I came to know that even with Xn→XX_n \rightarrow X in probability, it is possible not to have P{Xn∈(a,b)}→P{X∈(a,b)}P\{X_n \in (a,b)\}\rightarrow P\{X \in (a,b)\} for all intervals (a,b)(a,b). Can someone give me an example related to this?

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1 Answer

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It is even not true with almost sure convergence: choose for example Xn=1/nX_n=1/n, X=0X=0 and a=0a=0, b=1b=1.