## The Annals of Probability

### Almost Sure Convergence of Certain Slowly Changing Symmetric One- and Multi-Sample Statistics

#### Abstract

Let $X^{(i)}_j, i = 1,\ldots, k; j \in \mathbf{N}$, be independent $d$-dimensional random vectors which are identically distributed for each fixed $i = 1,\ldots, k$. We give a sufficient condition for almost sure convergence of a sequence $T_{n_1,\ldots, n_k}$ of statistics based on $X^{(i)}_j i = 1,\ldots, k; j = 1, \ldots, n_i$, which are symmetric functions of $X^{(i)}_1,\ldots, X^{(i)}_{n_i}$ for each $i$ and do not change too much when variables are added or deleted. A key auxiliary tool for proofs is the Efron-Stein inequality. Applications include strong limits for certain nearest neighbor graph statistics, runs and empty blocks.

#### Article information

Source
Ann. Probab., Volume 20, Number 2 (1992), 1086-1098.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176989819

Digital Object Identifier
doi:10.1214/aop/1176989819

Mathematical Reviews number (MathSciNet)
MR1159587

Zentralblatt MATH identifier
0759.62017

JSTOR