To understand limsup and liminf. Today I came across Lim Sup and Lim Inf in Frank Burk’s book. I checked out a wonderful article in Math overstack which said one can visually see lim sup

> n <- 1e+05
> x <- rpois(n, 12)
> plot(1:n, x, col = "blue", title = "LimSup example ")
> abline(h = 1:n, col = "grey")


  1. I have generated a million poisson independent random variables and E be the event that there is an infinite random subsequences Nk such that Xnk>nk

This probability is 0 according to Borel-Cantelli lemma. This means with probability one, there exists a number N such that for all n>N, Xn ⇐ n