Problem with truthfulness of equivalence.

Question

Let $X_n$ be a sequance of random variables. I wonder about this equivalence : $${\displaystyle {\overset {}{X_{n}\ {\xrightarrow {p}}\ 0\ \ }}} \Leftrightarrow E(X_n)\rightarrow0 $$

"$\Rightarrow$"

If ${\displaystyle {\overset {}{X_{n}\ {\xrightarrow {p}}\ 0\ \ }}}$ then for very big $n$ $(X_n)$ only take values very close to zero, so Expected value of $(X_n)$ is also arbitrarily close to zero. So it has to be : $E(X_n)\rightarrow0.$

"$\Leftarrow$"

Let's take $X_n$ ~ $U[-1,1]$. Then for any $n$, we have $E(X_n)=0, so \;E(X_n)\rightarrow0.$

But $P(|X_n|<\varepsilon) \neq1 $, so $X_n$ dosen't converge to $0$ in probability.

Am i thinning correctly ?

Answer 1

Take $X_n=nI(Z\in [0, 1/n])$ where $Z\sim \text{Unif}[0, 1]$. Then $X_n\to 0$ almost surely, and hence in probability yet $EX_n=1$ for all $n$ so $EX_n\not\to 0$.