Concentration inequality for $k$-dependent Bernoulli r.v.s

Question

Given $X_1,X_2, \cdots$ are iid $Ber(p)$, and we define $$Z_1 = X_1X_2\cdots X_k\\ Z_2 = X_2 X_3 \cdots X_{k+1}\\ \cdots$$ Is there a concentration inequality (like Hoeffding's inequality) for $$\mathbb{P}\bigg(\sum_{i=1}^n Z_i - \mathbb{E}\Big[\sum_{i=1}^n Z_i\Big]\geq t\bigg) \leq e^{(\cdots)}?$$

Answer 1

Theorem 1.16 of this paper https://arxiv.org/abs/1507.06871 applies to your situation. To summarize, the authors establish a concentration inequality for block factors of iid, of which your example is a special case. More precisely, a $k$-block factor of iid is a process $(Z_i)$ for which there exists a function $F\colon \mathbb R^k\to\mathbb R$ and an iid sequence $(X_i)$ such that $Z_i=F(X_i,X_{i+1},\ldots,X_{i+k-1})$. In your case, $F(x_1,\ldots,x_k)=x_1\cdots x_k$.