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Suppose $X_n$, $n=1,2,... $ and $X$ are random variables defined on probability space $(\Omega,\mathcal{F},\mathbb{ P } )$. They have bounded first moments. Assume that $X_n \geq 0$ almost surely, $\mathbb{E}X_n = 1$ and $\mathbb{E}(X_n \log X_n) \leq 1$. Assume that for every bounded random variable $Y$, $\mathbb{E}(X_nY)\rightarrow \mathbb{E}(XY) $ as $n \rightarrow \infty$. Show that: $X_n \rightarrow X$ in probability and $\mathbb{E}(X\log X) \leq 1$.

In this problem, we know that $X_n,X \in L^{1}(\Omega,\mathcal{F},\mathbb{ P })$, and $X_n$ converges weakly to $X$ in $L^{1}(\Omega,\mathcal{F},\mathbb{ P })$, we need to deduce that $X_n$ converges in probability. Currently, I have no idea of this problem. Could anyone give me some hints?

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  • $\begingroup$ Are you explicitly asked to prove that $X_n \to X$ in probability or is it something you think you need in order to prove the second part (namely $\mathbb E[X \log X] \le 1$)? I'm asking because you can directly prove the second part without the first. $\endgroup$ – Stefano Nov 10 '16 at 0:15
  • $\begingroup$ Yes, the problem directly asked me to prove that $\mathbb{E}(X log X) \leq 1 $, but I was given a hint that first prove converge in probability and then prove the final consequence, so I was stuck here for a long time. $\endgroup$ – Miskel Nov 10 '16 at 1:16

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