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Summary: Representations for the rate of convergence of
birth-death processes
Erik A. van Doorn
Faculty of Mathematical Sciences
University of Twente
P.O. Box 217, 7500 AE Enschede, The Netherlands
E-mail: doorn@math.utwente.nl
May 4, 2001
Abstract. We display some representations for the rate of convergence of a
birth-death process, which are useful for obtaining upper and lower bounds.
The expressions are brought to light by exploiting the spectral representation
for the transition probabilities of a birth-death process and results from the
theory of orthogonal polynomials.
Keywords and phrases: birth-death process, decay parameter, duality, orthog-
onal polynomials, rate of convergence, spectral measure
2000 Mathematics Subject Classification: Primary 60J80
1 Introduction
In a recent paper Kartashov [14] has shown that the rate of convergence to
stationarity of an ergodic birth-death process on the nonnegative integers with
birth rates n and death rates µn is bounded below by the quantity
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