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Some Results on Two-Sided LIL Behavior

Uwe Einmahl and Deli Li
The Annals of Probability
Vol. 33, No. 4 (Jul., 2005), pp. 1601-1624
Stable URL: http://www.jstor.org/stable/3481740
Page Count: 24
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Some Results on Two-Sided LIL Behavior
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Abstract

Let {X,Xn;n≥ 1} be a sequence of i.i.d. mean-zero random variables, and let Sn=Σ i=1 nXi,n≥ 1. We establish necessary and sufficient conditions for having with probability $1,0<\text{lim sup}_{n\rightarrow \infty}|S_{n}|/\sqrt{nh(n)}<\infty $, where h is from a suitable subclass of the positive, nondecreasing slowly varying functions. Specializing our result to h(n)=( log log n)p, where p > 1 and to h(n)=( log n)r, r > 0, we obtain analogues of the Hartman-Wintner LIL in the infinite variance case. Our proof is based on a general result dealing with LIL behavior of the normalized sums {Sn/cn;n≥ 1}, where cn is a sufficiently regular normalizing sequence.

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