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Moment conditions for the existence and nonexistence of optimal stopping rules for $ S\sb{n}/n\sp{1}$

Authors: Y. S. Chow and Jack Cuzick
Journal: Proc. Amer. Math. Soc. 75 (1979), 300-307
MSC: Primary 60G40; Secondary 62L15
MathSciNet review: 532155
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Abstract: Sufficient conditions of moment type are given for the existence of optimal stopping rules for $ {S_n}/n$. These results make use of a generalization of a result of Dini. In general it is shown that necessary conditions of moment type do not exist. In particular, we exhibit a symmetric random variable with $ E{X^ + }{({\log ^ + }X)^{1 + \varepsilon }} = \infty $ for which an optimal rule exists. However under regularity conditions, we show that the sufficient condition given is very nearly necessary. Some results on exponentials of regularly varying functions are of independent interest.

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  • [M] Michael J. Klass, Properties of optimal extended-valued stopping rules for 𝑆_{𝑛}/𝑛, Ann. Probability 1 (1973), 719–757. MR 0426140
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Keywords: Optimal stopping, Dini's theorem, regular variation
Article copyright: © Copyright 1979 American Mathematical Society

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