Distribution Convegence Rate of Continvous Random Variables Functions

Suppose continuous random variables ξ_1n，…ξ_rn are mutually independent and distributiuon convegence to ξ₁，…,ξ_r and η_n=φ(ξ_1n，…ξ_rn) are continuous random variables,F_ξin(x),F_ηn(x),F_ξi(x),F_η(x) are the corresponding distribution functions. Under some conditions, we prove that sup_x|F_ξn(x)-F_*n(x)|≤d f c \sqrtn if sup_x|F_ξin(x)-F_ξi(x)|≤\fracL\sqrtn,where c and L are constants, i=1,… r. Especially whenφ(x₁，…，x_r)＝x₁²＋…＋x or φ(x₁，…，x_r)＝x₁²＋…＋x_k²/x₁²＋…＋x_r² the result above is correct.