I have a question regarding Fatou's Lemma and a sequence of random variables converging almost surely. Fatou's Lemma states
If $\forall n \in \mathbb{N}, \,\, X_{n} \ge 0$ and $\displaystyle X = \liminf_{n \rightarrow \infty} X_{n}$, then $\displaystyle\mathbb{E}[ \liminf_{n \rightarrow \infty}\: X_{n}] \le \liminf_{n \rightarrow \infty}\: \mathbb{E}[ X_{n}]$
Suppose we also know that $X_{n} \rightarrow X$ almost surely. How can we connect this to the requirements of Fatou's Lemma? It seems to me that the Lemma asks for pointwise convergence, a wholly different beast.
Best Answer
Suppose you know that $X_n \to X$ a.s. To avoid confusion, let's write $Y$ for $\liminf X_n$. Since for a convergent sequence, the limit and liminf are equal, we have $X = Y$ a.s., so $E[X] = E[Y]$, and by Fatou $E[Y] \le \liminf E[X_n]$.