Weak convergence problem without a condition

Question

I have a problem that I am not sure so I need your help.
If we have ${X_n} \Rightarrow X$ and $P(X=0)=0$ then we can show $X_n^{-1} \Rightarrow X^{-1}$ ($0^{-1}$ is defined as $0$) but can we prove it without the condition $P(X=0)=0$? Any comments would be appreciated!

Kavi Rama Murthy · Accepted Answer · 2019-12-12T12:48:05.633

1

Clearly false without the condition $P(X=0)=0$. Take $X_n =\frac 1 n, X=0$ for a counter-example.

edited Dec 12 '19 at 12:48

answered Dec 12 '19 at 12:40

Kavi Rama Murthy

311,013

So you mean without this condition, ${X_n}$ will never converge weakly to $X=0$ even if $n$ go to infinity? Can you explain a bit more detail? Thank you very much! – Thang Pham Dec 12 '19 at 12:56
Just think of the corresponding question for sequences of real numbers. – Kavi Rama Murthy Dec 12 '19 at 13:14
Thanks for your explanation! I will check it. – Thang Pham Dec 12 '19 at 19:52

Weak convergence problem without a condition

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