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A simple random walk and an associated asymptotic behaviour of the Bessel functions

Published online by Cambridge University Press:  24 October 2008

J. Keilson
J. Keilson
Affiliation:
Applied Research LaboratorySylvania Electronic Systems100 First Avenue, Waltham 54, Mass., U.S.A.
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Extract

We consider a random walk defined in the following way. We have a set of states indexed by n where n takes on all negative and positive integral values and zero. When we are at state n, there is a probability per unit time λ of going to n + 1, and a probability per unit time λ of going to n − l. Let us start out at n = 0, and study Wn(t), the probability of being at n at time t. Continuity of probability requires that whence since G(s, 0) = 1, we have It follows from the well-known result .

Type
Research Notes
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 58 , Issue 4 , October 1962 , pp. 708 - 709
Copyright
Copyright © Cambridge Philosophical Society 1962

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