Expected Value of Product of Sequences of Random Variables

Question

I am trying to prove the following:

Let $X_{n}$ be a sequence of random variables converging in probability to some random variable $X$. Furthermore $P(|Xn|>k)=0$ for all n and some $k>0$.

Let $Y_{n}$ be a sequence in $L^{1}(\Omega)$. Assume that there exists a real number $\lambda$ such that $E(Y_{n}) = \lambda$ for all n and $\sup |Y_{n}| \le \eta$ for some $ \eta \in L^{1}(\Omega) $.

Prove that if $X=c$ then $lim_{n \to \infty} E(Y_{n}X_{n})=c \lambda$.

How do I show this if the sequences are not independent?

Answer 1

Here's something to get you started. $$\begin{align}|E(Y_nX_n) - c \lambda| &= |E(Y_nX_n) - c E(Y_n)| = |E(Y_n(X_n - c))| \\ &\leq |E(Y_n(X_n - c)\mathbf{1}_{|X_n - c| \geq \epsilon})| + |E(Y_n(X_n - c)\mathbf{1}_{|X_n - c| < \epsilon})| \\ &\leq E(|\eta| |X_n - c|\mathbf{1}_{|X_n - c| \geq \epsilon}) + \epsilon |\lambda| \end{align}$$

Answer 2

First,

$$ \mathbb{E}[Y_nX_n]=\mathbb{E}[Y_n(X_n-c)]+c\lambda. $$

However, since $\{X_n\}$ is uniformly bounded (by your comment), the DCT implies that

$$ |\mathbb{E}[Y_n(X_n-c)]|\le \mathbb{E}\eta|X_n-c|\to 0 \quad\text{as } n\to \infty. $$