Suppose you want to show $sup_{x\in D}|f_n(x)|\to_p 0$, for $n\to \infty$, where $D\subset \mathbb R$ is a compact interval, $f$ is continuous depending on one or more random variables, and $\to_p$ means convergence in probability. For example, $f_n(x)=\sum_{i=1}^n(X_i-x)$ (this, however, is not the problem).
Because showing statements as the one above directly is rather difficult I was wondering if it is sufficient to show $sup_{x\in D}| Ef_n(x)|\to 0$ and $sup_{x\in D}| Var(f_n(x))|\to 0$. Where $E$ and $Var$ are the expectation and variance operator, respectively. If some of you know a good read on this I appreciate your suggestions. Thanks in advance. Cheers.
It's not enough. Let $X_i$ be independent uniform $[0,1]$ random variables and $f_i(x) = e^{i(X_i-x)^2}$. Then both $\sup_{x\in(0,1)}E(f_n(x))$ and $\sup_{x\in(0,1)}Var(f_n(x))$ converge to zero, but $\sup_{x\in(0,1)} f_n(x) = 1$ almost surely for every $n$.
Billingsley is a classic on this sort of thing. It's got all the examples of things like this you should watch out for. (Or you can always google "measure theory lecture notes", and get something for free.