Lagmental Vicfred

Slutsky's Theorem Combines Random and Deterministic Limits by Vicfred

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

A distributional limit survives addition or multiplication by a term converging in probability to a constant. The point is to make the formal expression readable enough to audit line by line.

Set-up

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.

$$ X_n\xrightarrow{d}X,\qquad Y_n\xrightarrow{\mathbf P}c $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ X_n+Y_n\xrightarrow{d}X+c,\qquad X_nY_n\xrightarrow{d}cX $$

The calculation

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ \frac{\sqrt n(\overline X_n-\mu)}{S_n}=\underbrace{\frac{\sqrt n(\overline X_n-\mu)}{\sigma}}_{\xrightarrow{d}N(0,1)}\underbrace{\frac{\sigma}{S_n}}_{\xrightarrow{\mathbf P}1}\xrightarrow{d}N(0,1) $$

What survives abstraction

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad X_n\xrightarrow{d}X,\qquad Y_n\xrightarrow{\mathbf P}c,\\[5pt] \mathsf{C}\;&:\quad X_n+Y_n\xrightarrow{d}X+c,\qquad X_nY_n\xrightarrow{d}cX. \end{aligned} $$

The boundary

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] X_n+Y_n\xrightarrow{d}X+c,\qquad X_nY_n\xrightarrow{d}cX \end{gathered}} $$

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 Sat 26 November 2011. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.