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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.

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Authors and Affiliations

  1. Departement Wiskunde, Vrije Universiteit Brussel, Pleinlaan 2, 1050, Brussel, Belgium

    Uwe Einmahl

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  1. Uwe Einmahl
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Correspondence to Uwe Einmahl.

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Dedicated to Sándor Csörgő on the occasion of his sixtieth birthday

Research partially supported by an FWO-Vlaanderen Grant.

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Einmahl, U. A Generalization of Strassen’s Functional LIL. J Theor Probab 20, 901–915 (2007). https://doi.org/10.1007/s10959-007-0091-0

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  • DOI: https://doi.org/10.1007/s10959-007-0091-0

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