While reading about random walks, I started thinking about this and got a headache:
Given a random process $\{X_n\}_{n\in \mathbb{Z}^+}$ with a real state space (i.e., $X_n$ takes on real numbers), what can you say about these two expressions?
$$\begin{align*}
\lim_{n\rightarrow\infty} \mathbb{E}\left[X_n\mathop{\big|}X_0=j\right] & \quad (1)
\\ \mathbb{E}\left[\lim_{n\rightarrow\infty} X_n\mathop{\big|} X_0 = j\right] & \quad(2)
\end{align*}$$
(or also considering omitting the conditional statement on $X_0$… I'm not sure whether it matters or not).
I did some Googling, and I found something called Fatou's lemma, but I have no background in measure theory, and most of the Wikipedia article is way over my head.
My questions
- When can you pass the limit in and out (equality), and when is it an inequality (Fatou's lemma)?
- Does anything change if $\{X_n\}$ is a Markov chain?
- What if it was a continuous process $\{X_t\}_{t\in\mathbb{R}^+}$ instead?
- How do you interpret (explain in words) what those two expressions mean and what the difference is between them?
- Is there any causal relationship between existence of one limit and the other? For example, does the existence of the limit in $(1)$ imply that the limits in $(2)$ all exist?
- Besides measure theory and probability theory, are there any other foundational topics/subjects/fields that I could look into to learn more about this?
I'm sorry if my questions don't make much sense or if I am using terminology incorrectly; I am not very familiar with this material, and would like to learn more about it. Any helpful information would be much appreciated.
Best Answer
Try Lebesgue dominated convergence theorem: if $X_n\to X$ almost surely with respect to the probability measure $\mathbb P_j(\ )=\mathbb P(\ \mid X_0=j)$, and if $|X_n|\leqslant Y$ almost surely and for every $n$, with $\mathbb E_j(Y)$ finite, then $\lim\limits_{n\to\infty}\mathbb E_j(X_n)=\mathbb E_j(X)$, that is, $$\lim\limits_{n\to\infty}\mathbb E(X_n\mid X_0=j)=\mathbb E\left(\lim\limits_{n\to\infty}X_n\,{\Large\mid}\, X_0=j\right).$$