Quantifying closeness of distributions of sums and maxima when tails are fat

Let X1, X2,…, Xn be n independent, identically distributed, non negative random variables and put and Mn = ni=1 Xi. Let (X, Y) denote the uniform distanc distributions of random variables X and Y; i.e.We consider (Sn, Mn) when P(X1>x) is slowly varying and we provide bounds for the asymptotic behaviour of this quantity as n¿8, thereby establishing a uniform rate of convergence result in Darling's law for distributions with slowly varying tails.