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Limit theorems for suprema, threshold-stopped random variables and last exits of i.i.d. random variables with costs and discounting, with applications to optimal stopping

Part of: Stochastic processes Sequential methods Limit theorems

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, 30332, USA.
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Abstract

For linear-cost-adjusted and geometric-discounted infinite sequences of i.i.d. random variables, point process convergence results are proved as the cost or discounting effect diminishes. These process convergence results are combined with continuous-mapping principles to obtain results on joint convergence of suprema and threshold-stopped random variables, and last-exit times and locations. Applications are made to several classical optimal stopping problems in these settings.

Keywords

OPTIMAL STOPPINGMAXIMA OF I.I.D. RANDOM VARIABLESDOMAINS OF ATTRACTIONLAST-EXIT TIMESEXTREME-VALUE THEORYPOINT PROCESSESPOISSON RANDOM MEASUREWEAK CONVERGENCE

MSC classification

Primary: 60G55: Point processes
Secondary: 60F05: Central limit and other weak theorems 60G40: Stopping times; optimal stopping problems; gambling theory 60G70: Extreme value theory; extremal processes 62L15: Optimal stopping
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 2 , June 1992 , pp. 241 - 266
Copyright
Copyright © Applied Probability Trust 1992 

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Footnotes

This author was supported in part by NSF grant DMS-88-01818.

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