## The Annals of Mathematical Statistics

### A Limit Theorem for Sums of Minima of Stochastic Variables

Ulf Grenander

#### 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.

#### Article information

Source
Ann. Math. Statist., Volume 36, Number 3 (1965), 1041-1042.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177700076

Digital Object Identifier
doi:10.1214/aoms/1177700076

Mathematical Reviews number (MathSciNet)
MR176514

Zentralblatt MATH identifier
0142.14901

JSTOR