On limiting behavior of sums and maxima when tails are slowly varying (Q1324848)

Suppose that \(\{X_ i: i \geq 1\}\) are independent, nonnegative random variables with a corresponding distribution functions (d.f.) \(F_ i\). The d.f.s \(F_ i\) are always assumed to be continuous. Put \[ \overline {F_ i} (x)=1-F_ i(x),\;S_ n=\sum^ n_{i=1} X_ i, \quad M_ n=\max (X_ 1, \dots, X_ n), \quad S_ n^{(1)} = S_ n-M_ n. \] We shall prove the following two theorems: 1. Let \(F_ i(x)\) be slowly varying for every \(i \geq 1\). If there exist two distribution functions \(F^*(x)\) and \(F_ *(x)\) such that \(F_ *(x) \leq F_ i(x) \leq F^*(x)\), for every \(i \geq 1\) and \(x \in \mathbb{R}\), \(\overline F_ *(x)\) and \(\overline F^*(x)\) are slowly varying at infinity, and \(\overline F_ *(x)/ \overline F^*(x) \to 1\), as \(x \to+\infty\), then \(\mathbb{E}(S_ n^{(1)}/M_ n) \to 0\) as \(n \to \infty\). 2. If \(S_ n^{(1)}/M_ n \to 0\), \(n \to+\infty\), in probability, and there exist d.f.s \(F_ *(x)\) and \(F^*(x)\) such that \(F_ *(x) \leq F_ i(x) \leq F^*(x)\), for every \(i \geq 1\) and \(x \in \mathbb{R}\), and \(\overline F_ *(x)/ \overline F^*(x) \to 1\) as \(x \to + \infty\), then \(F^*(x)\) and \(F_ *(x)\) are slowly varying at infinity.