The Annals of Mathematical Statistics

On the Convergence of Empiric Distribution Functions

J. R. Blum

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Abstract

Let $\mu$ be a probability measure on the Borel sets of $k$-dimensional Euclidean space $E_k.$ Let ${X_n}, n = 1, 2, \cdots,$ be a sequence of $k$-dimensional independent random vectors, distributed according to $\mu.$ For each $n = 1, 2, \cdots$ let $\mu_n$ be the empiric distribution function corresponding to $X_1, \cdots, X_n,$ i.e., for every Borel set $A \epsilon E_k,$ we define $\mu_n(A)$ to be the proportion of observations among $X_1, \cdots, X_n$ which fall in $A.$ Let $\alpha$ be the class of Borel sets in $E_k$ defined below. The object of this paper is to prove that $P{\lim_{n\rightarrow\infty}} \sup_{A \epsilon \mathscr{a} \|\mu_n(A) - \mu(A)\| = 0} = 1.$

Article information

Source
Ann. Math. Statist., Volume 26, Number 3 (1955), 527-529.

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177728499

Digital Object Identifier
doi:10.1214/aoms/1177728499

Mathematical Reviews number (MathSciNet)
MR70871

Zentralblatt MATH identifier
0065.11303

JSTOR
links.jstor.org

Citation

Blum, J. R. On the Convergence of Empiric Distribution Functions. Ann. Math. Statist. 26 (1955), no. 3, 527--529. doi:10.1214/aoms/1177728499. https://projecteuclid.org/euclid.aoms/1177728499


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