Strong Approximations for Partial Sums of Random Variables Attracted to a Stable Law

Abstract

Let Xi, i = 1, 2,…, be i.i.d. symmetric random variables in the domain of attraction of a symmetric stable distribution Gα with 0 < α < 2. Let Yi, i = 1, 2, …, be i.i.d. symmetric stable random variables with the common distribution Gα. It is known that under certain conditions the sequences {Xi} and {Yi} can be reconstructed on a new probability space without changing the distribution of each such that a.s. as n ∞, where α ≦ γ < 2 (see Stout [10]). We will give a second approximation by partial sums of i.i.d. stable (with characteristic exponent α*, α < α* ≦ 2) random variables Ui, i = 1, 2,…, n, and we will obtain strong upperbounds for the differences .