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.

  • 2
    $\begingroup$ Do you forget something? Why should that be true? Also is that almost sure convergence? $\endgroup$ – user251257 Oct 9 '16 at 23:34
  • 1
    $\begingroup$ 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

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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.