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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Stop rule inequalities for uniformly bounded sequences of random variables
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by Theodore P. Hill and Robert P. Kertz PDF
Trans. Amer. Math. Soc. 278 (1983), 197-207 Request permission

Abstract:

If ${X_{0}},{X_1},\ldots$ is an arbitrarily-dependent sequence of random variables taking values in $[0,1]$ and if $V({X_0},{X_1},\ldots )$ is the supremum, over stop rules $t$, of $E{X_t}$, then the set of ordered pairs $\{ (x,y):x = V({X_0},{X_1},\ldots ,{X_n})$ and $y = E({\max _{j \leqslant n}}{X_j})$ for some ${X_0},\ldots ,{X_n}\}$ is precisely the set \[ {C_n} = \{ (x,y):x \leqslant y \leqslant x ( {1 + n (1 - {x^{1/n}})} );0 \leqslant x \leqslant 1\} ;\] and the set of ordered pairs $\{ (x,y):x = V({X_{0}},{X_1},\ldots )$ and $y = E({\sup _n}\;{X_n})$ for some ${X_0},{X_1},\ldots \}$ is precisely the set \[ C = \bigcup \limits _{n = 1}^\infty {{C_n}} .\] As a special case, if ${X_0},{X_1},\ldots$ is a martingale with $E{X_0} = x$, then $E({\max _{j \leqslant n}} X) \leqslant x + nx(1 - {x^{1 / n}})$ and $E({\sup _n}\;{X_n}) \leqslant x - x\ln \;x$, and both inequalities are sharp.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 278 (1983), 197-207
  • MSC: Primary 60G40; Secondary 60G42, 62L15
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0697070-7
  • MathSciNet review: 697070