Lagmental Vicfred

The Weak Law Follows from Variance Decay by Vicfred

tags: mathematics, limit theorems, Borel--Cantelli, central limit theorem

For iid variables with finite variance, the sample mean converges in probability to the common mean. A small computation will anchor the general statement before the abstraction takes over.

Objects and notation

For events \(A_n\), the notation \(A_n\ \mathrm{i.o.}\) means infinitely many occur. For sums \(S_n=X_1+\cdots+X_n\), different normalizations lead to laws of large numbers or central limits.

$$ \overline X_n=\frac1n\sum_{i=1}^{n}X_i,\qquad\mathbf E X_i=\mu,\quad\operatorname{Var}X_i=\sigma^2 $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ \overline X_n\xrightarrow{\mathbf P}\mu $$

Push the symbols

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ \mathbf P(|\overline X_n-\mu|\ge\varepsilon)\le\frac{\operatorname{Var}(\overline X_n)}{\varepsilon^2}=\frac{\sigma^2}{n\varepsilon^2}\to0 $$

Structural reading

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad \overline X_n=\frac1n\sum_{i=1}^{n}X_i,\qquad\mathbf E X_i=\mu,\quad\operatorname{Var}X_i=\sigma^2,\\[5pt] \mathsf{C}\;&:\quad \overline X_n\xrightarrow{\mathbf P}\mu. \end{aligned} $$

A hypothesis worth keeping

Convergence almost surely, in probability, in distribution, and in \(L^p\) are distinct. One implication cannot be reversed without extra hypotheses.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \overline X_n\xrightarrow{\mathbf P}\mu \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Sun 15 April 2018. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.