pproximation of stable law in Wasserstein-1 distance by Stein.s method

Let n.., let .n,1,.,.n,n be a sequence of independent random variables with ...n,i=0 and ..|.n,i|<. for each i, and let . be an .-stable distribution having characteristic function e.|.|. with ..(1,2a.Denote Sn=.n,1+.+.n,n and its distribution by .(Sn), we bound the Wasserstein-1 distance of .(Sn) and . essentially by an L1discrepancy between two kernels. More precisely, we prove the following inequality: dW(.(Sn),.).C[.i=1n.N.N.....(t,N)n.Ki(t,N)....dt+.N,n], where dW is the Wasserstein-1 distance of probability measures, ..(t,N) is the kernel of a decomposition of the fractional Laplacian ..2, Ki(t,N) is a K function (Normal Approximation by Stein.s Method (2011) Springer) with a truncation and .N,n is a small remainder.The integral term .i=1n.N.N.....(t,N)n.Ki(t,N)....dt can be interpreted as an L1discrepancy.As an application, we prove a general theorem of stable law convergence rate when .n,i are i.i.d. and the distribution falls in the normal domain of attraction of ..To test our results, we compare our convergence rates with those known in the literature for four given examples, among which the distribution in the fourth example is not in the normal domain of attraction of ..