Given a nonnegative random variable $X$, can we prove $\dfrac{\mathbb{E}(X\mathbf{1}_{\{X\leq n\}})}{n}\to0$ as $n\to\infty$ or not?
Asked
Active
Viewed 50 times
-1
-
Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Another User Sep 16 '22 at 16:36
1 Answers
0
$\forall\,m,n\in\mathbb{N}$, we have $$ \frac{\mathbb{E}(X\mathbf{1}_{\{X\leq n\}})}{n}=\frac{\mathbb{E}(X\mathbf{1}_{\{m<X\leq n\}})}{n}+\frac{\mathbb{E}(X\mathbf{1}_{\{X\leq m\}})}{n}\leq\mathbb{P}(m<X\leq n)+\frac{m}{n}\mathbb{P}(X\leq m). $$ Fix $m$, let $n\to\infty$, and then let $m\to\infty$.