## The Annals of Statistics

### Strong Law of Large Numbers for Measures of Central Tendency and Dispersion of Random Variables in Compact Metric Spaces

Harald Sverdrup-Thygeson

#### Abstract

Given a sample of independent random variables $Z_1, Z_2, \cdots, Z_n$ with identical distribution $p$ on a compact metric space $(M, d)$, a measure of central tendency is a sample centroid (of order $r > 0$) defined as a point $\hat{X}_n$ in $M$ satisfying $\frac{1}{n} \sum^n_{i=1} d^r(\hat{X}_n, Z_i) = \inf_{x \in M} \frac{1}{n} \sum^n_{i=1} d^r(x, Z_i).$ A (population) centroid of $Z$ is any point $x^\ast$ in $M$ such that $\int_M d^r(x^\ast, z) dp(z) = \inf_{x \in M} \int_M d^r(x, z) dp(z).$ The quantity $\frac{1}{n} \sum^n_{i=1} d^r(\hat{X}_n, Z_i)$ itself is called the sample variation, whereas $\int_M d^r(x^\ast, z) dp(z)$ is the variation of $Z$. This paper establishes almost sure convergence for the sample centroid and variation to the corresponding population values for all orders $r > 0$. Convergence is also proved for the case when the sample centroid is restricted to be one of the sample values.

#### Article information

Source
Ann. Statist., Volume 9, Number 1 (1981), 141-145.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176345340

Digital Object Identifier
doi:10.1214/aos/1176345340

Mathematical Reviews number (MathSciNet)
MR600540

Zentralblatt MATH identifier
0445.60025

JSTOR