## The Annals of Probability

### Some Extensions of the LIL Via Self-Normalizations

#### Abstract

We study some generalizations of the LIL when self-normalizations are used. Two particular results proved are: (1) an extension of the Kolmogorov-Erdos test for partial sums of symmetric i.i.d. random variables having finite second moments; this result eliminates distinctions required when nonrandom normalizers are used and $E(X^2I(|X| > t))$ is not $O((L_2t)^{-1})$, and (2) an extension of a universal bounded LIL of Marcinkiewicz to nonsymmetric random variables. An interesting corollary of this work is a short new proof of the classical LIL avoiding truncation methods.

#### Article information

Source
Ann. Probab., Volume 19, Number 1 (1991), 380-395.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176990551

Digital Object Identifier
doi:10.1214/aop/1176990551

Mathematical Reviews number (MathSciNet)
MR1085343

Zentralblatt MATH identifier
0722.60028

JSTOR