0
$\begingroup$

I'm not sure my question is of research level, but I cannot find the answer in the existing reference. Let $\mu_n$ be a sequence of probability measures on $\mathbb R$ satisfying

$$\int_{\mathbb R}xd\mu_n~=~0,~~~ \int_{\mathbb R}x^+d\mu_n~=~C \mbox{ for all } n\ge 1.$$

Assume further $\mu_n$ converges weakly to some $\mu$. Could some one may provide an example such that

$$\int_{\mathbb R}xd\mu_n~\neq~0 \mbox{ or } \int_{\mathbb R}x^+d\mu_n~\neq~C?$$

Clearly, if $\mu_n$s are not required to be probability measures, there are several examples. But as for probability measures, I cannot find such an example. Many thx!

Improve this question
$\endgroup$
1
  • 1
    $\begingroup$ $\mu$ instead of $\mu_n$ in the 2nd display? $\endgroup$
    – Serguei Popov
    Commented Apr 5, 2016 at 17:05

1 Answer 1

Reset to default
4
$\begingroup$

If, for $n>2$, $\mu_n$ puts masses $n^{-1},n^{-1}, 1-2n^{-1}$ at $2n, -n, -\frac{n}{n-2}$ respectively (so that $\mu$ puts the unit mass at $-1$), doesn't it work?

Improve this answer
$\endgroup$

Not the answer you're looking for? Browse other questions tagged .