Sequence of expectation doesn't converge despite weak convergence

Question

Can someone give example where sequence of random variable $X_{n}$ converge in distribution to random variable $X$. $\sup E(X_{n})< \infty$ over all $n$, $\lim_{ n \to \infty} E(X_{n})\neq E(X)$?

Answer 1

Let $P[X_n=0] = 1 - \frac{1}{n}, \, P[X_n=n] = \frac{1}{n}$. Then clearly $E[X_0] = \sup E[X_n] = 1$, but $X_n \to 0$ in distribution, since the cumulative distribution functions $$ F_n(x) = P[X_n \leq x] = \left( 1-\frac{1}{n} \right) 1_{[0,\infty)}(x) + \frac{1}{n} 1_{[n, \infty)}(x)$$ converge to $$F(x) = 1_{[0, \infty)}(x),$$ the distribution function of $X \equiv 0$.

In order to get convergence of the expectations from that in distribution, you need an additional assumption like uniform integrability.