Quantifying closeness of distributions of sums and maxima when tails are fat (Q1825504)

Assume \(X_ 1,X_ 2,...,X_ n\) be n independent, identically distributed, non-negative random variables. Put \(S_ n=X_ 1+...+X_ n\) and \(M_ n=\max (X_ 1,...,X_ n)\). Let \[ \rho (U,V)=\sup | P(U\leq x)-P(V\leq x)| \] be the uniform distance between the distributions of U and V. The authors assume \(P(X_ 1>x)\) to be slowly varying and investigate \(\rho (S_ n,M_ n)\). They provide bounds for the asymptotic behavior of this quantity as \(n\to \infty\), thereby establishing a uniform rate of convergence result in Darling's law for distributions with slowly varying tails [\textit{D. A. Darling}, Trans. Am. Math. Soc. 73, 95-107 (1952; Zbl 0047.375)]. The proofs of the main bound depend heavily on an intricate study of slowly varying functions with or without remainder.