## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 1 (1981), 1-48.

### General One-Sided Laws of the Iterated Logarithm

#### Abstract

Let $\{X_i\}$ be a sequence of independent, identically distributed nondegenerate random variables and $S_n = \sum^n_{i = 1}X_i$. We consider the question for various centering sequences $\{\alpha_n\}$: when is it possible to find a positive, monotone sequence $\{\beta_n\}$ such that $\lim \sup \beta^{-1}_n (S_n - \alpha_n) = c$ a.s., $c$ a finite nonzero constant? If $\alpha_n = \operatorname{med} S_n$, we obtain a necessary and sufficient condition for this. An important corollary is a one-sided version of the Hartman-Wintner law of the iterated logarithm: if $E(X^+)^2 < \infty$, then it is always possible to find such a norming sequence. Explicit norming sequences are given which are easy to obtain. Necessary and sufficient conditions are also given for being able to find a norming sequence $\{\beta_n\}$ for the two-sided problem $(\lim \sup \beta^{-1}_n |S_n - \alpha_n| = c$ a.s.) when $\alpha_n = ES_n$ and $\alpha_n = 0$. The two-sided problem with $\alpha_n = \operatorname{med} S_n$ was solved by Kesten. The one-sided problem remains open for $\alpha_n = ES_n$ and $\alpha_n = 0$. Examples are given which illustrate the advantage of considering different centering sequences. A one-sided version of Strassen's converse to the law of the iterated logarithm is also given: if $\lim \sup S_n/ \sqrt{2n \log \log n} = 1$ a.s., then $EX = 0, EX^2 = 1$.

#### Article information

**Source**

Ann. Probab., Volume 9, Number 1 (1981), 1-48.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994508

**Digital Object Identifier**

doi:10.1214/aop/1176994508

**Mathematical Reviews number (MathSciNet)**

MR606797

**Zentralblatt MATH identifier**

0462.60030

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

**Keywords**

Law of the iterated logarithm domains of attraction exponential bounds truncation one-sided large values for $S_n$

#### Citation

Pruitt, William E. General One-Sided Laws of the Iterated Logarithm. Ann. Probab. 9 (1981), no. 1, 1--48. doi:10.1214/aop/1176994508. https://projecteuclid.org/euclid.aop/1176994508