Lagmental Vicfred

The First Borel--Cantelli Lemma Needs No Independence by Vicfred

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

If the sum of event probabilities is finite, only finitely many of the events occur almost surely. This is a compact note, but the quantifiers and hypotheses stay on the page.

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.

$$ \sum_{n\ge1}\mathbf P(A_n)<\infty $$

I read the first line as input and the second as output. The symbols \(\forall\) and \(\exists\) are not interchangeable, and neither may be upgraded silently to \(\Longleftrightarrow\).

$$ \mathbf P(A_n\ \mathrm{i.o.})=0 $$

The calculation

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ A_n\ \mathrm{i.o.}=\bigcap_{N\ge1}\bigcup_{n\ge N}A_n,\qquad\mathbf P\!\left(\bigcup_{n\ge N}A_n\right)\le\sum_{n\ge N}\mathbf P(A_n)\to0 $$

What survives abstraction

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad \sum_{n\ge1}\mathbf P(A_n)<\infty,\\[5pt] \mathsf{C}\;&:\quad \mathbf P(A_n\ \mathrm{i.o.})=0. \end{aligned} $$

The boundary

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \mathbf P(A_n\ \mathrm{i.o.})=0 \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Tue 23 November 2010. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.