A note on general sliding window processes

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 $Z_i=f(U_i,\mathrm{.}\mathrm{.}\mathrm{.},U_{i+k-1})$. We show that for all $\mathrm{\ell }$, there is a positive probability, uniform in $f$, for $Z_1,\mathrm{.}\mathrm{.}\mathrm{.},Z_{\mathrm{\ell }}$ 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.