# Random Variables converge weakly in $L^{1}(\Omega,\mathcal{F},\mathbb{ P } )$

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?

• 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. – Stefano Nov 10 '16 at 0:15
• 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. – Miskel Nov 10 '16 at 1:16