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On some problems concerning a restricted random walk

Published online by Cambridge University Press:  14 July 2016

Ora Engelberg
Ora Engelberg*
Affiliation:
Columbia University, New York
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Extract

A particle that is located at the position (0, 0) in the (x, y) plane at the moment x = 0 is subjected to random impulses at the moments of time x = 1, 2, …, a + b. As the result of each impulse, the particle may be displaced either by a positive or by a negative step with respect to the y-axis; it is given that the particle is restricted by the condition that it has exactly a positive and b negative displacements. Hence in the (x, y) plane the path of the particle at each impulse is depicted by a shift of a unit to the right and by one unit upward or downward; that is the particle's path starts at (0, 0) and terminates at (a + b, a – b) after a + b steps. We assume that all possible different paths are equally probable, and assume also, without loss of generality, that a ≧ b.

Type
Research Papers
Information
Journal of Applied Probability , Volume 2 , Issue 2 , December 1965 , pp. 396 - 404
Copyright
Copyright © Applied Probability Trust 

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References

[1] Dvoretzky, A. and Motzkin, T. (1947) A problem of arrangements. Duke Math. J. 14, 305–313.Google Scholar
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[3] Feller, W. (1957) An Introduction to Probability Theory and its Applications 1 (2nd ed.) John Wiley, New York.Google Scholar
[4] Engelberg, O. (1965) Generalizations of the ballot problem. Zeit. Wahrscheinlichkeitstheorie 3, 271–275.CrossRefGoogle Scholar