The Annals of Probability

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

N. Henze and B. Voigt

Full-text: Open access

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


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