ThmDex – An index of mathematical definitions, results, and conjectures.

Result R3493 on D5074: Empirical probability distribution function

Empirical distribution function converges in probability to the true distribution function

Formulation 0

Let $X_1, X_2, X_3, \ldots \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that

(i)	$X_1, X_2, X_3 \ldots$ is an D3358: I.I.D. random collection

Let $x \in \mathbb{R}$ be a D993: Real number.

Then \begin{equation} \frac{1}{N} \sum_{n = 1}^N I_{\{ X_n \leq x \}} \overset{p}{\longrightarrow} \mathbb{P}(X_1 \leq x) \quad \text{ as } \quad N \to \infty \end{equation}

Formulation 1

Let $X_1, X_2, X_3, \ldots \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that

(i)	$X_1, X_2, X_3 \ldots$ is an D3358: I.I.D. random collection
(ii)	$F : \mathbb{R} \to [0, 1]$ is a D205: Probability distribution function for $X_1$

Let $x \in \mathbb{R}$ be a D993: Real number.

Then \begin{equation} \frac{1}{N} \sum_{n = 1}^N I_{\{ X_n \leq x \}} \overset{p}{\longrightarrow} F(x) \quad \text{ as } \quad N \to \infty \end{equation}

Proofs

Proof 0

Let $X_1, X_2, X_3, \ldots \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that

(i)	$X_1, X_2, X_3 \ldots$ is an D3358: I.I.D. random collection

Let $x \in \mathbb{R}$ be a D993: Real number.

The random indicator $I_{\{ X_n \leq x \}}$ has expectation \begin{equation} \mathbb{E}(I_{\{ X_n \leq x \}}) = 0 \cdot \mathbb{P}(X > x) + 1 \cdot \mathbb{P}(X_n \leq x) = \mathbb{P}(X_1 \leq x) \end{equation} The claim is thus a consequence of R2410: I.I.D. real weak law of large numbers under finite second absolute moments. $\square$