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Problem of the Week #252 - Jun 27, 2017

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Euge

MHB Global Moderator
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Jun 20, 2014
1,892
Here is this week's POTW:

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Let $(X_n)$ be a sequence of $L^1$ random variables on a probability space $(\Omega, \Bbb P)$. Let $f$ be a continuous, nondecreasing function from $[0,\infty)$ onto itself such that

1. $\Bbb E[f(|X_n|)]$ is uniformly bounded

2. $\dfrac{f(x)}{x}\to \infty$ as $x\to \infty$

Show that $(X_n)$ is uniformly integrable.
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Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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Euge

MHB Global Moderator
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Jun 20, 2014
1,892
Due to the difficulty some may have had with this problem, I'm extending the deadline one more week.
 
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Euge

MHB Global Moderator
Staff member
Jun 20, 2014
1,892
No one answered this week's problem. You can read my solution below.



Set $M = \sup_{m\in \Bbb N} E[f(\lvert X_n\rvert)]$. Given $\epsilon > 0$, choose $\delta > 0$ such that for all $x$, $x \ge \delta$ implies $f(x) > \frac{x}{\epsilon}$. For all $n\in \Bbb N$,

$$E[\lvert X_n\rvert I_{\lvert X_n\rvert \ge \delta}] \le E[\epsilon f(\lvert X_n\rvert)I_{\lvert X_n\rvert \ge \delta}] \le \epsilon M$$

Since $\epsilon$ was arbitrary, $(X_n)$ is uniformly integrable.
 
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