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Markov Processes Involving q-Stirling Numbers

Published online by Cambridge University Press:  01 June 1997

D. CRIPPA
Affiliation:
Institute for Theoretical Computer Science, ETH–Zurich, CH–8092 Zurich, Switzerland (e-mail: simon@inf.ethz.ch)
K. SIMON
Affiliation:
Institute for Theoretical Computer Science, ETH–Zurich, CH–8092 Zurich, Switzerland (e-mail: simon@inf.ethz.ch)
P. TRUNZ
Affiliation:
Institute for Theoretical Computer Science, ETH–Zurich, CH–8092 Zurich, Switzerland (e-mail: simon@inf.ethz.ch)

Abstract

In this paper we consider the Markov process defined by

P1,1=1, Pn,[lscr ]=(1−λn,[lscr ]) ·Pn−1,[lscr ]n,[lscr ]−1 ·Pn−1,[lscr ]−1

for transition probabilities λn,[lscr ]=q[lscr ] and λn,[lscr ]=qn−1. We give closed forms for the distributions and the moments of the underlying random variables. Thereby we observe that the distributions can be easily described in terms of q-Stirling numbers of the second kind. Their occurrence in a purely time dependent Markov process allows a natural approximation for these numbers through the normal distribution. We also show that these Markov processes describe some parameters related to the study of random graphs as well as to the analysis of algorithms.

Type
Research Article
Copyright
1997 Cambridge University Press

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