The law of the iterated logarithm on subsequences-characterizations

Abstract

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:.