On the other LIL for variables without finite variance

Authors:
R. P. Pakshirajan and M. Sreehari

Journal:
Theor. Probability and Math. Statist. **107** (2022), 159-171

MSC (2020):
Primary 60F15; Secondary 60G50, 60J65

DOI:
https://doi.org/10.1090/tpms/1179

Published electronically:
November 8, 2022

Full-text PDF

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Additional Information

Abstract: In this paper we give a simpler proof of Jain’s [Z. Wahrsch. Verw. Gebiete 59 (1982), no. 1, 117–138] result concerning the Other Law of the Iterated Logarithm for partial sums of a class of independent and identically distributed random variables with infinite variance but in the domain of attraction of a normal law. Jain’s result is less restrictive than ours but depends heavily on the techniques of Donsker and Varadhan in the theory of Large deviations. Our proof involves elementary properties of slowly varying functions. We assume that the distribution of random variables $X_n$ satisfies the condition that $\lim _{ x\rightarrow \infty } \frac {\log H(x)}{(\log x)^\delta } = 0$ for some $0 <\delta < 1/2$, where $H(x)=\mathsf E\left (X_1^2 I(|X_1|\le x)\right )$ is a slowly varying function. The condition above is not very restrictive.

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Additional Information

**R. P. Pakshirajan**

Affiliation:
227, 18th Main, 6th Block, Koramangala, Bengaluru -560095, India

Email:
vainatheyarajan@yahoo.in

**M. Sreehari**

Affiliation:
6-B, Vrundavan Park, New Sama Road, Vadodara 390008, India

Email:
msreehari03@yahoo.co.uk

Keywords:
Chung’s LIL,
domain of non-normal attraction,
Skorohod embedding,
slowly varying function,
Karamata representation

Received by editor(s):
June 10, 2021

Accepted for publication:
December 23, 2021

Published electronically:
November 8, 2022

Article copyright:
© Copyright 2022
Taras Shevchenko National University of Kyiv