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November, 1982 Moments and Error Rates of Two-Sided Stopping Rules
Adam T. Martinsek
Ann. Probab. 10(4): 935-941 (November, 1982). DOI: 10.1214/aop/1176993715


For $X_1, X_2,\cdots$ i.i.d., $EX_1 = \mu \neq 0, S_n = X_1 + \cdots + X_n$, the asymptotic behavior of moments and error rates of the two-sided stopping rules $\inf \{n \geq 1: |S_n| > cn^\alpha\}, c > 0, 0 \leq \alpha < 1$, is considered. Convergence of (normalized) moments of all orders as $c \rightarrow \infty$ is obtained, without the higher moment assumptions needed in the one-sided case of extended renewal theory (Gut, 1974), and in a more general setting than just the i.i.d. case. Necessary and sufficient conditions are given for convergence of series involving the error rates, in terms of the moments of $X_1$.


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Adam T. Martinsek. "Moments and Error Rates of Two-Sided Stopping Rules." Ann. Probab. 10 (4) 935 - 941, November, 1982.


Published: November, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0496.60040
MathSciNet: MR672294
Digital Object Identifier: 10.1214/aop/1176993715

Primary: 60G40
Secondary: 60G50 , 62L10

Keywords: delayed sums , error rates of sequential tests , moment convergence , Stopping rules , uniform integrability

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 4 • November, 1982
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