Bounds for the accuracy of Poissonian approximations of stable laws
Stable laws \(G_{\alpha }\) admit a well-known series representation of the type \[ G_{\alpha } = {\mathcal L} \;\Big (\sum ^{\infty }_{j=1} \Gamma _1^{-1/\alpha } X_j\Big ), \quad 0 < \alpha <2, \] where \(\Gamma _1, \Gamma _2,\ldots \) are the successive times of jumps of a standard Poisson process and \(X_1,X_2,\ldots \), denote i.i.d.\ random variables, independent of \(\Gamma _1,\Gamma _2,\ldots .\) We investigate the rate of approximation of \(G_{\alpha }\) by distributions of partial sums \(S_n = \sum ^n_{j=1} \Gamma ^{-1/\alpha }_j X_j\), and we obtain (asymptotically) optimal bounds for the variation of \(G_{\alpha } - {\mathcal L} \;(S_n)\). These results complement and improve results of A. Janicki and P. Kokoszka, and M. Ledoux and V. Paulauskas. Bounds for the concentration function of \(S_n\) are also proved.
scientific article; zbMATH DE number 1134788