I have a possibly basic question, which I am not sure on whether or not it is true.
Suppose we have a sequence of identically distributed, but not necessarily independent random variables $X_n$ on a bounded metric space $M$ such that $X_n\to N(0,\sigma^2)$ in distribution. Can we conclude that $\sigma^2=\lim_{n\to\infty} Var(X_n)$?
I tried seeing if it holds by using the Portmanteau theorem, but f(x)=x^2 is not bounded so this can not be applied. Does anyone have any idea?
Edit: Note that if we place no restriction on the limiting distribution and allow independence, then the answer is false. Indeed, suppose $X_n=n$ with probability $1/n$ and $0$ otherwise. Then $X_n\to 0$ in probability, hence in distribution, and $\mathbb{E}(X_n^2)\to \infty\not=0$.