Let $X_n \ge 0$ be a sequence of random variables (which are not necessarily independent) satisfying the following conditions:
- $E(X_n) = \mu_n$ where $l \le \mu_n \le u$ for some constants $l, u < \infty$ (but we don't know whether $\mu_n$ converges to a constant).
- $\lim_{n \rightarrow \infty} V(X_n) \rightarrow 0$ as $n \rightarrow \infty$.
By Chebyshev's inequality, we know that $|X_n - \mu_n| \rightarrow 0$ in probability.
Can we say something probabilistic about $X_n$ being in the interval $[l,u]$ with high probability?
Thanks!
EDIT: I've erased a part where I asked whether we can say $X_n \rightarrow \xi$ for $\xi \in [l,u]$, which is not true, as per saz's comment.