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

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Result R5399 on D4198: Random average series
Subresult of R2368: I.I.D. real strong law of large numbers with the identity index sequence

Sample average of I.I.D. integrable random real numbers converges to expectation almost surely

Formulation 0

Let

$X_1, X_2, X_3, \ldots \in \text{Random}(\mathbb{R})$ 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)	$\begin{equation} \mathbb{E} \|X_1\| < \infty \end{equation}$

Then

$\begin{equation} \lim_{N \to \infty} \frac{1}{N} \sum_{n = 1}^N X_n \overset{a.s.}{=} \mathbb{E} X_1 \end{equation}$

Formulation 1

Let

$X_1, X_2, X_3, \ldots \in \text{Random}(\mathbb{R})$ 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)	$\begin{equation} \mathbb{E} \|X_1\| < \infty \end{equation}$

Then

$\begin{equation} \mathbb{P} \left( \lim_{N \to \infty} \frac{1}{N} \sum_{n = 1}^N X_n = \mathbb{E} X_1 \right) = 1 \end{equation}$

Subresults

▶	R5393: I.I.D. real empirical distribution measure converges to a probability for a fixed Borel set

Proofs

Proof 0

Let

$X_1, X_2, X_3, \ldots \in \text{Random}(\mathbb{R})$ 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)	$\begin{equation} \mathbb{E} \|X_1\| < \infty \end{equation}$

$\square$