Consider $n$ i.i.d. positive random variables $X_1, \cdots, X_n$ which are regularly varying with tail index $\alpha > 0$. Let $M_n$ denote the sample maximum and $S_n = \sum X_i$. We know from Bingham and Teugels (1981) that \begin{equation} \tag{1} \lim_{n \to \infty} \mathbb{E}\, \left( \frac{M_n}{S_n} \right) = 1-\alpha \end{equation} if, and only if, $\alpha \in (0, 1)$.

In general, even if (1) holds we have $\lim_{n \to \infty}\mathbb{E}\, \left( M_n - (1-\alpha) S_n \right)\neq 0$. The question is, what can we say instead? Suppose for instance I am interested in \begin{equation} \tag{2} \mathbb{E}\,f\left(\frac{1}{n} S_n \right), \end{equation} where $f$ is assumed to render the expectation finite. Is there any way I can leverage (1) and substitute it into (2) such that the problem converts to one about the distribution of $M_n$?



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