Prove that $\lim_{n\rightarrow \infty} \mathbb{E}(YX_n) = 0$ for uncorrelated $(X_n)$ with $0$ mean, $\rm{Var}(X_i)=1, Cov(X_i,X_j)=0$ and bounded $Y$

Question

Let $(X_n)$ be a sequence of uncorrelated random variables (i.e., $\operatorname{Cov}(X_i,X_j) = 0$ for i , j) with expectation $0$ and variance $1$. Prove that for any bounded random variable $Y$, we have $\lim_{n\rightarrow\infty} E[X_nY] = 0$. Hint: With $\alpha_n := E[X_nY]$, first consider $E(Y-\sum_{k=1}^na_kX_k)^2$.

I've derived that $E(Y-\sum_{k=1}^na_kX_k)^2 = E(Y^2) - \sum_{k=1}^n a_k^2$ and I know that from Cauchy Schwartz that $\operatorname{Cov}(Y,X_n) = E(X_nY)\leq \operatorname{Var}(Y)\leq \infty$ but I don't know how to proceed.

Answer 1

We have $0\leq E(Y-\sum_{k=1}^na_kX_k)^2 = E(Y^2) - \sum_{k=1}^n a_k^2$ and this holds for all $n$. Hence, $\sum\limits_{k=1}^{\infty} a_k^{2} \leq EY^{2} <\infty$. Convergence of the series $\sum_k a_k^{2}$ implies that $a_k \to 0$ which is what want to prove.