Lebesgue Dominated Convergence Theorem Application

Question

Let $(X_i)_{i\geq 1}$ be a sequence of random variables and $W=|X_1|+\Sigma_{i=1}^∞ |X_{i+1}-X_{i}|$. If $E(W)<\infty$, show that $\lim_{x\to\infty} E(X_n) = E(\lim_{x\to\infty} X_n)$.

I'm thinking of using Lebesgue Dominated Convergence Theorem, but I don't know how to start it.

Answer 1

For every $n$, you have that $\vert X_n\vert \leq \sum_{i=1}^{n-1}\vert X_{i+1}-X_i\vert+\vert X_1\vert$. And so, since $\mathbb{E}\big( \vert X_n \vert \big)\leq E(W)$ and $W\in L^1$, you have a dominating function for the $\{ X_n \}$.