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Let X 1, …, X n , … be a sequence of independent identically distributed (i.i.d.) random variables with a common distribution function F( x). Introduce the empirical distribution function: \(\displaystyle F_{n}(x)=\frac {1}{n}\sum _{i=1}^{n} I \lbrace X_{i}< x\rbrace \) where I{⋅} is the indicator function. By strong law of large numbers (SLLN) for any fixed point x ∈ R, we have \(\displaystyle F_{n}(x)\rightarrow F(x) \ \ a.s., \ \ as \ \ n\rightarrow \infty . \) Glivenko-Cantelli theorem states that this convergence is uniform.

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Glivenko-Cantelli Theorems

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  • 发表日期：

    2025年

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  • Olimjon Sh. Sharipov

摘要

Let X 1, …, X n , … be a sequence of independent identically distributed (i.i.d.) random variables with a common distribution function F( x). Introduce the empirical distribution function: \(\displaystyle F_{n}(x)=\frac {1}{n}\sum _{i=1}^{n} I \lbrace X_{i}< x\rbrace \) where I{⋅} is the indicator function. By strong law of large numbers (SLLN) for any fixed point x ∈ R, we have \(\displaystyle F_{n}(x)\rightarrow F(x) \ \ a.s., \ \ as \ \ n\rightarrow \infty . \) Glivenko-Cantelli theorem states that this convergence is uniform.