Bounds for the accuracy of Poissonian approximations of stable laws

Bentkus V, Götze F, Paulauskas V. Bounds for the accuracy of Poissonian approximations of stable laws. STOCHASTIC PROCESSES AND THEIR APPLICATIONS. 1996;65(1):55-68.Stable laws G(alpha) admit a well-known series representation of the type [GRAPHICS] where Gamma(1), Gamma(2), ... are the successive times of jumps of a standard Poisson process, and X(1), X(2), ..., denote i.i.d. random variables, independent of Gamma(1), Gamma(2), ... We investigate the rate of approximation of G(alpha) by distributions of partial sums S-n = Sigma(j=1)(n) Gamma(j)(-1/alpha)X(j), and we get (asymptotically) optimal bounds for the variation of G(alpha)-L(S-n). The results obtained complement and improve the results of A. Janicki and P. Kokoszka, and M. Ledoux and V. Paulauskas. Bounds for the concentration function of S-n are also proved