Suppose we are given a probability space $(\Omega, \mathcal{F},P)$ s.t. r.v.s $X$ and $(Y_i)_{i=1}^\infty$ are $\mathcal{F}$-measurable. The relevant filtration is given by $\mathcal{F}_n=\sigma((Y_i)_{i=1}^n,X)\subset \mathcal{F}$. Also let $M_n=\sum_{i=1}^n f(Y_i,X)$ for some measurable $f$. The paper I read claims that under certain conditions on $f$, $M_n$ is $(\mathcal{F}_n)$-martingale under conditional probability $P(\cdot\mid X)$. However, I'm not sure how to interpret the martingale condition in this case, i.e.
$$\mathbb{E}_{P(\cdot\mid X)}[f(Y_n,X)\mid \mathcal{F}_{n-1}]=0.$$
Edit:
I found a related concept - conditional martingale - described in this paper. It seems to match my case. However, there is no much literature on the topic...