Limit distribution for sums of the reciprocal of a positive power of independent random variables

Abstract

Let X₁, X₂,…. be a sequence of independent random variables of fn the probability density function of Xn for n = 1,2, ….. In a paper entitled “Limit distributions for sums of reciprocals of independent random variable” J.M. Shapiro investigates the limit distribution function of [ดูสมการในบทคัดย่อ] for some suitably chosen constants An(r) and Bn(r) where r is a positive real number greater than ½. In this work, we deal with the problem in the case where r ≤ ½. Conditions are found so that for some suitably chosen constants An(r) and Bn(r) , n = 1,2,….. , the distribution functions of [ดูสมการในบทคัดย่อ] where r ≤ ½. converges to this normal law, and a possible choice of those constants An(r) and Bn(r) is given.