ABSTRACT
1 The distribution function F of a real random variable X is defined by F(x) = P{X ≤ x}. For a sample of n values of X, the sample distribution function Kn is defined by Kn (x) = N(x)/n, where N(x) = number of sample values ≤ x. It is a step function. The Glivenko–Cantelli theorem (published in 1933) states that, with probability one, sup x | K n ( x ) − F ( x ) | → 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351076777/1531b062-750f-426f-9035-7dcbd680db0f/content/eq642.tif"/> as n → ∞. This is the existence theorem for Statistics as a branch of Applied Mathematics.
