## The Annals of Probability

### Some Iterated Logarithm Results for Sums of Independent Two-dimensional Random Variables

Shey Shiung Sheu

#### 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.

#### Article information

Source
Ann. Probab., Volume 2, Number 6 (1974), 1139-1151.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176996502

Digital Object Identifier
doi:10.1214/aop/1176996502

Mathematical Reviews number (MathSciNet)
MR358949

Zentralblatt MATH identifier
0323.60035

JSTOR