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August, 1984 The Finite Mean LIL Bounds are Sharp
Michael J. Klass
Ann. Probab. 12(3): 907-911 (August, 1984). DOI: 10.1214/aop/1176993240


Let $X, X_1, X_2, \cdots$ be i.i.d. nonconstant mean zero random variables and put $S_n = X_1 + \cdots + X_n$. Let $K(y) > 0$ satisfy $yE\{|X/K(y)|^2 \wedge |X/K(y)|\} = 1$ (for $y > 0$). Then let $a_n = (\log \log n)K(n/\log \log n)$ and $L = \lim \sup_{n\rightarrow\infty}S_n/a_n.$ It is known that $L$ is finite iff $P(X_n > a_n \text{i.o.}) = 0$. When $L < \infty$, it is also known that $1 \leq L \leq 1.5$ and that it is possible for $L$ to equal one. In this paper we construct an example for which $L = 1.5$.


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Michael J. Klass. "The Finite Mean LIL Bounds are Sharp." Ann. Probab. 12 (3) 907 - 911, August, 1984.


Published: August, 1984
First available in Project Euclid: 19 April 2007

zbMATH: 0546.60029
MathSciNet: MR744246
Digital Object Identifier: 10.1214/aop/1176993240

Primary: 60F15
Secondary: 60G50

Keywords: Generalized or universal law of the iterated logarithm , sums of i.i.d. random variables

Rights: Copyright © 1984 Institute of Mathematical Statistics

Vol.12 • No. 3 • August, 1984
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