Summary
IfX takes values in a Banach spaceB and is in the domain of attraction of a Gaussian law onB, thenX satisfies the compact law of the iterated logarithm (LIL) with respect to a regular normalizing sequence {γ n } iffX satisfies a certain integrability condition. The integrability condition is equivalent to the fact that the maximal term of the sample {‖X 1‖, ‖X 2‖,..., ‖X n‖} does not dominate the partial sums {S n}, and here we examine the precise influence of these maximal terms and its relation to the compactLIL. In particular, it is shown that if one deletes enough of the maximal terms there is always a compactLIL with non-trivial limit set.
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Supported in part by NSF Grant MCS-8219742
Work done while visiting the University of Wisconsin, Madison, with partial support by NSF Grant MCS-8219742
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Kuelbs, J., Ledoux, M. Extreme values and the law of the iterated logarithm. Probab. Th. Rel. Fields 74, 319–340 (1987). https://doi.org/10.1007/BF00699094
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DOI: https://doi.org/10.1007/BF00699094