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June, 1976 Strong Limit Theorems for Certain Arrays of Random Variables
R. J. Tomkins
Ann. Probab. 4(3): 444-452 (June, 1976). DOI: 10.1214/aop/1176996092


A lemma concerning real sequences is proved and applied to sequences of random variables $(\mathrm{rv}) X_1, X_2\cdots$ to determine conditions under which $\lim\sup_{n\rightarrow\infty} b_n^{-1} \sum^n_{m=1} f(m/n)X_m < \infty$ a.s. for all $f$ in a particular collection of absolutely continuous functions and for nondecreasing positive real sequences $\{b_n\}$. Theorems in the case $b_n = (2n \log \log n)^\frac{1}{2}$ are proved for generalized Gaussian rv, for equinormed multiplicative systems and for certain martingale difference sequences.


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R. J. Tomkins. "Strong Limit Theorems for Certain Arrays of Random Variables." Ann. Probab. 4 (3) 444 - 452, June, 1976.


Published: June, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0339.60020
MathSciNet: MR402885
Digital Object Identifier: 10.1214/aop/1176996092

Primary: 60F15
Secondary: 60G10 , 60G99

Keywords: generalized Gaussian random variables , Law of the iterated logarithm , Martingale difference sequence , multiplicative systems , Strong law of large numbers

Rights: Copyright © 1976 Institute of Mathematical Statistics

Vol.4 • No. 3 • June, 1976
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