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On the occupancy problem for a regime-switching model

Part of: Markov processes

Published online by Cambridge University Press:  04 May 2020

Michael Grabchak ,
Mark Kelbert  and
Quentin Paris
Michael Grabchak*
Affiliation:
University of North Carolina Charlotte
Mark Kelbert*
Affiliation:
National Research University Higher School of Economics
Quentin Paris*
Affiliation:
National Research University Higher School of Economics
*
*Postal address: Department of Mathematics and Statistics, Charlotte, NC, USA. Email address: [email protected]
**Postal address: National Research University Higher School of Economics (HSE), Faculty of Economics, Department of Statistics and Data Analysis, Moscow, Russia. Email address: [email protected]
***Postal address: National Research University Higher School of Economics (HSE), Faculty of Computer Science, School of Data Analysis and Artificial Intelligence & HDI Lab, Moscow, Russia. Email address: [email protected]
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Abstract

This article studies the expected occupancy probabilities on an alphabet. Unlike the standard situation, where observations are assumed to be independent and identically distributed, we assume that they follow a regime-switching Markov chain. For this model, we (1) give finite sample bounds on the expected occupancy probabilities, and (2) provide detailed asymptotics in the case where the underlying distribution is regularly varying. We find that in the regularly varying case the finite sample bounds are rate optimal and have, up to a constant, the same rate of decay as the asymptotic result.

Keywords

occupancy problemregime switchingMarkov chainregular variation

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 53 - 77
Copyright
© Applied Probability Trust 2020

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