Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T14:37:08.122Z Has data issue: false hasContentIssue false
  • English
  • Français

Correlated random walk

Published online by Cambridge University Press:  24 October 2008

J. Gillis
J. Gillis
Affiliation:
Weizmann Institute of ScienceIsrael Institute for Advanced StudyPrincetonNew Jersey
Get access Rights & Permissions [Opens in a new window]

Abstract

Random walk on a d-dimensional lattice is investigated such that, at any stage, the probabilities of the step being in the various possible directions depend upon the direction of the previous step. The motion may be characterized by a generating function which is here derived. The generating function is then used to obtain some general properties of the walk. Certain special cases are considered in greater detail. The existence of recurrent points is investigated in particular, and the probability of returning to the origin after 2n steps. This latter function is evaluated asymptotically for the cases d = 1 and d = an even integer.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 51 , Issue 4 , October 1955 , pp. 639 - 651
Copyright
Copyright © Cambridge Philosophical Society 1955

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

REFERENCES

(1)Feller, W.An introduction to probability theory and its applications, vol. 1 (New York, 1950).Google Scholar
(2)Frèchet, M. and Shohat, J.A proof of the generalized second-limit theorem in the theory of probability. Trans. Amer. math. Soc. 33 (1931), 533–43.Google Scholar
(3)Goldstein, S.On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. 4 (1951), 129–56.Google Scholar
(4)Klein, G.A generalization of the random-walk problem. Proc. Roy. Soc. Edinb. A, 63 (1952), 268–79.Google Scholar
(5)Polya, G.Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straszennetz. Math. Ann. 84 (1921), 149–60.Google Scholar
(6)Watson, G. N.Asymptotic expansions of hypergeometric functions. Trans. Comb. phil. Soc. 22 (1912–1923), 277–308.Google Scholar
(7)Whittaker, E. T. and Watson, G. N.Modern analysis, 4th ed. (Cambridge, 1940).Google Scholar