A Note on Tail Triviality for Determinantal Point Processes

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.


Goldman's Transference Principle
We review some definitions. See [4] for more details.
Let E be a locally compact Polish space (equivalently, a locally compact second countable Hausdorff space). Let µ be a Radon measure on E, i.e., a Borel measure that is finite on compact sets. Let N (E) be the set of Radon measures on E with values in N ∪ {∞}. We give N (E) the vague topology generated by the maps ξ → f dξ for continuous f with compact support; then N (E) is Polish. The corresponding Borel σ-field of N (E) is generated by the maps ξ → ξ(A) for Borel A ⊆ E.
Let X be a simple point process on E, i.e., a random variable with values in N (E) such that X({x}) ∈ {0, 1} for all x ∈ E. We call X determinantal if for some measurable K : E 2 → C and all k ≥ 1, the function (x 1 , . . . , x k ) → det[K(x i , x j )] i,j≤k is a k-point intensity function of X. In this case, we denote the law of X by P K .
We consider only K that are locally square integrable (i.e., |K| 2 µ 2 is Radon), are Hermitian (i.e., K(y, x) = K(x, y) for all x, y ∈ E), and are positive semidefinite. In this case, K defines a positive semidefinite integral operator (Kf )(x) := K(x, y)f (y) dµ(y) on functions f ∈ L 2 (µ) with compact support. We consider K as defined only up to changes on a µ 2 -null set. For every Borel A ⊆ E, we denote by µ A the measure µ restricted to Borel subsets of A and by K A the compression of K to A, i.e., K A f := (Kf ) A for f ∈ L 2 (A, µ A ). If X has law P K , then the restriction of X to A has law P K A . The operator K is locally trace-class, i.e., for every compact A ⊆ E, the compression K A is trace class, having a spectral decomposition The following extends Goldman's transfer principle from trace-class operators, as given in [4,Section 3.6], to locally trace-class operators: Theorem 1.1. Let µ be a Radon measure on a locally compact Polish space, E. Let K be a locally trace-class positive contraction on L 2 (E, µ). Let A i ; i ≥ 1 be a partition of E into precompact Borel subsets of E. Then there exists a denumerable set F with a partition B i ; i ≥ 1 and a positive contraction Q on 2 (F ) such that the joint distribution of the random variables X(A i ) ; i ≥ 1 for X ∼ P K equals the joint P Q -distribution of the random variables X(B i ) ; i ≥ 1 for X ∼ P Q . Moreover, we can choose Q to be unitarily equivalent to K.
Proof. For each i, fix an orthonormal basis w i,j ; j < n i for the subspace of L 2 (E, µ) of functions that vanish outside A i . Here, n i ∈ N ∪ {∞}. Define B i := {(i, j) ; j < n i } and F := i B i . Let T be the isometric isomorphism (i.e., unitary map) from L 2 (E, µ) to 2 (F ) that sends w i,j to 1 {(i,j)} . Define Q := T KT −1 , so Q is unitarily equivalent to K. Note that for all φ ∈ L 2 (E) and all i ≥ 1, Then K Em and Q Fm are unitarily equivalent trace-class operators. If φ k,m ; k ≥ 1 are orthonormal eigenvectors of K Em , so that K Em = k λ Em k φ k,m ⊗ φ k,m , then Q Fm = k λ Em k T φ k,m ⊗ T φ k,m . Furthermore, for all φ, ψ ∈ L 2 (E, µ) and all i ≥ 1, we have (1 Ai φ, ψ) L 2 (E,µ) = (T 1 Ai φ, T ψ) 2 (F ) = (1 Bi T φ, T ψ) 2 (F ) . Thus, [4,Theorem 3.4] shows that the P K Em -distribution of X(A i ) ; i ≤ m equals the P Q Fm -distribution of X(B i ) ; i ≤ m . But these are precisely the P Kdistribution of X(A i ) ; i ≤ m and the P Q -distribution of X(B i ) ; i ≤ m , respectively. Because these are equal for all m ≥ 1, the desired result follows.

Tail Triviality: Deduction from the Discrete Case
For a Borel set A ⊆ E, let F (A) denote the σ-field on N (E) generated by the functions ξ → ξ(B) for Borel B ⊆ A. The tail σ-field is the intersection of F (E \ A) over all compact A ⊆ E; it is said to be trivial when each of its events has probability 0 or 1. For a collection A of Borel subsets of E, write G (A ) for the σ-field generated by the functions ξ → ξ(B) for B ∈ A . Theorem 2.1 (conjectured by [4], proved by [5,1]). If K is a locally trace-class positive contraction, then P K has a trivial tail σ-field. m ) converges in L 1 to 1 A as m → ∞. In particular, if A is a tail event, then there is a sequence m n → ∞ such that P(A | G (n) mn ) converges in L 1 to 1 A as n → ∞. It follows that A belongs to the completion of the σ-field n≥k G (n) mn for each k ≥ 1. Now let A be a tail event and m n ; n ≥ 1 be such a sequence. Let C := C k ; k ≥ 1 be the parts of the partition of E generated by {A mn,i ; A mn,i ∩ D (n) = ∅, n ≥ 1, i ≥ 1}. Write H n := G {C k ; k ≥ n} . Then A belongs to the completion of the σ-field H n for each n ≥ 1, whence P(A | n≥1 H n ) = lim n→∞ P(A | H n ) = 1 A a.s. by Lévy's downwards theorem. By Theorem 1.1, there is a partition B k ; k ≥ 1 of a denumerable set F and a positive contraction Q on 2 (F ) such that the P Q -distribution of X(B k ) ; k ≥ 1 equals the P K -distribution of X(C k ) ; k ≥ 1 . Let H n := G {B k ; k ≥ n} . Then n≥1 H n is contained in the tail σ-field B finite F (F \ B). Since the latter is trivial by [3,Theorem 7.15], so is the former. Therefore, so is n≥1 H n , whence A has probability 0 or 1.