0
$\begingroup$

Consider a real-valued random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$.

Let $a\in \mathbb{R}$.

Let $\{f_n(X, a)\}_n$ be a sequence of real-valued random functions.

Consider a non-random function $f:\mathbb{R}\rightarrow \mathbb{R}$ continuous on the entire domain and such that $\int_{-\infty}^\infty |f(a)|da<\infty$

Assume $\lim_{n\rightarrow \infty} E_{\mathbb{P}}(f_n(X,a))-f(a)=0$ $\forall a\in \mathbb{R}$

Could you help me to show that all these assumptions imply $\lim_{n\rightarrow \infty}\sup_{a} |E_{\mathbb{P}}(f_n(X,a))-f(a)|=0$?

$\endgroup$
0
$\begingroup$

This may be not true: take $f_n(X,a):=f(a)+a/n$: the assumptions are satisfied but $\sup_{a} |E_{\mathbb{P}}(f_n(X,a))-f(a)|$ is infinite.

$\endgroup$

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.