## The Annals of Probability

### An Almost Sure Invariance Principle for the Partial Sums of Infima of Independent Random Variables

#### Abstract

Let $\{X_n, n \geqslant 1\}$ be a sequence of independent random variables uniformly distributed on the unit interval. Put $X^\ast_n = \inf(X_1, X_2,\cdots, X_n)$ and $S_n = X^\ast_1 + X^\ast_2 + \cdots + X^\ast_n, n \geqslant 2, S_1 = 0$. The aim of this note is to give an almost sure invariance principle for $S_n$. Next we extend the given results to the case when $X_n, n \geqslant 1$, are not uniformly distributed but bounded, and moreover, to sums $\hat{S}_n = X^{(m)}_m + X^{(m)}_{m+1} +\cdots + X^{(m)}_n$, where $X^{(m)}_j$ is the $m$th order statistic of $(X_1, X_2,\cdots, X_j)$.

#### Article information

Source
Ann. Probab., Volume 7, Number 6 (1979), 1036-1045.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176994896

Digital Object Identifier
doi:10.1214/aop/1176994896

Mathematical Reviews number (MathSciNet)
MR548897

Zentralblatt MATH identifier
0423.60031

JSTOR