Suppose I have a sequence of random variables $X_n$ s.t. $\lim_{n\to\infty}X_n=c$ a.s. For example, we could take the $X_n$ to be the sample mean from some well-behaved distribution.
If I have a way of generating values for the $X_i$ and want to estimate the value of $c$, then, say, $X_{10^5}$ is likely to be a decent approximation for $c$. I could, for example, plot a graph of values for $X_1,\dots,X_{10^5}$ and hopefully it should be pretty apparent what the value of $c$ approximately is.
My question is if instead I have $\limsup_{n\to\infty}X_n=c$ a.s. (or with $\liminf$), how can I approximate $c$?
In this case, $X_{10^5}$ is of course not necessarily a good approximation. Would something like $\max_{i\leq10^5}X_i$ be better (or $\min$ if we have $\liminf$)?