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The law of the iterated logarithm on subsequences-characterizations

Published online by Cambridge University Press:  22 January 2016

Michel Weber
Michel Weber*
Affiliation:
Université Louis Pasteur, Uer de Mathématiques et Informatique, 7, rue René Descartes, 67084 Strasbourg Cedex, France
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Let be any increasing sequence of integers and M> 1; we connect to them in a very simply way, an increasing unbounded function φ: → R+ . Let also X1, X2 , · · · be a sequence of i.i.d. random vectors with value in euclidian space Rm . We prove that the cluster set of the sequence almost surely coincides with the unit ball of Rm , if, and only if, the covariance matrix of X1 is the identity matrix of Rm and EX1 is the zero vector of Rm . We define a functional A on the set of increasing sequences of integers as follows:

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Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 118 , June 1990 , pp. 65 - 97
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1990

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