Rate of Convergence to a Stable Law

We consider the normalized sum of n independent identically distributed random variables Xi; i = 1; : : : ; n with the common probability distribution function F (x) and probability density q(y) = F (y). We suppose that the distribution function Fn(y) of the normalized sum converges as n ! 1 to a stable distribution or stable law. The only stable distributions are the normal distributions and the Levy distributions. We shall study the rate at which the density qn(y)= F 0 n(y) converges to the density q(y) of a stable law, which has been studied extensively for general F (x). The book of Christoph and Wolf [1] and the recent paper by Juozulynas and Paulauskas [2] contain results most closely related to those in this paper, and give relevant references. See also Petrov [3]. By assuming that F (x) has a probability density p(x) with a very speci c tail behavior, we study how the rate of convergence depends on the parameters describing the tail. We compare and contrast our results with those of [1] and [2] in sections 2 and 3. Levy distributions have algebraically decaying densities. Therefore they often arise in nature describing the e ects due to randomly placed sources producing slowly decaying elds. For example, Chandrasekhar [4] obtained the Holtsmark distribution, with tails decaying as the 5=2 power, for the magnitude of the gravitational force due to a random