## Annals of Probability

- Ann. Probab.
- Volume 5, Number 6 (1977), 861-874.

### Iterated Logarithm Laws for Asymmetric Random Variables Barely with or Without Finite Mean

Michael Klass and Henry Teicher

#### Abstract

One-sided iterated logarithm laws of the form $\lim \sup (1/b_n) \sum^n_1 X_i = 1$, a.s. and $\lim \sup (1/b_n) \sum^n_1 X_i = -1$, a.s. are obtained for asymmetric independent and identically distributed random variables, the first when these have a vanishing but barely finite mean, the second when $E|X|$ is barely infinite. In both cases, $\lim \inf (1/b_n) \sum^n_1 X_i = -\infty$, a.s. The constants $b_n/n$ are slowly varying, decreasing to zero in the first case and increasing to infinity in the second. Although defined via the distribution of $|X|, b_n$ represents the order of magnitude of $E|\sum^n_1 X_i|$ when this is finite. Corresponding weak laws of large numbers are established and related to Feller's notion of "unfavorable fair games" and in the process a theorem playing the same role for the weak law as Feller's generalization of the strong law is proved.

#### Article information

**Source**

Ann. Probab., Volume 5, Number 6 (1977), 861-874.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995656

**Digital Object Identifier**

doi:10.1214/aop/1176995656

**Mathematical Reviews number (MathSciNet)**

MR445588

**Zentralblatt MATH identifier**

0372.60042

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

**Keywords**

Law of the iterated logarithm weak law of large numbers slowly varying unfavorable fair game

#### Citation

Klass, Michael; Teicher, Henry. Iterated Logarithm Laws for Asymmetric Random Variables Barely with or Without Finite Mean. Ann. Probab. 5 (1977), no. 6, 861--874. doi:10.1214/aop/1176995656. https://projecteuclid.org/euclid.aop/1176995656