一类随机变量函数的分布收敛速度

Distribution Convegence Rate of Continvous Random Variables Functions

摘要: 设连续型随机变量ξ_1n，…ξ_rn相互独立且分布收敛于ξ₁，…,ξ_r,y＝φ(x₁，…，x_r)是R_r，到R^１的连续函数, η_n=φ(ξ_1n，…ξ_rn)和η=φ(ξ₁，…,ξ_r)都为连续型随机变量，F_ξin(x),F_ηn(x),F_ξi(x),F_η(x)为相应的分布函数本文讨论并证明了:如果sup_x|F_ξin(x)-F_ξi(x)|≤\fracL\sqrtn,i=1,...,r,L为常数．那么在一定条件下；存在常数c使\sup _x\left|F_\eta_n(x)-F_\eta(x)\right| \leq \fracc\sqrtn,特别地φ(x₁，…，x_r)＝x₁²＋…＋x_r²和φ(x₁，…，x_r)＝x₁²＋…＋x_k²/x₁²＋…＋x_r²时，上述结论成立。

Abstract: 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.