Translator Disclaimer
December, 1974 Some Iterated Logarithm Results for Sums of Independent Two-dimensional Random Variables
Shey Shiung Sheu
Ann. Probab. 2(6): 1139-1151 (December, 1974). DOI: 10.1214/aop/1176996502

Abstract

Let $Z_i = (X_i, Y_i), i \geqq 1$, be independent two-dimensional random variables, defined on a probability triple $(\Omega, \mathscr{A}, P)$, such that $E(X_i) = E(Y_i) = E(X_i Y_i) = 0, E(X_i^2) < \infty, E(Y_i^2) < \infty$ for all $i$. The purpose of this paper is to investigate the limit points of $\{(S_n(\omega)/L(n), T_n(\omega)/M(n)), n = 1,2,\cdots\}$, where $\omega \in \Omega, S_n = \sum^n_{i=1} X_i, T_n = \sum^n_{i=1} Y_i, L(n) = \lbrack 2E(S_n^2) \log \log E(S_n^2) \rbrack^{\frac{1}{2}}, M(n) = \lbrack 2E(T_n^2) \log \log E(T_n^2) \rbrack^{\frac{1}{2}}$. The author will show the limit sets are the closed unit disk almost surely under some general conditions. An example with all limit points lying on the two axes with probability one will be constructed.

Citation

Download Citation

Shey Shiung Sheu. "Some Iterated Logarithm Results for Sums of Independent Two-dimensional Random Variables." Ann. Probab. 2 (6) 1139 - 1151, December, 1974. https://doi.org/10.1214/aop/1176996502

Information

Published: December, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0323.60035
MathSciNet: MR358949
Digital Object Identifier: 10.1214/aop/1176996502

Subjects:
Primary: 60F15
Secondary: 60G50

Rights: Copyright © 1974 Institute of Mathematical Statistics

JOURNAL ARTICLE
13 PAGES


SHARE
Vol.2 • No. 6 • December, 1974
Back to Top