Show that $\{X_n\}_{n\ge 1}$ is a submartingale with respect to $\{F_n\}_{n\ge 1}$, where $X_n=\left(Z_1+Z_2+...+Z_n\right)^2$

Question

I am trying to do the following exercise from a past exam paper and I am really stuck in it. I know the theory and can prove other cases, but I am not too sure about this one. Any help would be greatly appreciated, thanks a lot!

QUESTION Suppose that $X_n=\left(Z_1+Z_2+...+Z_n\right)^2$, for every $n\ge 1$, where the $Z_i$'s are independent and identically distributed standard normal random variables. Show that $\{X_n\}_{n\ge 1}$ is a submartingale with respect to $\{F_n\}_{n\ge 1}$, where $F_n=\sigma\left(Z_1, Z_2,..., Z_n\right)$ for every $n\ge 1$.

What I know

So I need to show that $$\mathbb{E}[X_{n+1}|F_n]\ge X_n$$

I tried with the following but got lost. $$\mathbb{E}[X_{n+1}|F_n]\ge X_n \implies \mathbb{E}[X_{n+1}-X_n|F_n]\ge 0$$ Then we have, if we let $S_n=\sum_{i=1}^n Z_i$, $$\mathbb{E}[X_{n+1}-X_n|F_n]=\mathbb{E}[Z_{n+1}\left(2S_n+Z_{n+1}\right)|F_n]$$

But I am stuck here. How do I prove this? Thanks a lot!

Answer 1

$$X_{n+1}=(Z_1+\cdots+Z_n+Z_{n+1})^2=X_n+2Z_{n+1}(Z_1+\cdots+Z_n)+Z_{n+1}^2.$$

Now use the fact that expectation is linear, and that $Z_{n+1}$ is independent of $F_n$, along with the fact that $E[Z_{n+1}]=0$, $E[Z_{n+1}^2]>0$.

Answer 2

Denote $S_n = Z_1 + \cdots + Z_n$. $S_n$ is easily shown to be a martingale. Now $$ \mathbb E[X_{n+1} \mid \mathcal F_n] =\mathbb E[S_{n+1}^2 \mid \mathcal F_n] \geq (\mathbb E[S_{n+1} \mid \mathcal F_n])^2 = S_n^2 = X_n $$ where the inequality follows from Jensen's inequality.