## The Annals of Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60G50: Sums of independent random variables; random walks

**Keywords**

Law of the Iterated logarithm limit set sums of independent random variables

#### Citation

Sheu, Shey Shiung. Some Iterated Logarithm Results for Sums of Independent Two-dimensional Random Variables. Ann. Probab. 2 (1974), no. 6, 1139--1151. doi:10.1214/aop/1176996502. https://projecteuclid.org/euclid.aop/1176996502