Joint dissociation and the corner tail of an array #
A dissociated array has no global randomness left to remember: the events readable from the entries
X (i, j) with both indices arbitrarily large are almost surely trivial.
The proof is Kolmogorov's. Fix a cutoff n past which the entries are measurable and read the
array from there on. The corners [n, n + k] × [n, n + k] and the tail block
[n + k + 1, ∞) × [n + k + 1, ∞) have disjoint index sets, so joint dissociation makes each corner
independent of the tail block, hence of arrayTail X, which the tail blocks contain. The corners
increase to the σ-algebra of all entries with both indices at least n, that is to
arrayTailFamily X n, which contains arrayTail X. So the array tail is independent of everything
readable above the cutoff, in particular of itself, and is therefore trivial.
The same argument run along the diagonal — corners {(i, i) : n ≤ i ≤ n + k} against the square
block over [n + k + 1, ∞) — makes the tail of the diagonal process arrayDiag X trivial as well,
and it asks only the diagonal entries beyond the cutoff to be measurable.
Two disjointness conditions per pair of blocks is what dissociation asks for, and arrayTail is
built to supply them: it is the corner tail, cutting both index axes at n. The row tail
⨅ n, blockSigma X ([n, ∞) × ℕ) is genuinely not trivial for a dissociated array — an array whose
rows are all one common i.i.d. random path is separately dissociated, and its row tail carries the
whole path.
The converse holds for a jointly exchangeable array: if the corner tail is trivial, the array is jointly dissociated.
The ergodic form of the Aldous--Hoover representation is the dissociated one, and this is the zero-one law separating it from the general form, together with its converse.
Main results #
TauCeti.Probability.JointlyDissociated.indep_arrayTailFamily_arrayTail— the array tail of a jointly dissociated array is independent of everything readable above the cutoff, and in particularTauCeti.Probability.JointlyDissociated.indep_arrayTail_selfof itself;TauCeti.Probability.JointlyDissociated.measure_eq_zero_or_one_of_arrayTail— the zero-one law: the tail σ-algebra of a jointly dissociated array is trivial;TauCeti.Probability.JointlyDissociated.indep_tailProcess_arrayDiag_selfandTauCeti.Probability.JointlyDissociated.measure_eq_zero_or_one_of_tailProcess_arrayDiag— the same argument along the diagonal, giving a trivial tail for the diagonal process under measurability of the diagonal entries alone;TauCeti.Probability.jointlyDissociated_of_forall_arrayTail_measure_eq_zero_or_one— the converse: a jointly exchangeable array with trivial corner tail is jointly dissociated;TauCeti.Probability.jointlyDissociated_iff_forall_arrayTail_measure_eq_zero_or_one— the two together, joint dissociation ↔ corner-tail triviality for a coordinatewise measurable jointly exchangeable array under a zero-or-probability measure.
References #
- D. Aldous, "Representations for partially exchangeable arrays of random variables", Journal of Multivariate Analysis 11 (1981), 581--598.
- O. Kallenberg, Probabilistic Symmetries and Invariance Principles, Springer, 2005, Chapter 7.
The converse adapts the representation-free strategy of Graphon/RelRestrictionIndependence.lean
(VertexTailTrivial.isDissociated and its private Lévy-downward factorization step) in
cameronfreer/graphon (Apache 2.0) at commit 175911f9d2e053f2a33d966658dfce0e4ae2811d; the
array-level assembly — cylinders of the block σ-algebras, the window shift by joint
exchangeability, and the π-system passage to Indep — is developed here. No material is adapted
from cameronfreer/exchangeability, which treats exchangeable sequences rather than exchangeable
arrays.
The tail σ-algebra of a jointly dissociated array is independent of everything readable
above the cutoff. The corners above n are independent of the tail and increase to the tail
family at n. Only entries beyond that cutoff need to be measurable.
The tail σ-algebra of a jointly dissociated array is independent of itself, the special
case of JointlyDissociated.indep_arrayTailFamily_arrayTail in which the left σ-algebra is cut
down to the array tail.
The zero-one law for a dissociated array. Every event in the tail σ-algebra of a jointly
dissociated array has probability 0 or 1.
The tail cuts both index axes, as dissociation requires: for the row tail alone the statement is false, an array all of whose rows are one common i.i.d. random path being dissociated with a row tail that carries the whole path.
The tail σ-algebra of the diagonal of a jointly dissociated array is independent of
itself. This is the corner argument of JointlyDissociated.indep_arrayTail_self run along the
diagonal: the diagonal corners {(i, i) : n ≤ i ≤ n + k} are read by the square block over
[n, n + k], the diagonal tail by the square block over [n + k + 1, ∞)
(tailProcess_arrayDiag_le_arrayTail), and the corners exhaust the diagonal tail. Only the
diagonal entries beyond the cutoff need to be measurable.
The diagonal of a jointly dissociated array has a trivial tail. Every event in the tail
σ-algebra of the diagonal process has probability 0 or 1. Unlike the zero-one law for the
array tail, this needs only the diagonal entries beyond the cutoff to be measurable.
From tail triviality back to dissociation #
Corner-tail triviality implies joint dissociation. A coordinatewise measurable, jointly
exchangeable array whose corner tail is μ-trivial is jointly dissociated.
Joint dissociation ↔ array-tail triviality for a coordinatewise measurable jointly exchangeable array under a zero-or-probability measure.