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Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)



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
Published electronically: November 8, 2022
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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

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

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