INEQUALITIES OF CORRELATION TYPE FOR SYMMETRIC STABLE RANDOM VECTORS

A. L. Koldobsky and S. J. Montgomery–Smith

Abstract. We point out a certain class of functions f and g for which random variables f(X_{1}, …, X_{m}) and g(X_{m+1}, …, X_{k}) are non-negatively correlated for any symmetric jointly stable random variables X_{i}. We also show another result that is related to the correlation problem for Gaussian measures of symmetric convex sets.

1. Introduction

For 0 < q ≤ 2, let Y be a symmetric q-stable random vector in ℝ^{n} with characteristic function

(1)  ϕ(θ) = exp(−∥ ∑_{i=1}^{n} θ_{i}s_{i}∥^{q}), θ ∈ ℝ^{n},

where s_{1}, …, s_{n} ∈ L_{q}([0, 1]), and the norm is taken from the space L_{q}([0, 1]).

For any k ∈ ℕ, and any choice of vectors ξ_{1}, …, ξ_{k} ∈ ℝ^{n}, the inner products X_{1} = (Y, ξ_{1}), …, X_{k} = (Y, ξ_{k}) are symmetric q-stable random variables. The random variables X_{1}, …, X_{k} are jointly q-stable with zero mean, and we say that they are ℝ^{n}-generated in case we need to emphasize the dimension of the vector Y.

In this article, we show that, for any m < k, and any even continuous positive definite functions f and g on ℝ^{m} and ℝ^{k−m} respectively, the random variables f(X_{1}, … X_{m}) and g(X_{m+1}, … X_{k}) are non-negatively correlated, i.e.

(2) 𝔼(f(X_{1}, …, X_{m}) g(X_{m+1}, …, X_{k})) ≥ 𝔼f(X_{1}, … X_{m}) 𝔼g(X_{m+1}, …, X_{k}),

where 𝔼 stands for the expectation.

Inequality (2) reminds one of some results related to the concept of associated random variables. Recall that random variables X_{1}, …, X_{k} are said to be associated if, for any choice of non-decreasing (in each variable) functions f and g on ℝ^{k}, the random variables f(X_{1}, …, X_{k}) and g(X_{1}, …, X_{k}) are non-negatively correlated whenever the expectations exist. Pitt (1982) proved that jointly Gaussian

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 60E15. Secondary 60E07, 52A20, 42A82.

Key words and phrases. Stable random vector, Gaussian random vector, correlation, Fourier transform, positive definite function, convex set.

random variables are associated if and only if the correlation between each pair is non-negative. Lee, Rachev and Samorodnitsky (1990) generalized this result to the case of jointly q-stable random variables by giving a necessary and sufficient condition in terms of the spectral measure. Inequality (2) points out a special class of functions f and g for which the correlation between f(X) and g(X) is non-negative independently of relations between the jointly q-stable random variables X_{i}. For other results related to association of random variables, see Joag-dev, Perlman and Pitt (1983), and Suquet (1994).

Another celebrated result of Pitt (1977) shows that, for any jointly Gaussian ℝ^{2}-generated random variables X_{1}, …, X_{k}, inequality (2) holds if f and g are the indicator functions of cubes in ℝ^{m} and ℝ^{k−m}, namely, for each t > 0,

(3)  P ( max_{1≤i≤k} |X_{i}| < t) ≥ P ( max_{1≤i≤m} |X_{i}| < t) P ( max_{m+1≤i≤k} |X_{i}| < t).

In other words, the quantity in the left-hand side is minimal (subject to the given marginal distributions) if for each choice of i, j with 1 ≤ i ≤ m and m+1 ≤ j ≤ k the random variables X_{i} and X_{j} are independent, that is to say, b_{ij} = Cov(X_{i}, X_{j}) = 0. An equivalent formulation of the same fact is that, for any symmetric convex sets F and G in ℝ^{2}, μ(F ∩ G) ≥ μ(F)μ(G), where μ is a symmetric Gaussian measure in ℝ^{2}. The question of whether the same is true for symmetric convex sets in ℝ^{n} ( and, correspondingly, for ℝ^{n}-generated Gaussians) remains open (see Schlumprecht, Schechtman and Zinn (1994) for a historical survey and partial results).

In Section 3, we consider the quantity in the left-hand side of (3) as a function of the m(k − m) variables b_{i,j}, and prove that, for every dimension n, this function has a local minimum at the origin. Note that, to solve the problem completely, one has to prove that the function has global minimum at the origin.

2. A correlation inequality for positive definite functions of stable variables

In order to prove inequality (2) we need the following simple fact.

