Electronic Communications in Probability

A note on general sliding window processes

Noga Alon and Ohad Noy Feldheim

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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.

Article information

Electron. Commun. Probab., Volume 19 (2014), paper no. 66, 7 pp.

Accepted: 22 September 2014
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G07: General theory of processes
Secondary: 60C05: Combinatorial probability

k-factor d-dependent de Bruijn Ramsey

This work is licensed under a Creative Commons Attribution 3.0 License.


Alon, Noga; Feldheim, Ohad Noy. A note on general sliding window processes. Electron. Commun. Probab. 19 (2014), paper no. 66, 7 pp. doi:10.1214/ECP.v19-3341. https://projecteuclid.org/euclid.ecp/1465316768

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