## The Annals of Probability

- Ann. Probab.
- Volume 20, Number 2 (1992), 660-674.

### The Law of the Iterated Logarithm for Independent Random Variables with Multidimensional Indices

Deli Li, M. Bhaskara Rao, and Xiangchen Wang

#### Abstract

Let $X_{\bar n}, \bar{n} \in \mathbb{N}^d$, be a field of independent real random variables, where $\mathbb{N}^d$ is the $d$-dimensional lattice. In this paper, the law of the iterated logarithm is established for such a field of random variables. Theorem 1 brings into focus a connection between a certain strong law of large numbers and the law of the iterated logarithm. A general technique is developed by which one can derive the strong law of large numbers and the law of the iterated logarithm, exploiting the convergence rates in the weak law of large numbers in Theorem 2. In Theorem 3, we use Gaussian randomization techniques to obtain the law of the iterated logarithm which generalizes Wittmann's result.

#### Article information

**Source**

Ann. Probab., Volume 20, Number 2 (1992), 660-674.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989798

**Digital Object Identifier**

doi:10.1214/aop/1176989798

**Mathematical Reviews number (MathSciNet)**

MR1159566

**Zentralblatt MATH identifier**

0753.60029

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60G60: Random fields 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60G50: Sums of independent random variables; random walks

**Keywords**

Gaussian randomization law of the iterated logarithm multidimensional indices rates of convergence strong law of large numbers type 2 Banach spaces weak law of large numbers

#### Citation

Li, Deli; Rao, M. Bhaskara; Wang, Xiangchen. The Law of the Iterated Logarithm for Independent Random Variables with Multidimensional Indices. Ann. Probab. 20 (1992), no. 2, 660--674. doi:10.1214/aop/1176989798. https://projecteuclid.org/euclid.aop/1176989798