Weaker than martingale condition

Question

Let $\mathcal{F}_n$ be a filtration and $S_n$ be a sequence such that $\mathbb{E}[S_n-S_{n-1}|\mathcal{F}_{n-2}]=0$ for all $n$. This condition is similar to the martingale condition but the conditional expectation is with respect to $\mathcal{F}_{n-2}$ instead of $\mathcal{F}_{n-1}$. Clearly if $S_n$ is a martingale with respect to $\mathcal{F}_{n}$ then $\mathbb{E}[S_n-S_{n-1}|\mathcal{F}_{n-2}]=0$. Has this type of sequences been studied in the martingale literature? Any comments/suggestions would be greatly appreciated.

Answer 1

Let $D_n:=S_n-S_{n-1}$ for $n\geqslant 2$ and $D_1=S_1$. It is temping to want to work with martingales in order to study the properties of $\sum_{n=1}^ND_n$ and one can have the feeling that we are not too far away.

The idea is to add and substract an appropriated term. Indeed, define $$ D'_n=D_n-\mathbb E\left[D_n\mid\mathcal F_{n-1}\right]+\mathbb E\left[D_{n+1}\mid\mathcal F_{n}\right], $$ $$ C_n=\mathbb E\left[D_{n+1}\mid\mathcal F_{n}\right]. $$ Then $D_n=D'_n+C_{n-1}-C_n$ and $\left(D'_n\right)_{n\geqslant 1}$ is a martingale differences sequences with respect to the filtration $\left(\mathcal{F}_n\right)_{n\geqslant 0}$. Moreover, $$ \sum_{n=1}^ND_n=\sum_{n=1}^ND'_n+C_0-C_N.$$

Note that this can be easily generalized to the case where there exists a $k_0>1$ such that for each $n$, $\mathbb E\left[S_n-S_{n-1}\mid\mathcal F_{n-k_0}\right]=0$ almost surely.

More generally, it is possible to work with sequences $(S_n,\geqslant 1)$ by giving a control on the norms or the tails of the random variables $\mathbb E\left[S_n-S_{n-1}\mid\mathcal F_{n-k}\right]=0$, $k,n\geqslant 1$. The keyword to find results in the literature in this direction is martingale approximation.