Let $(\Omega,\mathscr{F},P)$ be a probability space and $\{X_n\}_{n\geq 1}$ be a stochastic process. Assume each $X_n$ only takes two values $0$ or $1$, i.e., $X_n:\Omega\rightarrow \{0,1\}$. Of course if $\{X_n\}$ are independently and identically distributed, then this is just Bernoulli trial.
Instead, assume this process is not necessarily i.i.d. But it satisfies $$P\big(X_{n+1}=1\big|X_1,X_2,\cdots, X_n\big)=P\big(X_{n+1}=1\big|\sum_{l=1}^n X_l\big).$$ That is the distribution of $X_{n+1}$ given $X_1,\cdots,X_n$ only depends on the sum of $X_1,\cdots,X_n$.
I think if we write $S_n=\sum_{l=1}^n X_n$, then $\{S_n\}$ is just an inhomogeneous Markov chain, with initial state $S_0\equiv 0$. My question is whether there is a formal name for this process, or whether there is some study on the asymptotic behavior of this process. Thanks!