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How to estimate the rate function of a cumulative process

Part of: Limit theorems Special processes

Published online by Cambridge University Press:  14 July 2016

Ken Duffy  and
Anthony P. Metcalfe
Ken Duffy*
Affiliation:
National University of Ireland
Anthony P. Metcalfe*
Affiliation:
University College Cork
*
∗Postal address: Hamilton Institute, National University of Ireland, Maynooth, County Kildare, Ireland. Email address: [email protected]
∗∗Postal address: Boole Centre for Research in Informatics, University College Cork, County Cork, Ireland.
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Abstract

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Consider two sequences of bounded random variables, a value and a timing process, that satisfy the large deviation principle (LDP) with rate function J(⋅,⋅) and whose cumulative process satisfies the LDP with rate function I(⋅). Under mixing conditions, an LDP for estimates of I constructed by transforming an estimate of J is proved. For the case of a cumulative renewal process it is demonstrated that this approach is favourable to a more direct method, as it ensures that the laws of the estimates converge weakly to a Dirac measure at I.

Keywords

Estimating large deviationscumulative processrenewal process

MSC classification

Primary: 60F10: Large deviations
Secondary: 60K05: Renewal theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 4 , December 2005 , pp. 1044 - 1052
Copyright
© Applied Probability Trust 2005 

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