Asymptotic Normality of Sums of Minima of Random Variables

Abstract

Let $x_1, x_2,\cdots$ be independent and positive random variables with the common distribution function $F$. We show that if $\int^1_0|F(x) - x/b| \times x^{-2}dx < \infty$ for some $0 < b < \infty$, then $\sum^n_{k=1} \min(x_1,\cdots, x_k)$ is asymptotically normal with expectation $b \log n$ and variance $b^2 2 \log n$.