## The Annals of Probability

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

### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 62G10: Hypothesis testing

**Keywords**

Almost sure convergence Efron-Stein inequality nearest neighbors geometric probability runs empty blocks

#### Citation

Henze, N.; Voigt, B. Almost Sure Convergence of Certain Slowly Changing Symmetric One- and Multi-Sample Statistics. Ann. Probab. 20 (1992), no. 2, 1086--1098. doi:10.1214/aop/1176989819. https://projecteuclid.org/euclid.aop/1176989819