Hostname: page-component-6898cfcb55-kwdml Total loading time: 0 Render date: 2026-11-11T05:04:02.070Z Has data issue: false hasContentIssue false
  • English
  • Français

The correlated random walk

Published online by Cambridge University Press:  14 July 2016

Eric Renshaw and
Robin Henderson
Eric Renshaw*
Affiliation:
University of Edinburgh
Robin Henderson*
Affiliation:
University of Edinburgh
*
∗Postal address: Department of Statistics, University of Edinburgh, James Clerk Maxwell Building, The King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, U.K.
∗Postal address: Department of Statistics, University of Edinburgh, James Clerk Maxwell Building, The King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, U.K.
Get accessRights & Permissions [Opens in a new window]

Abstract

A one-dimensional random walk is studied in which, at each stage, the probabilities of continuing in the same direction or of changing direction are p and 1 – p, respectively. Exact expressions are derived for the n-step transition probabilities, and various limiting distributions are investigated.

Keywords

CORRELATED RANDOM WALKOCCUPATION PROBABILITIESDIFFUSIONBESSEL FUNCTIONS

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 18 , Issue 2 , June 1981, pp. 403 - 414
Copyright
Copyright © Applied Probability Trust 

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.)

Article purchase

Temporarily unavailable

References

Abramowitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions. Dover, New York.Google Scholar
Bartlett, M. S. (1975) Appendix to the paradox of probability in physics. In Probability, Statistics and Time. Chapman and Hall, London.Google Scholar
Cane, V. R. (1967) Random walks and physical processes. Bull. Internat. Stat. Inst. 42, 622–640.Google Scholar
Cane, V. R. (1975) Diffusion models with relativity effects. In Perspectives in Probability and Statistics: Papers in Honour of M. S. Bartlett on the Occasion of his Sixty-fifth Birthday, ed. Gani, J., Distributed by Academic Press, London for the Applied Probability Trust, Sheffield, 263–273.Google Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. 1, 2nd edn. Wiley, New York.Google Scholar
Flory, P. J. (1962) Principles of Polymer Chemistry. Cornell University Press, Ithaca, New York.Google Scholar
Gillis, J. (1955) Correlated random walk. Proc. Camb. Phil. Soc. 51, 639–651.Google Scholar
Goldstein, S. (1951) On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. 4, 129–156.Google Scholar
Gorostiza, L. G. (1973) An invariance principle for a class of d-dimensional polygonal random functions. Trans. Amer. Math. Soc. 177, 413–445.Google Scholar
Mohan, C. (1955) The gambler's ruin problem with correlation. Biometrika 42, 486–493.Google Scholar
Montroll, E. W. (1964) Random walks on lattices. Proc. Symp. Appl. Math. of Amer. Math. Soc. 16, 193–220.Google Scholar
Moyal, J. E. (1950) The momentum and sign of fast cosmic ray particles. Phil. Mag. 41, 1058–1077.Google Scholar
Proudfoot, A. D. and Lampard, D. G. (1972) A random walk problem with correlation. J. Appl. Prob. 9, 436–440.Google Scholar
Rosen, B. (1967) On the central limit theorem for sums of dependent random variables. Z. Wahrscheinlichkeitsth. 7, 48–82.Google Scholar
Skellam, J. G. (1973) The formulation and interpretation of mathematical models of diffusionary processes in population biology. In The Mathematical Theory of the Dynamics of Biological Populations, ed. Bartlett, M. S. and Hiorns, R. W., Academic Press, London, 63–85.Google Scholar