Abstract
We consider a sequence of independent and identically distributed positive stochastic variables $x_1, x_2, x_3, \cdots$ with the distribution function $F(x)$. Let $y_n$ be the smallest of the values taken by the $n$ first of these variables and $S_n = y_1 + y_2 + \cdots + y_n$. It is then shown that $S_n/\log n$ tends in probability to the value $F = \lim_{t\downarrow 0} t/F(t)$ assumed to exist as a finite or infinite number.
Citation
Ulf Grenander. "A Limit Theorem for Sums of Minima of Stochastic Variables." Ann. Math. Statist. 36 (3) 1041 - 1042, June, 1965. https://doi.org/10.1214/aoms/1177700076
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