## The Annals of Statistics

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60B99: None of the above, but in this section

**Keywords**

Strong law of large numbers compact metric space central tendency dispersion centroid

#### Citation

Sverdrup-Thygeson, Harald. Strong Law of Large Numbers for Measures of Central Tendency and Dispersion of Random Variables in Compact Metric Spaces. Ann. Statist. 9 (1981), no. 1, 141--145. doi:10.1214/aos/1176345340. https://projecteuclid.org/euclid.aos/1176345340