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April, 1975 Iterated Logarithm Results for Weighted Averages of Martingale Difference Sequences
R. J. Tomkins
Ann. Probab. 3(2): 307-314 (April, 1975). DOI: 10.1214/aop/1176996401


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.


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R. J. Tomkins. "Iterated Logarithm Results for Weighted Averages of Martingale Difference Sequences." Ann. Probab. 3 (2) 307 - 314, April, 1975.


Published: April, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0302.60019
MathSciNet: MR372972
Digital Object Identifier: 10.1214/aop/1176996401

Primary: 60F15
Secondary: 26A45 , 60G45 , 60G50

Keywords: function of bounded variation , Independent random variables , Law of the iterated logarithm , Martingales , Strong law of large numbers

Rights: Copyright © 1975 Institute of Mathematical Statistics


Vol.3 • No. 2 • April, 1975
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