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Embedding submartingales in wiener processes with drift, with applications to sequential analysis

Published online by Cambridge University Press:  14 July 2016

W. J. Hall
W. J. Hall*
Affiliation:
Stanford University and University of North Carolina
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Summary

Skorokhod (1961) demonstrated how the study of martingale sequences (and zero-mean random walks) can be reduced to the study of the Wiener process (without drift) at a sequence of random stopping times. We show how the study of certain submartingale sequences, including certain random walks with drift and log likelihood ratio sequences, can be reduced to the study of the Wiener process with drift at a sequence of stopping times (Theorem 4.1). Applications to absorption problems are given. Specifically, we present new derivations of a number of the basic approximations and inequalities of classical sequential analysis, and some variations on them — including an improvement on Wald's lower bound for the expected sample size function (Corollary 7.5).

Type
Research Papers
Information
Journal of Applied Probability , Volume 6 , Issue 3 , December 1969 , pp. 612 - 632
Copyright
Copyright © Applied Probability Trust 

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References

Anderson, T. W. (1960) A modification of the sequential probability ratio test to reduce the sample size. Ann. Math. Statist. 31, 165–197.Google Scholar
Armitage, P. (1957) Restricted sequential prodecures. Biometrika 44, 9–26.Google Scholar
Chow, Y. S., and Robbins, Herbert (1963) A renewal theorem for random variables which are dependent or non-identically distributed. Ann. Math. Statist. 34, 390–395.Google Scholar
Cox, D. R., and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
De Finetti, Bruno (1939) La teoria del rischio e il problema della “rovina dei giocatori”. G. 1st. Ital. Attuari 10, 41–51.Google Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Dubins, Lester E., and Savage, Leonard J. (1965) How to Gamble if You Must: Inequalities for Stochastic Processes. McGraw-Hill, New York (Section 8.7).Google Scholar
Dvoretzky, A., Kiefer, J., and Wolfowitz, J. (1953). Sequential decision problems for problems with continuous time parameter-testing hypotheses. Ann. Math. Statist. 24, 254–264.CrossRefGoogle Scholar
Hall, W. J. (1968a) On Wald's equations for Wiener processes. Techn. Report No. 32, NSF Grant GP-5705, Department of Statistics, Stanford University.Google Scholar
Hall, W. J. (1968b) On the Skorokhod embedding theorem. Techn. Report No. 33, NSF Grant GP-5705, Department of Statistics, Stanford University. To appear in J. Appl. Prob. 7 (1970).Google Scholar
Skorokhod, A. V. (1961) Translated from the Russian in 1965 as: Studies in the Theory of Random Processes. Addison-Wesley, Reading, Mass. (Chapter 7).Google Scholar
Wald, Abraham (1947) Sequential Analysis. Wiley, New York.Google Scholar