On asymptotic independence of the partial sums of positive and negative parts of independent random variables

Extract

The aim of this study is an investigation of the joint limiting distribution of the sequence of partial sums of the positive parts and negative parts of a sequence of independent identically distributed random variables. In particular, let {X_n} be a sequence of independent identically distributed random variables with common distribution function F, assume F is in the domain of attraction of a stable distribution with characteristic exponent α, 0 < α ≦ 2, and let {B_n} be normalizing coefficients for F. Let us denote X_n⁺ = X_nI_{[Xn > 0]} and X_n⁻ = − X_nI_[Xn<0], so that X_n = X_n⁺ - X_n⁻. Let F⁺ and F⁻ denote the distribution functions of X₁⁺ and X₁⁻ respectively, and let S_n⁽⁺⁾ = X₁⁺ + · · · + X_n⁺, S_n^(-) = X₁⁻ + · · · + X_n⁻. The problem considered here is to find under what conditions there exist sequences of real numbers {a_n} and {b_n} such that the joint distribution of (B_n^-1S_n⁽⁺⁾ + a_n, B_n^-1S_n^(-) + b_n) converges to that of two independent random variables (U, V). As might be expected, different results are obtained when α < 2 and when α = 2. When α < 2, there always exist such sequences so that the above is true, and in this case both U and V are stable with characteristic exponent a, or one has such a stable distribution and the other is constant. When α = 2, and if 0 < ∫ x²dF(x) < ∞, then there always exist such sequences such that the above convergence takes place; both U and V are normal (possibly one is a constant), and if neither is a constant, then U and V are not independent. If α = 2 and ∫ x²dF(x) = ∞, then at least one of F⁺, F⁻ is in the domain of partial attraction of the normal distribution, and a modified statement on the independence of U and V holds. Various specialized results are obtained for α = 2.