## The Annals of Probability

- Ann. Probab.
- Volume 6, Number 1 (1978), 162-168.

### On the Law of the Iterated Logarithm

#### Abstract

Let $X_1, X_2,\cdots$ be a sequence of independent random variables, each with zero mean and finite variance. Define $S_n = \sum^n_{i=1} X_i, s_n^2 = E(S_n^2), t_n^2 = 2 \log \log s_n^2$ and $\Lambda = \lim \sup_{n\rightarrow\infty} S_n/(s_n t_n)$. Suppose that $|X_n| \leqq c_n s_n$ a.s. for all $n$ and some real sequence $\{c_n\}$ and that $s_n \rightarrow \infty$, and let $\nu = \lim \sup_{n\rightarrow\infty} t_nc_n$. By Kolmogorov's law of the iterated logarithm, $\Lambda = 1$ if $\nu = 0$. Egorov proved that $0 \leqq \Lambda < \infty$, in every case. In this paper it will be shown that, if $\nu < \infty$, then $0 < \Lambda \leqq 1 + \sum^\infty_{j=3} \nu^{j-2}/j!$. A similar result for certain classes of unbounded random variables will also be presented.

#### Article information

**Source**

Ann. Probab., Volume 6, Number 1 (1978), 162-168.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995622

**Digital Object Identifier**

doi:10.1214/aop/1176995622

**Mathematical Reviews number (MathSciNet)**

MR501300

**Zentralblatt MATH identifier**

0376.60032

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

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

**Keywords**

Law of the iterated logarithm exponential bounds independent random variables

#### Citation

Tomkins, R. J. On the Law of the Iterated Logarithm. Ann. Probab. 6 (1978), no. 1, 162--168. doi:10.1214/aop/1176995622. https://projecteuclid.org/euclid.aop/1176995622