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If we have that $X_n$, $X$ are random variables, how can I show that

$$ \lim_{n\to\infty} X_n = X \iff \liminf_{n\to\infty} X_n = X $$

Here the limit implies almost sure convergence.

I know that this amounts to showing:

$$\omega\in\left\{\lim_{n\to\infty} X_n= X\right\}\iff \omega\in\bigcup_{n=1}^\infty\bigcap_{k=n}^\infty\{|X_n-X|<\varepsilon \}, \text{ for all }\varepsilon > 0.$$

The forward direction is simple because I know that if the limit of $X_n$ exists and equals to $X$, then the limit infimum and supremum is equal. However, I am not sure how to get the backward direction, that if the limit infimum exists and equals to $X$, then the limit exists also? Thanks.

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    $\begingroup$ Do you forget something? Why should that be true? Also is that almost sure convergence? $\endgroup$ – user251257 Oct 9 '16 at 23:34
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    $\begingroup$ The converse is false even for a deterministic sequences. $\endgroup$ – Sangchul Lee Oct 9 '16 at 23:40
  • $\begingroup$ You need additional conditions for the converse (for an arbitrary sequence, the limsup or liminf is not necessarily the limit should it exist. But for a monotone sequence...) $\endgroup$ – Batman Oct 10 '16 at 0:19
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There are different modes of convergence of random variables. You should be carful when you write $$ \lim_{n\to\infty}X_n=X. $$

I assuming you are talking about the set $$ A=\{\omega\in \Omega\mid \lim_n X_n(\omega)=X(\omega)\}. $$ and the set $$ B=\{\omega\in \Omega\mid \liminf_n X_n(\omega)=X(\omega)\}. $$ Then $A\subset B$, but in general we don't have $B\subset A$. Consider for instance $$ X_n(\omega)=(-1)^n,\quad X(\omega)=-1 $$ for all $\omega\in\Omega$ when $\Omega$ is assumed to be the underlying sample space.

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  • $\begingroup$ I interpret the limit infimum of sets as being the event of some event happening always after some $n$ and the limit supremum as being the event infinitely many events occur. It seems like here the limit supremum is $1$ while the limit infimum is $-1$. However, it doesn't appear to be the case that after some $n$, $-1$ occurs each and everytime. How can I reconcile these definitions of limit infimum for sets and real numbers? $\endgroup$ – user321627 Oct 29 '16 at 7:21

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