk with absorbing barriers. Metron 37, 149–163.Google Scholar
Nain, R. B. and Sen, K. (1980) Transition probability matrices for correlated walks. J. Appl. Prob. 17, 253–258.Google Scholar
Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models. The Johns Hopkins University Press, Baltimore.Google Scholar
Proudfoot, A. D. and Lampard, D. G. (1972) A random walk problem with correlation. J. Appl. Prob. 9, 436–440.Google Scholar
Ramaswami, V. and Latouche, G. (1986) A generalized class of Markov process with explicit matrix-geometric solution. OR Spektrum 8, 209–218.Google Scholar
Renshaw, E. and Henderson, R. (1981) The correlated random walk. J. Appl. Prob. 18, 403–414.Google Scholar
Roerdink, J. B. T. M. (1985) A note on the asymptotic properties of correlated random walks. J. Appl. Prob. 22, 951–956.CrossRefGoogle Scholar
Seth, A. (1963) The correlated unrestricted random walk. J. R. Statist. Soc. B 25, 394–400.Google Scholar