Exercise: Let $X_i$ be a sequence of independent random variables with $\mathbb{E}[X_i] = 0$ and $\text{Var}(X_i) = \sigma_i^2$. Show that the sequence
$$ S_n = \sum_{i=1}^{n} (X_i^2 - \sigma_i^2) $$
is a martingale with respect to the sequence $X_i$.
Attempted solution: For $S_n$ to be a martingale with respect to the sequence $X_i$, we need to see the following:
- $S_n$ is $(X_1, \dots X_n)$-measurable and $\mathbb{E}[|S_n|] < \infty$
- $\mathbb{E}[S_{n+1} | X_1, \dots X_n ] = S_n $
There's a proposition that states: Let the $\sigma$-algebra $\mathcal{A}$ be generated by a finite partition $\mathcal{P} = \{A_1, \dots A_n \}$. Then the function $Y$ is $\mathcal{A}$-measurable if and only if $Y$ may be written as $$Y(\omega)= \sum_{i=1}^{n}y_i\textbf{1}_{A_i}(\omega),$$ some $y_i, 1 \leq i \leq n$; in other words, $Y$ is constant on each set $A_i$.
I don't know how to translate this in a rigorous way.
For $\mathbb{E}[|S_n|]$,
$$ \mathbb{E}[|S_n|] = \mathbb{E}[|\sum^n(X_i^2-\sigma_i^2)|] \leq \sum^n | \mathbb{E}[X_i^2-\sigma_i^2]| = \sum^n | \mathbb{E}\big[X_i^2 -\mathbb{E}[X_i^2] + \mathbb{E}[X_i]^2 \big] | = \sum^n | \mathbb{E}\big[X_i^2 -\mathbb{E}[X_i^2] \big] | = \dots$$
And at last, need to show that
$$ \mathbb{E}[S_{n+1} - S_n] = 0$$.