Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-22T05:42:33.395Z Has data issue: false hasContentIssue false
  • English
  • Français

The general correlated random walk

Published online by Cambridge University Press:  14 July 2016

Anyue Chen  and
Eric Renshaw
Anyue Chen*
Affiliation:
University of Hong Kong
Eric Renshaw*
Affiliation:
University of Strathclyde
*
∗Postal address: Department of Statistics, University of Hong Kong, Pokfulam Road, Hong Kong.
∗∗Postal address: Department of Statistics and Modelling Science, Livingstone Tower, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we provide further results on the general d-dimensional correlated random walk. In particular, we prove that the n-step characteristic function of any correlated random walk satisfies a recurrence formula which enables both it and the total characteristic function to be obtained. Some examples are then considered.

Keywords

CHARACTERISTIC FUNCTIONMOMENTSRECURRENCE
Type
Research Article
Information
Journal of Applied Probability , Volume 31 , Issue 4 , December 1994 , pp. 869 - 884
Copyright
Copyright © Applied Probability Trust 1994 

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

Barber, M. N. and Ninham, B. W. (1970) Random and Restricted Walks. Gordon and Breach, New York.Google Scholar
Chen, A. Y. and Renshaw, E. (1992) The Gillis-Domb-Fisher correlated random walk. J. Appl. Prob. 29, 792–813.CrossRefGoogle Scholar
Chung, K. L. and Fuchs, W. H. J. (1951) On the distribution of values of sums of random variables. Mem. Amer. Math. Soc. 6, 1–12.Google Scholar
Daley, D. J. (1979) Lattice valued random walks with Markov chain dependent steps. Math. Proc. Camb. Phil. Soc. 86, 115–126.CrossRefGoogle Scholar
Domb, C. and Fisher, M. E. (1958) On random walks with restricted reversals. Proc. Camb. Phil. Soc. 54, 48–59.CrossRefGoogle Scholar
Flory, P. J. (1962) Principles of Polymer Chemistry. Cornell University Press, Ithaca, NY.Google Scholar
Gantmacher, F. R. (1959) The Theory of Matrices. Vol. I. Chelsea, New York.Google Scholar
Gillis, J. (1955) Correlated random walk. Proc. Camb. Phil. Soc. 51, 639–651.CrossRefGoogle Scholar
Goldstein, S. (1951) On diffusion by discontinuous movements and on the telegraph equation. Quart. J. Mech. 4, 129–156.CrossRefGoogle Scholar
Henderson, R. and Renshaw, E. (1980) Spatial stochastic models and computer simulation applied to the study of tree root systems. Compstat 80, 389–395. Physica Verlag, Vienna.Google Scholar
Henderson, R., Ford, E. D. and Renshaw, E. (1983a) Morphology of the structural root system of Sitka spruce, 2: Computer simulation of rooting patterns. Forestry 56, 137–153.CrossRefGoogle Scholar
Henderson, R., Ford, E. D., Renshaw, E. and Deans, J. D. (1983b) Morphology of the structural root system of Sitka spruce, 1: analysis and quantitative description. Forestry 56, 121–135.CrossRefGoogle Scholar
Henderson, R., Renshaw, E. and Ford, E. D. (1984) A correlated random walk model for two-dimensional diffusion. J. Appl. Prob. 21, 233–246.CrossRefGoogle Scholar
Horn, R. A. and Johnson, C. A. (1985) Matrix Analysis. Cambridge University Press.CrossRefGoogle Scholar
Iossif, G. (1986) Return probabilities for correlated random walks. J. Appl. Prob. 23, 201–207.CrossRefGoogle Scholar
Jain, G. C. (1971) Some results in a correlated random walk. Canad. Math. Bull. 14, 341–347.CrossRefGoogle 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.CrossRefGoogle Scholar
Jakeman, E. and Renshaw, E. (1987) Correlated random walk model for scattering. J. Opt. Soc. Amer. A4, 1206–1212.CrossRefGoogle Scholar
Klein, G. (1952) A generalization of the random walk problem. Proc. Roy. Soc. Edin. A63, 268–279.Google Scholar
Lal, R. and Bhat, U. N. (1989) Some explicit results for correlated random walks. J. Appl. Prob. 26, 757–766.CrossRefGoogle Scholar
Lancaster, P. (1969) Theory of Matrices. Academic Press, London.Google Scholar
Nain, R. B. and Sen, Kanwar (1980) Transition probability matrices for correlated random walks. J. Appl. Prob. 17, 253–258.CrossRefGoogle Scholar
Renshaw, E. (1985) Computer simulation of Sitka spruce: spatial branching models for canopy growth and root structure. IMA J. Math. Appl. Med. Biol. 2, 183–200.CrossRefGoogle Scholar
Renshaw, E. (1991) Modelling Biological Populations in Space and Time. Cambridge University Press.CrossRefGoogle Scholar
Renshaw, E. and Henderson, R. (1981) The correlated random walk. J. Appl. Prob. 18, 403–414.CrossRefGoogle Scholar
Seth, A. (1963) The correlated unrestricted random walk. J. R. Statist. Soc. B25, 394–400.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., pp. 63–85. Academic Press, London.Google Scholar
Zhang, Y. L. (1992) Some problems on a one-dimensional correlated random walk with various types of barriers. J. Appl. Prob. 29, 196–201.CrossRefGoogle Scholar