The Annals of Probability

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

H. Hebda-Grabowska and D. Szynal

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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

Permanent link to this document
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
links.jstor.org

Subjects
Primary: 60B10: Convergence of probability measures
Secondary: 60F15: Strong theorems

Keywords
Invariance principle infima law of the iterated logarithm Brownian motion

Citation

Hebda-Grabowska, H.; Szynal, D. An Almost Sure Invariance Principle for the Partial Sums of Infima of Independent Random Variables. Ann. Probab. 7 (1979), no. 6, 1036--1045. doi:10.1214/aop/1176994896. https://projecteuclid.org/euclid.aop/1176994896


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