Bounds for the accuracy of Poissonian approximations of stable laws

[摘要] 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.

[发布日期] 1996-12-13

[关键词] stable laws;Poissonian representation;convergence in variation;convergence rates;Berry-Esseen bounds;concentration functions