The Annals of Probability

Strong Limit Theorems for Certain Arrays of Random Variables

R. J. Tomkins

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Abstract

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.

Article information

Source
Ann. Probab., Volume 4, Number 3 (1976), 444-452.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996092

Digital Object Identifier
doi:10.1214/aop/1176996092

Mathematical Reviews number (MathSciNet)
MR402885

Zentralblatt MATH identifier
0339.60020

JSTOR
links.jstor.org

Subjects
Primary: 60F15: Strong theorems
Secondary: 60G10: Stationary processes 60G99: None of the above, but in this section

Keywords
Law of the iterated logarithm strong law of large numbers multiplicative systems generalized Gaussian random variables martingale difference sequence

Citation

Tomkins, R. J. Strong Limit Theorems for Certain Arrays of Random Variables. Ann. Probab. 4 (1976), no. 3, 444--452. doi:10.1214/aop/1176996092. https://projecteuclid.org/euclid.aop/1176996092


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