Correlation Measures

We study a class of Borel probability measures, called correlation measures. Our results are of two types: first, we give explicit constructions of non-trivial correlation measures; second, we examine some of the properties of the set of correlation measures. In particular, we show that this set of measures has a convexity property. Our work is related to the so-called Gaussian correlation conjecture.


Introduction
In this article, we study a class of Borel probability measures on R d , which we call correlation measures.Our work is related to the so-called Gaussian correlation conjecture; to place our results in context, we will review this important conjecture.
Given x, y ∈ R d , let (x, y) and x denote the canonical inner product and norm on R d , respectively.As is customary, given A, B ⊂ R d and t ∈ R, we will write tA = {ta : a ∈ A} and A + B = {a + b : a ∈ A, b ∈ B}; the set A is said to be symmetric provided that −A = A and convex provided that tA + (1 − t)A ⊂ A for each t ∈ [0, 1].Let C d denote the set of all closed, convex, symmetric subsets of R d , and let γ d be the standard Gaussian measure on R d , that is, The Gaussian correlation conjecture states that for each pair of sets A, B ∈ C d , d ≥ 1.For d = 1, this conjecture is trivially true, and Pitt [5] has shown that it is true for d = 2.For d ≥ 3, the conjecture remains unsettled, but a variety of partial results are known.Borell [1] establishes (1.1) for sets A and B in a certain class of (not necessarily convex) sets in R d , which for d = 2 includes all symmetric, convex sets.The conjecture can be reformulated as follows for each 1 ≤ k < n.Khatri [4] and Šidák [7,8] have shown that (1.2) is true for k = 1.In part, the paper of Das Gupta, Eaton, Olkin, Perlman, Savage, and Sobel [2] generalizes the results of Khatri and Šidák for elliptically contoured distributions.
The recent paper of Schechtman, Schlumprecht and Zinn [6] sheds new light on the Gaussian correlation conjecture.Their results are of two types: first, they show that the conjecture is true whenever the sets satisfy additional geometric restrictions (additional symmetry, centered ellipsoids); second, they show that the conjecture is true provided that the sets are not too large.
Here is the central question of this article: to what extent is the correlation inequality (1.1) a Gaussian result?In other words, are there any non-trivial probability measures on R d satisfying (1.1)?We answer the question in the affirmative.
We will call a Borel probability measure λ on R d a correlation measure provided that for each pair of sets A, B ∈ C d ; we will denote the set of all correlation measures on R d by M d .
In section 2 we give sufficient conditions for membership in M d and show that M d contains non-trivial elements for each d ≥ 2. In section 3, we examine some properties of correlation measures.In particular, we show that non-trivial correlation measures have unbounded support, and that M d has a certain convexity property.Using this convexity property, we construct an element of M 2 based on a model introduced by Kesten and Spitzer [3].Our results can thus be roughly summarized as:

Measures
Correlation property bounded support no (except in dimension 1) exponential tail (including Gaussian) unknown heavy tail some examples known The correlation measures that we construct in section 2 are heavy-tailed, with the measure of the complement of the ball of radius r decaying only as a power of r.As our result of section 3 demonstrates, the measure of the complement of the ball of radius r must be positive for each r ≥ 0. Thus it is natural to ask whether there is a minimal rate with which the measure of the complement of the ball of radius r approaches 0. Perhaps the Gaussian measures lie close to, or on, the "boundary" of M d , which may account for the difficulty of the Gaussian correlation conjecture.

The construction of correlation measures
Throughout this section, µ will denote a spherically-symmetric, Borel probability measure on R d .For r ≥ 0, let The main result of this section is Theorem 2.2, which gives sufficient conditions on F for µ to be a correlation measure; through this result, we produce explicit, nontrivial correlation measures.
The proof of Theorem 2.2 rests on a geometric fact, which we describe presently.Let The number h = h(S) is called the half-width of S; when h = 0, S is a hyperplane of dimension d − 1.Let S d denote the set of all symmetric slabs in R d , and, for A ∈ C d , let

