Abstract
We prove that every negatively associated sequence of Bernoulli random variables with “summable covariances” has a trivial tail σ-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, in general, they do not 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.
Acknowledgments
The authors would like to thank Jeffrey Steif for his comments on earlier versions of this paper and suggesting the example at the end of the Introduction, which shows that tail triviality does not follow from pairwise negative correlations even under strong stationarity. We also wish to thank Yuzhou Gu who pointed out a mistake in an earlier version of this paper. Finally, we thank the anonymous referees for their insightful comments and suggestions, which significantly improved this article.
Citation
Kasra Alishahi. Milad Barzegar. Mohammadsadegh Zamani. "On tail triviality of negatively dependent stochastic processes." Ann. Probab. 51 (4) 1548 - 1558, July 2023. https://doi.org/10.1214/23-AOP1626
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