, Volume 20, Issue 4, pp 901-915
Date: 12 May 2007

A Generalization of Strassen’s Functional LIL

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Abstract

Let X 1,X 2,… be a sequence of i.i.d. mean zero random variables and let S n denote the sum of the first n random variables. We show that whenever we have with probability one, lim sup  n→∞|S n |/c n =α 0<∞ for a regular normalizing sequence {c n }, the corresponding normalized partial sum process sequence is relatively compact in C[0,1] with canonical cluster set. Combining this result with some LIL type results in the infinite variance case, we obtain Strassen type results in this setting.

Dedicated to Sándor Csörgő on the occasion of his sixtieth birthday
Research partially supported by an FWO-Vlaanderen Grant.