Domains of Attraction for Reciprocals of Powers of Random Variables

Abstract

The problem of finding to which domain of attraction the reciprocal of a positive power of a random variable belongs is considered. Under general conditions,, it is shown that this reciprocal always belongs to the domain of attraction of some stable law. The characteristic exponent' of this stable law is given in terms of the power to which the original random variable is raised. Special attention is given to the case where the positive power is one. In this case; the problem is that of finding the limit distribution of normed sums of reciprocals of a sequence of independent identically distributed random variables. Under general conditions, the limit distribution is shown to be the Cauchy law, and the exact norming constants are found.