TITLE: Sequence which converges in probability but neither almost surely nor in $L^p$
QUESTION [7 upvotes]: Is it possible to find a sequence of r.v.'s $\{X_n\}$ that converge in probability but not almost surely nor in $L^p$?
Does anyone know of a single example that illustrates this point? I've been trying to think of one for ages and I've not come up with anything. I'm sure such a sequence must exist, but can't think what it could be.

REPLY [7 votes]: $X_n$ independent such that $P(X_n=n)=\frac{1}{n}$ and $P(X_n=0)=1-\frac{1}{n}$.
Then $\forall \varepsilon>0$, $P(|X_n|>\varepsilon)=\frac{1}{n} \to 0$ but by second Borel Cantelli $P(\limsup |X_n|>\varepsilon)=1$ and $E|X_n|^p=n^{p-1}$ that doesn't converge to 0 $\forall p\geq1$.