Electronic Communications in Probability

A note on general sliding window processes

Noga Alon and Ohad Noy Feldheim

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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465316768

Digital Object Identifier
doi:10.1214/ECP.v19-3341

Mathematical Reviews number (MathSciNet)
MR3262072

Zentralblatt MATH identifier
1300.60043

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

Keywords
k-factor d-dependent de Bruijn Ramsey

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

Citation

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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