The Annals of Probability

Iterated Logarithm Results for Weighted Averages of Martingale Difference Sequences

R. J. Tomkins

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Let $(X_n, \mathscr{F}_n, n \geqq 1)$ be a martingale difference sequence with $E(X_n^2 \mid \mathscr{F}_{n-1}) = 1$ a.s. This paper presents iterated logarithm results involving $\lim \sup_{n\rightarrow\infty} \sum^n_{m=1} f(m/n)X_m/(2n \log \log n)^{\frac{1}{2}}$, where $f$ is a continuous function on [0, 1]. For example, it is shown that the above limit superior equals the $L_2$-norm of $f$ if the $X_n$'s are uniformly bounded and $f$ is a power series with radius in excess of one. These results generalize (and correct the proof of) a previous theorem due to the author. A generalization of the strong law of large numbers is also established.

Article information

Ann. Probab., Volume 3, Number 2 (1975), 307-314.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60F15: Strong theorems
Secondary: 60G45 26A45: Functions of bounded variation, generalizations 60G50: Sums of independent random variables; random walks

Law of the iterated logarithm martingales independent random variables function of bounded variation strong law of large numbers


Tomkins, R. J. Iterated Logarithm Results for Weighted Averages of Martingale Difference Sequences. Ann. Probab. 3 (1975), no. 2, 307--314. doi:10.1214/aop/1176996401.

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