Suppose that $X_n$ is a sequence of random variables such that $X_n \rightarrow X$ either almost surely or in $L^p$ or both.

Now consider the sequence of conditional random variables $f(X_n) | X_n \in B$ for some measurable set $B$ and some nice function $f$ (we can assume it's Lipschitz to make everything easy). Do I still preserve convergence properties for this conditional random variable, i.e. does $f(X_n) | X_n \in B \rightarrow f(X) | X \in B$ in some sense (a.s. or in $L^p)$?

I'm stuck verifying this by using the definition of almost sure convergence or $L^p$ convergence.

For example, regarding almost sure convergence, I have to check that $P(\lim_{n\rightarrow \infty} f(X_n)|X_n \in B = f(X)|X\in B)$ which looks strange from a notation perspective because of the conditional. Or should I resort to using conditional expectation?

  • 2
    $\begingroup$ You cannot define $f(X_n)|X_n$ as random variables. You use conditioning only for distributions and expectations. So almost sure convergence does not even make sense here. $\endgroup$ – Kavi Rama Murthy Aug 17 '18 at 5:38
  • $\begingroup$ Perhaps you are really trying to show $E[f(X_n)|X_n \in B] \rightarrow E[f(X)|X\in B]$? Of course this is not always true, take $B= \mathbb{R}$, $X=0$, $f(x)=x$, then the statement is equivalent to showing $E[X_n]\rightarrow 0$ whenever $X_n\rightarrow 0$ with prob 1, which is not always true. $\endgroup$ – Michael Aug 17 '18 at 6:27

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