The Annals of Probability

The Finite Mean LIL Bounds are Sharp

Michael J. Klass

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Abstract

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 $L = 1.5$.

Article information

Source
Ann. Probab., Volume 12, Number 3 (1984), 907-911.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993240

Digital Object Identifier
doi:10.1214/aop/1176993240

Mathematical Reviews number (MathSciNet)
MR744246

Zentralblatt MATH identifier
0546.60029

JSTOR
links.jstor.org

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

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

Citation

Klass, Michael J. The Finite Mean LIL Bounds are Sharp. Ann. Probab. 12 (1984), no. 3, 907--911. doi:10.1214/aop/1176993240. https://projecteuclid.org/euclid.aop/1176993240


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