Lagmental Vicfred

The Second Borel--Cantelli Lemma Uses Independence by Vicfred

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

Independent events whose probabilities have divergent sum occur infinitely often almost surely. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.

The data

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.

$$ A_n\ \text{independent},\qquad\sum_n\mathbf P(A_n)=\infty $$

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.

$$ \mathbf P(A_n\ \mathrm{i.o.})=1 $$

Derivation

The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.

$$ \mathbf P\!\left(\bigcap_{n=N}^{M}A_n^c\right)=\prod_{n=N}^{M}(1-\mathbf P(A_n))\le\exp\!\left(-\sum_{n=N}^{M}\mathbf P(A_n)\right)\to0 $$

Invariant content

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.

$$ \begin{aligned} \mathsf{D}\;&:\quad A_n\ \text{independent},\qquad\sum_n\mathbf P(A_n)=\infty,\\[5pt] \mathsf{C}\;&:\quad \mathbf P(A_n\ \mathrm{i.o.})=1. \end{aligned} $$

Scope

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \mathbf P(A_n\ \mathrm{i.o.})=1 \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Mon 13 September 2010. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.