On Tail Triviality of Negatively Dependent Stochastic Processes

TL;DR

This paper proves tail triviality for negatively associated Bernoulli sequences with summable covariances and extends the result to Gaussian processes, revealing new classes of negatively associated measures.

Contribution

It generalizes tail triviality results to broader classes of negatively associated processes, including Gaussian and Gaussian threshold processes, and introduces negatively associated Gaussian vectors outside the strongly Rayleigh class.

Findings

Negatively associated Bernoulli sequences with summable covariances have trivial tail sigma-fields.

Gaussian and Gaussian threshold processes are tail trivial even without summable covariances.

Constructed negatively associated Gaussian vectors that are not strongly Rayleigh.

Abstract

We prove that every negatively associated sequence of Bernoulli random variables with "summable covariances" has a trivial tail sigma-field. A corollary of this result is the tail triviality of strongly Rayleigh processes. This is a generalization of a result due to Lyons which establishes tail triviality for discrete determinantal processes. We also study the tail behavior of negatively associated Gaussian and Gaussian threshold processes. We show that these processes are tail trivial though they do not in general satisfy the summable covariances property. Furthermore, we construct negatively associated Gaussian threshold vectors that are not strongly Rayleigh. This identifies a natural family of negatively associated measures that is not a subset of the class of strongly Rayleigh measures.