Bounds for the accuracy of Poissonian approximations of stable laws

Stable law Gz admit a well-known series representation of the type Γ1, Γ2, … are the successive times of jumps of a standard Poisson process, and X1, X2, …, denote i.i.d. random variables, independent of Γ1, Γ2, … We investigate the rate of approximation of Gα by distributions of partial sums Sn = ∑nj = 1 Γ−1αjXj, and we get (asymptotically) optimal bounds for the variation of Gα − L(Sn). 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 Sn are also proved.