Quantifying Closeness of Distributions of Sums and Maxima When Tails Are Fat

摘要

Let X(1),X(2),...,X(n) be n independent, identically distributed, non-negative random variables and put S(n)=sumX(i),i=1 ton and M(n)=V(X(i)),i=1 to n. Let rho(X,Y) denote the uniform distance between the distributions of random variables X and Y. The authors consider rho(S(n),M(n)) when P(X(1)>x) is slowly varying and provide bounds for the asymptotic behavior of this quantity as n approaches infinity, thereby establishing a uniform rate of convergence resulting in Darling's law for distributions with slowly varying tails.