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
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.