A note on general sliding window processes

Let f : R → [r] = {1, 2, . . . , r} be a measurable function, and let {Ui}i∈N be a sequence of i.i.d. random variables. Consider the random process Zi = f(Ui, ..., Ui+k−1). We show that for all q, there is a positive probability, uniform in f , that Z1 = Z2 = ... = Zq. A continuous counterpart is that if f : R → R, and Ui and Zi are as before, then there is a positive probability, uniform in f , for Z1, ..., Zq to be monotone. We prove these theorems, give upper and lower bounds for this probability, and generalize to variables indexed on other lattices. The proof is based on an application of combinatorial results from Ramsey theory to the realm of continuous probability.