Let take $(X_n)$ be a martingale associated to a given filtration $\left(\mathcal{F}_n\right)$. Set $Y_n=X_{n+1}-X_{n}$.
What we can say about the distribution of $\sum X_n Y_n$ ? Can we get an analogy of Hoeffding's inequality for this special case?
My question is vague but any idea is welcomed.
Thanks in advanced
Best Answer
It turns out that the sequence $(X_nY_n)_{n\geqslant 1}$ is a martingale difference sequence with respect to the filtration $(\mathcal F_{n+1})_{n\geqslant 1}$, provided that each $X_n$ is square integrable. Indeed, observe that $X_nY_n$ is $\mathcal F_{n+1}$-measurable, as a product of two of such random variables. Moreover, Cauchy-Schwarz inequality show that $X_nY_n$ is integrable and the pull-out property of conditional expectation show that $\mathbb E\left[X_nY_n\mid\mathcal F_{n+1-1}\right]=0$.
Then one can use Hoeffding's inequality if $X_n$ is bounded, and more generally all the inequalities for martingales to find upper bounds for the distribution function of $\sum_{n=1}^N X_nY_n$. However, it seems too ambitious to hope for an explicit computation of the distribution function.