I came to know that even with $X_n \rightarrow X$ in probability, it is possible not to have $P\{X_n \in (a,b)\}\rightarrow P\{X \in (a,b)\}$ for all intervals $(a,b)$. Can someone give me an example related to this?
2026-04-04 12:36:52.1775306212
property related to convergence of random variables
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It is even not true with almost sure convergence: choose for example $X_n=1/n$, $X=0$ and $a=0$, $b=1$.