A note on tail triviality for determinantal point processes

Russell Lyons

doi:10.1214/18-ecp175

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.

Highlights

We give a very short proof that determinantal point processes have a trivial tail σ-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
Osada and Osada relied on the earlier result of Lyons [3] that the conjecture held in the discrete case, as does the present short proof
In the discrete case and under the restrictive assumption that the spectrum of K is contained in the open interval (0, 1), Shirai and Takahashi [7] proved that the tail σ-field is trivial

Summary

Introduction

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. We give a very short proof that determinantal point processes have a trivial tail σ-field. Osada and Osada relied on the earlier result of Lyons [3] that the conjecture held in the discrete case, as does the present short proof.

Results

Conclusion