Questions Concerning Martingales and Uniform Integrability

Question

This is Exercise 5.16 from Lawler's Introduction to Stochastic Processes:

Let $X_1,X_2,\cdots$ be independent, identically distributed random variables with mean $\mu$. Let $T$ be a stopping time with respect to $X_1,X_2,\cdots$ with $\Bbb{E}(T)<\infty$.

1. Let $$Y=\sum_{n=1}^{\infty}|X_n|I\{T \geq n\},$$ where $I$ denotes the indicator function. Show that $\Bbb{E}(Y)<\infty$.

2. Let $T_n=\min\{n,T\}$ and $$M_n=X_1+\cdots+X_{T_n}-\mu T_n.$$ Explain why $M_n$ is a uniformly integrable martingale.

3. Prove Wald's equation, $$\Bbb{E}\left(\sum_{n=1}^T X_n\right)=\mu \Bbb{E}(T).$$

4. Suppose $\{\mathcal{F}_n\}$ is a filtration such that $X_n$ is $\mathcal{F}_n$-measurable and for $m>n$, $X_m$ is independent of $\mathcal{F}_n$ (i.e., $X_m$ is independent of every $\mathcal{F}_n$-measurable random variable.) Suppose that $T$ is a stopping time with respect to $\{\mathcal{F}_n\}$. (In other words, more information than $X_1,\cdots,X_n$) is used to determine whether to stop at time $n$. However, any additional information used is independent of $X_{n+1},X_{n+2},\cdots$). Show that (1) through (4) still hold.

I can finish the first three parts, but I am completely confused by the last part's description. Is there any relationship between the fourth part and the others? (I mean what's the differences from the ordinary conditions) And a way to figure it out? Thanks for any help!

Answer 1

Parts 1, 2 and 3 where done under the assumption that $\mathcal F_n$ is the $\sigma$-algebra generated by $X_1,\dots,X_n$. Part 4 proposes to extend this to more general filtrations, with the described constraints (for example, $\mathcal F_n$ is generated by $Y,X_1,\dots,X_n$, where $Y$ is independent of $(X_i)_{i\geqslant 1}$. For example, for part one, we write $$ \mathbb EY=\sum_{n=1}^{+\infty}\mathbb E\left[\lvert X_n\rvert\mathbf 1\{T\geqslant n\}\right], $$ then $$ \mathbb E\left[\lvert X_n\rvert\mathbf 1\{T\geqslant n\}\right]=\mathbb E\left[\mathbb E\left[\lvert X_n\rvert\mathbf 1\{T\geqslant n\}\mid\mathcal F_n\right]\right], $$ use the fact that $\lvert X_n\rvert$ is $\mathcal F_n$-measurable and that the event $\{T\geqslant n\}$ is $\mathcal F_{n-1}$-measurable hence independent of $\lvert X_n\rvert$.