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

  • Douglas P. Kennedy (a1) and Robert P. Kertz (a2)

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.

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Corresponding author

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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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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References

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[1] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.
[2] Chow, Y. S., Robbins, H., and Siegmund, D. (1971) Great Expectations: The Theory of Optimal Stopping. Houghton-Mifflin, New York.
[3] Chow, Y. S., and Teicher, H. (1978) Probability Theory: Independence, Interchangeability, Martingales. Springer-Verlag, New York.
[4] Gumbel, E. J. (1958) Statistics of Extremes. Columbia University Press, New York.
[5] Haan, L. De and Verkade, E. (1987) On extreme-value theory in the presence of a trend. J. Appl. Prob. 24, 6276.
[6] Hüsler, J. (1979) The limiting behaviour of the last exit time for sequences of independent, identically distributed random variables. Z. Wahrscheinlichkeitsth. 50, 159164.
[7] Kennedy, D. P. and Kertz, R. P. (1988) The asymptotic behavior of the reward sequence in the optimal stopping of i.i.d. random variables. Ann. Prob. To appear.
[8] Leadbetter, M. R., Lindgren, G., and Rootzén, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.
[9] Resnick, S. I. (1987) Extreme Values, Regular Variation, and Point Processes. Springer-Verlag, New York.
[10] Samuel-Cahn, E. (1984) Comparison of threshold stop rules and maximum for independent nonnegative random variables. Ann. Prob. 12, 12131216.
[11] Whitt, W. (1980) Some useful functions for functional limit theorems. Math. Operat. Res. 5, 6785.

Keywords

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

  • Douglas P. Kennedy (a1) and Robert P. Kertz (a2)

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