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# Moments and Error Rates of Two-Sided Stopping Rules

The Annals of Probability
Vol. 10, No. 4 (Nov., 1982), pp. 935-941
Stable URL: http://www.jstor.org/stable/2243549
Page Count: 7
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## Abstract

For X1, X2,⋯ i.i.d., EX1 = μ ≠ 0, Sn = X1 + ⋯ + Xn, 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 → ∞ 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 X1.

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