Asymptotic Normality of Sums of Minima of Random Variables

Let x₁, x₂,\⋯ be independent and positive random variables with the common distribution function F. We show that if \∫¹₀|F(x) - x/b| \× x⁻²dx < \∞ for some 0 < b < \∞, then \∑ⁿₖ₌₁ min(x₁,\⋯, xₖ) is asymptotically normal with expectation b log n and variance b² 2 log n.