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Limit theorems for threshold-stopped random variables with applications to optimal stopping

Published online by Cambridge University Press:  01 July 2016

Douglas P. Kennedy  and
Robert P. Kertz
Douglas P. Kennedy*
Affiliation:
University of Cambridge
Robert P. Kertz*
Affiliation:
Georgia Institute of Technology
*
∗Postal address: Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge, CB2 1SB, UK.
∗∗Postal address: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA.
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Abstract

The extremal types theorem identifies asymptotic behaviour for the maxima of sequences of i.i.d. random variables. A parallel theorem is given which identifies the asymptotic behaviour of sequences of threshold-stopped random variables. Three new types of limit distributions arise, but normalizing constants remain the same as in the maxima case. Limiting joint distributions are also given for maxima and threshold-stopped random variables. Applications to the optimal stopping of i.i.d. random variables are given.

Keywords

THRESHOLD STOPPING TIMESMAXIMA OF I.I.D. RANDOM VARIABLESEXTREMAL TYPES THEOREMDOMAINS OF ATTRACTIONEXTREME-VALUE THEORYPOISSON RANDOM MEASURESCONTINUOUS MAPPING THEOREM
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 2 , June 1990 , pp. 396 - 411
Copyright
Copyright © Applied Probability Trust 1990 

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Footnotes

This author is grateful to the School of Mathematics of the Georgia Institute of Technology, for support during the year 1987–1988.

Supported in part by NSF grants DMS-86–01153 and DMS-88–01818.

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