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July 2005 Some results on two-sided LIL behavior
Uwe Einmahl, Deli Li
Ann. Probab. 33(4): 1601-1624 (July 2005). DOI: 10.1214/009117905000000198


Let {X,Xn;n≥1} be a sequence of i.i.d. mean-zero random variables, and let Sn=∑i=1nXi,n≥1. We establish necessary and sufficient conditions for having with probability 1, 0<lim sup n→∞|Sn|/$\sqrt{nh(n)}$<∞, where h is from a suitable subclass of the positive, nondecreasing slowly varying functions. Specializing our result to h(n)=(loglogn)p, where p>1 and to h(n)=(logn)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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Uwe Einmahl. Deli Li. "Some results on two-sided LIL behavior." Ann. Probab. 33 (4) 1601 - 1624, July 2005.


Published: July 2005
First available in Project Euclid: 1 July 2005

zbMATH: 1078.60021
MathSciNet: MR2150200
Digital Object Identifier: 10.1214/009117905000000198

Primary: 60F15 , 60G50

Keywords: Cluster sets , Hartman–Wintner LIL , Law of the iterated logarithm , sums of i.i.d. random variables , super-slow variation , two-sided LIL behavior

Rights: Copyright © 2005 Institute of Mathematical Statistics


Vol.33 • No. 4 • July 2005
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