$X_n$ be a sequence of +ve $L^1$ r.v. converging a.s. to $X\in L^1$ and $E(X_n) \to E(X)$ . Then $E(X_nY)\to E(XY)$ for any bounded r.v. $Y$

Question

Let $\{X_n\}_{n\ge 1}$ be a sequence of positive random variables such that $E(|X_n|)<\infty,\forall n\ge 1$ and $X_n \to X$ almost surely, where $E(|X|)<\infty$. Also assume $\lim_{n \to \infty} E(X_n)=E(X)$.

Then how to show that for any bounded random variable $Y$, $\lim_{n\to \infty} E(YX_n)=E(YX)$ ?

Answer 1

Since $Y$ is bounded, $|YX_n|\le MX_n$ for some $M>0$ and $\mathsf{E}MX_n\to \mathsf{E}MX$. The result then follows from the generalized dominated convergence theorem.