On the Law of the Iterated Logarithm

.-The law of the iterated logarithm provides a family of bounds all of the same order such that with probability one only finitely many partial sums of a sequence of independent and identically distributed random variables exceed some members of the family, while for others infinitely many do so. In the former case, the total number of such excesses has therefore a proper probability distribution, and it is shown here that whenever the law applies this distribution possesses no moments of positive order. This result further elucidates the celebrated precision of this law of probability concerning fluctuations of sums of random variables.