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The Frobenius–Harper technique in a general recurrence model

Part of: Combinatorial probability Difference and functional equations Markov processes Difference equations Limit theorems

Published online by Cambridge University Press:  14 July 2016

Di Warren
Di Warren*
Affiliation:
University of Sydney
*
∗Postal address: 9 Trigg Avenue, Carlingford 2118, Australia.
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Abstract

We present a general recurrence model which provides a conceptual framework for well-known problems such as ascents, peaks, turning points, Bernstein's urn model, the Eggenberger–Pólya urn model and the hypergeometric distribution. Moreover, we show that the Frobenius-Harper technique, based on real roots of a generating function, can be applied to this general recurrence model (under simple conditions), and so a Berry–Esséen bound and local limit theorems can be found. This provides a simple and unified approach to asymptotic theory for diverse problems hitherto treated separately.

Keywords

Generating functionreal rootsinhomogeneous Markov chainascentspeaksturning pointsurn modelsBernsteinEggenberger–Pólyahypergeometric

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60C05: Combinatorial probability 60F05: Central limit and other weak theorems 39A10: Difference equations, additive
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 1 , March 1999 , pp. 30 - 47
Copyright
Copyright © Applied Probability Trust 1999 

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