A note on general sliding window processes

Abstract: Let $f:\mathbb{R}^k\to \mathbb{R}$ be a measurable function, and let ${U_i}{i\in\mathbb{N}}$ be a sequence of i.i.d. random variables. Consider the random process $Z_i=f(U{i},...,U_{i+k-1})$. We show that for all $\ell$, there is a positive probability, uniform in $f$, for $Z_1,...,Z_\ell$ to be monotone. We give upper and lower bounds for this probability, and draw corollaries for $k$-block factor processes with a finite range. The proof is based on an application of combinatorial results from Ramsey theory to the realm of continuous probability.