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A Compact Law of the Iterated Logarithm for Random Vectors in the Generalized Domain of Attraction of the Multivariate Gaussian Law

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Abstract

For a sequence of independent identically distributed random vectors, we prove that the limiting cluster set of the appropriately operator normed partial sums is, with probability one, the closed unit euclidean ball. The result is proved under the hypotheses that the law of the random vectors belongs to the Generalized Domain of Attraction of the multivariate Gaussian law and satisfy a mild integrability condition. The two conditions together are still weaker than finite second normed moment and are necessary and sufficient.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Saginaw Valley State University, University Center, Michigan, 48710

    Steven J. Sepanski

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  1. Steven J. Sepanski
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Sepanski, S.J. A Compact Law of the Iterated Logarithm for Random Vectors in the Generalized Domain of Attraction of the Multivariate Gaussian Law. Journal of Theoretical Probability 12, 757–778 (1999). https://doi.org/10.1023/A:1021632016642

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