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A note on general sliding window processes. (English) Zbl 1300.60043
Summary: Let $$f:\mathbb{R}^k\to[r]=\{1,2,\dots,r\}$$ 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 $$\{Z_i\}_{i \in \mathbb N}$$ defined by $$Z_i=f(U_{i},\dots{},U_{i+k-1})$$. We show that, for all $$q$$, there is a positive probability, uniform in $$f$$, that $$Z_1=Z_2=\dots=Z_q$$. A continuous counterpart is that, if $$f:\mathbb R^k \to \mathbb R$$, and $$U_i$$ and $$Z_i$$ are as before, then there is a positive probability, uniform in $$f$$, for $$Z_1,\dots,Z_q$$ 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.

##### MSC:
 60G07 General theory of stochastic processes 60C05 Combinatorial probability
##### Keywords:
$$k$$-factor; $$d$$-dependent; de Bruijn; Ramsey
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