Moment conditions for the existence and nonexistence of optimal stopping rules for $S_{n}/n^{1}$
HTML articles powered by AMS MathViewer
- by Y. S. Chow and Jack Cuzick PDF
- Proc. Amer. Math. Soc. 75 (1979), 300-307 Request permission
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
- 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.Β 159β177. MR 0383516
- Michael J. Klass, Properties of optimal extended-valued stopping rules for $S_{n}/n$, Ann. Probability 1 (1973), 719β757. MR 426140, DOI 10.1214/aop/1176996843 β, (1975), A survey of the ${S_n}/n$ problem, Statistical Inference and Related Topics, vol. 2, Academic Press, New York, pp. 179-201. Knopp, (1948), Theory and application of infinite series, Hafner, New York (reprinted 1971).
- Eugene Seneta, Regularly varying functions, Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin-New York, 1976. MR 0453936
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 75 (1979), 300-307
- MSC: Primary 60G40; Secondary 62L15
- DOI: https://doi.org/10.1090/S0002-9939-1979-0532155-1
- MathSciNet review: 532155