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

Let X i , i = 1, 2,…, be i.i.d. symmetric random variables in the domain of attraction of a symmetric stable distribution G α with 0 < α < 2. Let Y i , i = 1, 2, …, be i.i.d. symmetric stable random variables with the common distribution G α . It is known that under certain conditions the sequences { X i } and { Y i } can be reconstructed on a new probability space without changing the distribution of each such that \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_{i = 1}^n {(X_i - Y_i) = o(n^{1/\gamma})} $\end{document} a.s. as n → ∞, where α ≦ γ < 2 (see Stout [10]). We will give a second approximation by partial sums of i.i.d. stable (with characteristic exponent α*, α < α* ≦ 2) random variables U i , i = 1, 2,…, n , and we will obtain strong upperbounds for the differences \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_{i = 1}^n {(X_i - Y_i - U_i)} $\end{document} .