The Annals of Probability

Some results on two-sided LIL behavior

Uwe Einmahl and Deli Li

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Abstract

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.

Article information

Source
Ann. Probab. Volume 33, Number 4 (2005), 1601-1624.

Dates
First available in Project Euclid: 1 July 2005

Permanent link to this document
http://projecteuclid.org/euclid.aop/1120224592

Digital Object Identifier
doi:10.1214/009117905000000198

Mathematical Reviews number (MathSciNet)
MR2150200

Zentralblatt MATH identifier
1078.60021

Subjects
Primary: 60F15: Strong theorems 60G50: Sums of independent random variables; random walks

Keywords
Hartman–Wintner LIL law of the iterated logarithm super-slow variation two-sided LIL behavior sums of i.i.d. random variables cluster sets

Citation

Einmahl, Uwe; Li, Deli. Some results on two-sided LIL behavior. Ann. Probab. 33 (2005), no. 4, 1601--1624. doi:10.1214/009117905000000198. http://projecteuclid.org/euclid.aop/1120224592.


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