Little o-notation, convergence and random variables

Question

If $x_n$ is a deterministic sequence and $x_n \to x$ as $n \to \infty$, we can write $x_n = x+o(1)$, right? Can one do the same for random variables? Namely, if $X_n$ is a sequence of random variables and $X_n \to X_\infty$ a.s. as $n \to \infty$, can we write $X_n = X_\infty + o(1)?$ My intuition says no, but why exactly not? I suppose the difference $o(1)$ must be random?

Answer 1

There is nothing wrong in saying ht $X_n=X+o(1)$ almost surely but it should be remembered there is no uniformity w.r.t. $\omega$: $\sup_{\omega} |X_n(\omega)-X(\omega)|$ need not tend to $0$.