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A note on passage times and infinitely divisible distributions

Published online by Cambridge University Press:  14 July 2016

H. D. Miller
H. D. Miller*
Affiliation:
Imperial College, London
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Extract

Let X(t) be the position at time t of a particle undergoing a simple symmetrical random walk in continuous time, i.e. the particle starts at the origin at time t = 0 and at times T1, T1 + T2, … it undergoes jumps ξ1, ξ2, …, where the time intervals T1, T2, … between successive jumps are mutually independent random variables each following the exponential density e–t while the jumps, which are independent of the τi, are mutually independent random variables with the distribution . The process X(t) is clearly a Markov process whose state space is the set of all integers.

Type
Short Communications
Information
Journal of Applied Probability , Volume 4 , Issue 2 , August 1967 , pp. 402 - 405
Copyright
Copyright © Sheffield: Applied Probability Trust 

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References

Abramowitz, Milton and Stegun, Irene A. (Editors) (1965) Handbook of Mathematical Functions, Dover, New York.Google Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
Feller, W. (1966a) An Introduction to Probability Theory and its Applications. Vol. II, Wiley, London.Google Scholar
Feller, W. (1966b) Infinitely divisible distributions and Bessel functions associated with random walks. SIAM J. Appl. Math. 14, 864–875.Google Scholar
Keilson, J. (1962) The use of Green's functions in the study of bounded random walks with application to queueing theory. J. Math. and Phys. 41, 42–52.Google Scholar