Convergence in probability and almost surely
(WIP, private notes)

Fact 1

In a topological space, let $p_n$ be a sequence of points, $p$ another point, and assume the following holds: all subsequences of $p_n$ have a subsubsequence converging to $p$ . Then, $p_n$ converges to $p$ .

Indeed, if this were not the case, there would exist an open set containing $p$ but not infinitely many of the $p_n$ s, and no subsequence of those infinitely many $p_n$ s can converge to $p$ .

Fact 2

Convergence in almost surely implies convergence in probability.

(Not indeeded here.)

Fact 3

If $X_n$ is a sequence of random variables, $X$ another one (all on the same space), and any subsequence has a subsubsequence converging almost surely to $X$ , then $X_n$ converges itself in proba to $X$ .

Indeed,

any subsequence has a subsubsequence converging almost surely, hence also in proba by Fact 2;
apply Fact 1, using that convergence in proba is topological (actually, metric: there is a metric on random variables that metrizes that convergence).

Fact 4

Convergence almost surely is not topological:

The typewriter sequence is an example of a sequence of random variables that satisfy that any subsequence has a subsubsequence converging almost surely to zero. But it does not itself converge almost surely to zero. See https://en.wikipedia.org/wiki/Pointwise_convergence#Almost_everywhere_convergence.

Morale