@article{ECP3341,
author = {Noga Alon and Ohad Noy Feldheim},
title = {A note on general sliding window processes},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {19},
year = {2014},
keywords = {k-factor, d-dependent, de Bruijn, Ramsey},
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.},
pages = {no. 66, 1-7},
issn = {1083-589X},
doi = {10.1214/ECP.v19-3341},
url = {http://ecp.ejpecp.org/article/view/3341}}