Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [Y] Y. S. Chow and K. K. Lan, Optimal stopping rules for 𝑋_{𝑛}/𝑛 and 𝑆_{𝑛}/𝑛, Statistical inference and related topics (Proc. Summer Res. Inst. Statist. Inference for Stochastic Processes, Indiana Univ., Bloomington, Ind., 1974, Vol. 2; dedicated to Z. W. Birnbaum), Academic Press, New York, 1975, pp. 159–177. MR 0383516
  • [M] Michael J. Klass, Properties of optimal extended-valued stopping rules for 𝑆_{𝑛}/𝑛, Ann. Probability 1 (1973), 719–757. MR 0426140
  • 1. -, (1975), A survey of the $ {S_n}/n$ problem, Statistical Inference and Related Topics, vol. 2, Academic Press, New York, pp. 179-201.
  • [K] Knopp, (1948), Theory and application of infinite series, Hafner, New York (reprinted 1971).
  • [E] Eugene Seneta, Regularly varying functions, Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin-New York, 1976. MR 0453936

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60G40, 62L15

Retrieve articles in all journals with MSC: 60G40, 62L15

Additional Information

Keywords: Optimal stopping, Dini's theorem, regular variation
Article copyright: © Copyright 1979 American Mathematical Society