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Journal Article

# The Finite Mean LIL Bounds are Sharp

Michael J. Klass
The Annals of Probability
Vol. 12, No. 3 (Aug., 1984), pp. 907-911
Stable URL: http://www.jstor.org/stable/2243339
Page Count: 5

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Topics: Random variables, Logarithms

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Let X, X1, X2, ⋯ be i.i.d. nonconstant mean zero random variables and put Sn = X1 + ⋯ + Xn. Let $K(y) > 0$ satisfy $yE\{|X/K(y)|^2 \wedge |X/K(y)|\} = 1$ (for $y > 0$). Then let an = (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 ≤ L ≤ 1.5 and that it is possible for L to equal one. In this paper we construct an example for which L = 1.5.