Electronic Communications in Probability

A note on tail triviality for determinantal point processes

Russell Lyons

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Abstract

We give a very short proof that determinantal point processes have a trivial tail $\sigma $-field. This conjecture of the author has been proved by Osada and Osada as well as by Bufetov, Qiu, and Shamov. The former set of authors relied on the earlier result of the present author that the conjecture held in the discrete case, as does the present short proof.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 72, 3 pp.

Dates
Received: 17 July 2018
Accepted: 2 October 2018
First available in Project Euclid: 17 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1539763345

Digital Object Identifier
doi:10.1214/18-ECP175

Mathematical Reviews number (MathSciNet)
MR3866045

Zentralblatt MATH identifier
06964415

Subjects
Primary: 60K99: None of the above, but in this section
Secondary: 60G55: Point processes

Keywords
transference principle

Rights
Creative Commons Attribution 4.0 International License.

Citation

Lyons, Russell. A note on tail triviality for determinantal point processes. Electron. Commun. Probab. 23 (2018), paper no. 72, 3 pp. doi:10.1214/18-ECP175. https://projecteuclid.org/euclid.ecp/1539763345


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References

  • [1] Alexander I. Bufetov, Yanqi Qiu, and Alexander Shamov, Kernels of conditional determinantal measures and the Lyons–Peres conjecture, (2016), Preprint, arXiv:1612.06751.
  • [2] Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.
  • [3] Russell Lyons, Determinantal probability measures, Publ. Math. Inst. Hautes Études Sci. 98 (2003), no. 1, 167–212.
  • [4] Russell Lyons, Determinantal probability: Basic properties and conjectures, Proceedings of the International Congress of Mathematicians. Volume IV (Sun Young Jang, Young Rock Kim, Dae-Woong Lee, and Ikkwon Yie, eds.), Kyung Moon Sa, Seoul, 2014, Invited lectures, Held in Seoul, August 13–21, 2014, pp. 137–161.
  • [5] Hirofumi Osada and Shota Osada, Discrete approximations of determinantal point processes on continuous spaces: Tree representations and tail triviality, J. Stat. Phys. 170 (2018), no. 2, 421–435.
  • [6] Hirofumi Osada and Hideki Tanemura, Infinite-dimensional stochastic differential equations and tail $\sigma $-fields, (2014), Preprint, arXiv:1412.8674.
  • [7] Tomoyuki Shirai and Yoichiro Takahashi, Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes, J. Funct. Anal. 205 (2003), no. 2, 414–463.