It is immediate that ρ(A) ≤ h(A); in fact, since A is convex and symmetric, ρ(A) = h(A).
Since A is closed, A ⊃ B[0, ρ(A)]; since S d−1 is compact, there exists a symmetric slab of half-width h(A) containing A. We can summarize these findings as follows: Let σ be uniform surface measure on S d−1 , normalized so that σ(S d−1 ) = 1.Since µ is spherically symmetric, we can represent µ in polar form: for any Borel subset This special function may be expressed as where Let S be a symmetric slab of finite half-width h, and let p ≥ h (p > 0).Then, by symmetry and scaling, Here is the main result of this section.
Our next result uses Corollary 2.3 to demonstrate the existence of non-trivial correlation measures in each dimension d ≥ 2.

Theorem 2.4
For each L ≥ 1, there exists a differentiable, concave, increasing function Proof Let This makes F differentiable, concave, and increasing on [0, ∞).For r ≥ 1, the left-hand side of (2.11) is For r ≤ 1, the left-hand side of (2.11) is as was to be shown.
When L = 1, another solution to (2.11) is given by F (r) = (r/(1 + r)) 1/2 , for which the inequality (2.11) becomes an equality.This function F is thus the best possible solution to (2.11) in that sense.

Some properties of correlation measures
Let µ denote a Borel probability measure on R d .As is customary, let the support of µ (denoted by supp(µ)) be the intersection of the closed subsets of R d having full measure.
In other words, unless a correlation measure is supported on a one-dimensional subspace, it must have unbounded support.
Proof Let x 0 ∈ supp(µ) have maximal distance from 0. Without loss of generality we may assume that x 0 = e 1 = (1, 0, . . ., 0).For ∈ (0, 1), let Our next result shows that M d remains closed under certain convex combinations.Let µ and λ be Borel probability measures on R d .We will say that µ dominates λ (written µ λ)  In general, a linear combination of correlation measures need not be a correlation measure.
For example, let µ and λ be the centered Gaussian measures on R 2 with covariance matrices respectively.By the theorem of Pitt [5], µ and λ are correlation measures; however, the measure m = (µ + λ)/2 is not a correlation measure.To see this, let Then, by a calculation as in the proof of Theorem 3.2, m(A ∩ B) − m(A)m(B) < 0, which shows that m / ∈ M 2 .Theorem 3.2 be extended by induction:

and let {a
Dominating measures can be constructed through scaling.Given µ ∈ M d and s > 0, let µ s (A) = µ(sA) for each Borel subset of R d .If r ≥ s, then rA ⊃ sA for each A ∈ C d ; thus, µ r µ s .We will use this notion of domination through scaling in conjunction with Corollary 3.3 to construct elements of M 2 .
Let {S n : n ≥ 0} (S 0 = 0) be simple random walk on Z, and let {Y (k) : k ∈ Z} be a sequence of independent and identically distributed, two-dimensional, standard Gaussian random vectors.We will assume that the random walk and the Gaussian vectors are defined on a common probability space and generate independent independent σ-algebras.For n ≥ 0, let The process {Z n : n ≥ 0}, called random walk in random scenery, was introduced by Kesten and Spitzer [3], who investigated its weak limits.Theorem 3.4 For each n ≥ 0, the law of Z n is an element of M 2 .
Proof For n ≥ 0, let ζ n denote the law of Z n .For j ∈ Z and n ≥ 0, let The process {V n : n ≥ 0} is called the self-intersection local time of the random walk.Conditional on the σ-field generated by the random walk, Z n is a Gaussian random vector with covariance matrix V n times the identity matrix.Thus, for each Borel set A ∈ R 2 , By the theorem of Pitt [5], the measures {γ 2 (k −1/2 • ) : k ≥ 1} are in M 2 , and, by scaling, the measures can be ordered by domination; thus, by Corollary 3.3, ζ n is in M 2 , as was to be shown.
k = j)and observe that Z n = j∈Z j n Y (j).For n ≥ 0, let simpler form of this result can be obtained by strengthening the conditions on F .Let L 2 = 1 and, for d ≥ 3, let L d = K d .With this convention, dF (t), which, according to (2.5), is nonnegative.As such, µ(A ∩ B) ≥ µ(A)µ(B), as was to be shown.A