Bounds for the Distance Between the Distributions of Sums of Absolutely Continuous i.i.d. Convex-Ordered Random Variables with Applications

Abstract

Let X 1, X 2,… and Y 1, Y 2,… be two sequences of absolutely continuous, independent and identically distributed (i.i.d.) random variables with equal means E(X i)=E(Y i), i=1,2,… In this work we provide upper bounds for the total variation and Kolmogorov distances between the distributions of the partial sums ∑i=1 n X i and ∑i=1 n Y i. In the case where the distributions of the X is and the Y is are compared with respect to the convex order, the proposed upper bounds are further refined. Finally, in order to illustrate the applicability of the results presented, we consider specific examples concerning gamma and normal approximations.