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Some explicit results for an asymmetric two-dimensional random walk

Published online by Cambridge University Press:  24 October 2008

V. D. Barnett
V. D. Barnett
Affiliation:
Statistical Laboratory, University of Manchester
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Abstract

Three distinct methods are used to obtain exact expressions for various characteristics of a particular asymmetric two-dimensional random walk. The results obtained include, for the transient unrestricted walk, the probability of return to the starting-point and the average number of arrivals at the general lattice point; and, for a walk restricted within a rectangular absorbing barrier, the average number of arrivals at any accessible point and the absorption probabilities on the boundary. Whilst there is some duplication of results by using the three different methods of analysis, this is not extensive and provides a useful check on the results. Also the methods are of some general interest in themselves.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 59 , Issue 2 , April 1963 , pp. 451 - 462
Copyright
Copyright © Cambridge Philosophical Society 1963

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References

REFERENCES

(1)Polya, G., Üiber eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Strassennetz. Math. Ann. 84 (1921), 149–160.Google Scholar
(2)Domb, C., On multiple returns in the random walk problem. Proc. Cambridge Philos. Soc. 50 (1954), 586–591.Google Scholar
(3)Foster, F. G., and Good, I. J., On a generalization of Polya's random walk theorem. Quart. J. Math. Oxford Ser. (2), 4 (1953), 120–126.Google Scholar
(4)Chung, K. L., and Fuchs, W. H. J., On the distribution of values of sums of random variables. Mem. American Math. Soc. 6 (1951), 2–12.Google Scholar
(5)Montroll, E. W., Random walks in multidimensional spaces, especially on periodic lattices. J. Soc. Indust. Appl. Math. 4 (1956), 241–260.Google Scholar
(6)McCrea, W. H., and Whipple, F. J. W., Random paths in two and three dimensions. Proc. Roy. Soc. Edinburgh, Sect. A, 60 (1940), 281–298.Google Scholar
(7)Feller, W., An introduction to probability theory and its applications, vol. 1, 2nd ed. (Wiley; New York, 1959).Google Scholar
(8)Skellam, J. G., and Shenton, L. R., Distributions associated with random walk and recurrent events. J. Roy. Statist. Soc. Ser. B, 19 (1957), 64–118.Google Scholar
(9)Bailey, W. N., Generalised hypergeometric series (Cambridge, 1935).Google Scholar
(10)Whittaker, E. T., and Watson, G. N., Modern analysis, 4th ed. (Cambridge, 1927).Google Scholar
(11)Jahnke, E., Emde, F., and Lösch, F., Tables of higher functions, 6th ed. (McGraw-Hill; New York, 1960).Google Scholar
(12)Erdélyi, A. et al. , Higher transcendental functions, vol. 2 (McGraw-Hill; New York, 1953).Google Scholar