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Almost sure

This tiny blog-post is about almost sure convergence in probability theory. What are the great providers of almost sure convergence? The most natural answer is the Borel-Cantelli lemmas, the strong laws of large numbers, and the martingale convergence theorems. After years, I think that a great provider of almost sure convergence is the following:

  • if $X$ is a random variable taking values in $[0,+\infty]$ and such that $\mathbb{E}(X)<\infty$ then $$X<\infty\text{ almost surely.}$$

This elementary fact…

  • can be used to prove the first Borel-Cantelli lemma. Namely, if $(A_n)_n$ is a sequence of events such that $\sum_n\mathbb{P}(A_n)<\infty$ then this means by monotone convergence that $\mathbb{E}(\sum_n\mathbf{1}_{A_n})<\infty$ and thus $\sum_n\mathbf{1}_{A_n}<\infty$ almost surely, which means that $\mathbb{P}(\varliminf A_n^c)=1$, in other words $\mathbb{P}(\varlimsup A_n)=0$.
  • can be used to prove the strong law of large numbers for independent random variables bounded in $\mathrm{L}^4$. Namely if ${(X_n)}_n$ are independent centered random variables bounded in $\mathrm{L}^4$ then, setting $S_n=\frac{1}{n}(X_1+\cdots+X_n)$, we have $\mathbb{E}(S_n^4)=\mathcal{O}(n^{-2})$, and thus $\sum_n\mathbb{E}(S_n^4)<\infty$. By monotone convergence this gives $\mathbb{E}(\sum_nS_n^4)<\infty$, and thus, almost surely, $\sum_n S_n^4<\infty$, which implies that almost surely $\lim_{n\to\infty}S_n=0$.
  • can be placed at the heart of the proof via upcrossing of the Doob theorem of almost sure convergence of martingales bounded in $\mathrm{L}^1$. Idem for the theorem of convergence of martingales bounded in $\mathrm{L}^2$. The other martingale convergence theorems are corollaries.

Is it true that almost all almost sure statements in probability theory can be deduced from this elementary fact, directly or via its above mentioned consequences? Almost. Note however that the Strassen law of the iterated logarithm relies on both the first and second Borel-Cantelli lemmas. Another basic provider of almost sure convergence is the sigma-additivity of measures which gives that an at most countable intersection of almost sure events is almost sure.

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