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 = [logical and operator]ni=1 Xi. Let [varrho](X, Y) denote the uniform distanc distributions of random variables X and Y; i.e. . We consider [varrho](Sn, Mn) when P(X1>x) is slowly varying and we provide bounds for the asymptotic behaviour of this quantity as n-->[infinity], thereby establishing a uniform rate of convergence result in Darling's law for distributions with slowly varying tails.