A NONCLASSICAL LAW OF THE ITERATED LOGARITHM FOR I.I.D. SQUARE INTEGRABLE RANDOM VARIABLES

Abstract

We define a sequence of positive constants {b(n), n ≥ 1} which is monotonically approaching infinity and which is not asymptotically equivalent to log log n but is such that lim sup _{n → ∞} almost certainly for every sequence of i.i.d. random variables {X _n , n ≥ 1} with EX ₁ = 0 and EX ² ₁ = 1.

ACKNOWLEDGMENTS

This joint research between Oleg Klesov and Andrew Rosalsky has been funded in part by a grant from the National Research Council under the auspices of the Collaboration in Basic Science and Engineering Program. The authors are grateful to the National Research Council for their support. The contents of this publication do not necessarily reflect the views or policies of the National Research Council. The authors also wish to thank Professors Harry Kesten, Josef Steinebach, and N.M. Zinchenko for their interest in our work and for some helpful and interesting comments.