### Abstract

Let f: R^{k}→[r] = {1, 2,…,r} be a measurable function, and let {U_{i}}_{i∈N} be a sequence of i.i.d. random variables. Consider the random process {Z_{i}}_{i∈N} defined by Z_{i} = f(U_{i},…,U_{i+k-1}). We show that for all q, there is a positive probability, uniform in f, that Z_{1} = Z_{2} = … = Z_{q}. A continuous counterpart is that if f: R^{k} → R, and U_{i} and Z_{i} are as before, then there is a positive probability, uniform in f, for Z_{1},…,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.

Original language | English (US) |
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Journal | Electronic Communications in Probability |

Volume | 19 |

DOIs | |

State | Published - Sep 22 2014 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- D-dependent
- De Bruijn
- K-factor
- Ramsey

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## Cite this

*Electronic Communications in Probability*,

*19*. https://doi.org/10.1214/ECP.v19-3341