Suppose $X_1, X_2, X_3, \ldots$ is a sequence of random variables that converges almost surely to a random variable $X$.
How could I check for this experimentally? I don't need this to be some rigorously reliable test, I would just like to do this to better understand the ideas.
For concreteness, suppose $X$ is uniformly distributed on $[0,1]$ and $X_1, X_2, \ldots$ is some process that I can simulate, and we know that $X_n \to X$ almost surely. What is nagging me is that I could produce samples $x_1^1, x_2^1, x_3^1 \ldots, $ and produce samples $x_1^2, x_2^2, x_3^3, \ldots$, etc... and then hope that, averaged over each of the limits, we "land" almost anywhere in $[0,1]$. But how do I know where to stop within each individual sequence? Isn't it possible that no individual sample sequence converges?