Lemma 1. Let 0 < q ≤ 2, and ξ, η be any vectors from the space L_{q}([0, 1]). Then

exp(−∥ξ + η∥^{q}) + exp(−∥ξ − η∥^{q}) ≥ 2 exp(−∥ξ∥^{q} − ∥η∥^{q}).

Proof. A result of W. Orlicz (1933) (see also Clarkson (1936)) states that, for every 0 < q ≤ 2 and ξ, η ∈ L_{q},

∥ξ + η∥^{q} + ∥ξ − η∥^{q} ≤ 2(∥ξ∥^{q} + ∥η∥^{q}).

Now use the inequality relating the arithmetic and geometric means to obtain

exp(−∥ξ + η∥^{q}) + exp(−∥ξ − η∥^{q}) ≥ 2 exp(−∥ξ + η∥^{q}/2 − ∥ξ − η∥^{q}/2) ≥ 2 exp(−∥ξ∥^{q} − ∥η∥^{q}). □

Theorem 1. Let 0 < q ≤ 2 and X_{1}, …, X_{k} be jointly q-stable random variables. Then for any m < k and any even continuous positive definite functions f, g on ℝ^{m} and ℝ^{k−m} respectively, the random variables f(X_{1}, …, X_{m}) and g(X_{m+1}, …, X_{k}) are non-negatively correlated.

Proof. By Bochner’s theorem, f and g are the characteristic functions of finite measures μ and ν on ℝ^{m} and ℝ^{k−m} respectively. The measures μ and ν are symmetric because the functions f and g are even.

Let Y be the q-stable random vector in ℝ^{n} generating X_{1}, …, X_{k}, and let ξ_{1}, …, ξ_{k} ∈ ℝ^{n} be the vectors for which X_{1} = (Y, ξ_{1}), …, X_{k} = (Y, ξ_{k}). Denote by γ the distribution of the vector Y, so γ is a probability q-stable measure in ℝ^{n} with the characteristic function given by (1).

Using Fubini’s Theorem, we see that

𝔼(f(X_{1}, …, X_{m}) g(X_{m+1}, …, X_{k}))
= ∫_{ℝ^{n}} f((x, ξ_{1}), …, (x, ξ_{m})) g((x, ξ_{m+1}), …, (x, ξ_{k})) dγ(x)
= ∫_{ℝ^{n}} ( ∫_{ℝ^{m}} exp(−i(u_{1}(x, ξ_{1}) + ⋯ + u_{m}(x, ξ_{m}))) dμ(u_{1}, …, u_{m}) ×

∫_{ℝ^{k−m}} exp(−i(u_{m+1}(x, ξ_{m+1}) + ⋯ + u_{k}(x, ξ_{k}))) dν(u_{m+1}, …, u_{k})) dγ(x)

(4)

= ∫_{ℝ^{m}} ∫_{ℝ^{k−m}} ( ∫_{ℝ^{n}} exp ( − i(x, ∑_{j=1}^{k} u_{j}ξ_{j})) dγ(x)) dμ(u_{1}, …, u_{m}) dν(u_{m+1}, …, u_{k}).

Let α = ∑_{j=1}^{m} u_{j}ξ_{j}, β = ∑_{j=m+1}^{k} u_{j}ξ_{j} ∈ ℝ^{n}. Considering the coordinates of the vectors α and β as linear functions of the coordinates of u_{1}, …, u_{m} and u_{m+1}, …, u_{k}, respectively, and using (1) we see that the quantity in (4) is equal to (5)

I_{1} = ∫_{ℝ^{m}} ∫_{ℝ^{k−m}} exp(−∥ ∑_{j=1}^{n} α_{j}s_{j} + ∑_{j=1}^{n} β_{j}s_{j}∥^{q}) dμ(u_{1}, …, u_{m}) dν(u_{m+1}, …, u_{k}),

where the norm is taken from the space L_{q}([0, 1]). Denote by I_{2} the expression in (5) with minus instead of plus under the norm. Since the measure ν is symmetric, I_{1} = I_{2}. By Lemma 1,

(I_{1} + I_{2})/2 ≥ ∫_{ℝ^{m}} ∫_{ℝ^{k−m}} exp(−∥ ∑_{j=1}^{n} α_{j}s_{j}∥^{q}) ×

exp(−∥ ∑_{j=1}^{n} β_{j}s_{j}∥^{q}) dμ(u_{1}, …, u_{m}) dν(u_{m+1}, …, u_{k})

= ∫_{ℝ^{m}} exp(−∥ ∑_{j=1}^{n} α_{j}s_{j}∥^{q}) dμ(u_{1}, …, u_{m}) ×

∫_{ℝ^{k−m}} exp(−∥ ∑_{j=1}^{n} β_{j}s_{j}∥^{q}) dν(u_{m+1}, …, u_{k}).

Repeating all the calculations in the reverse order we show that the latter quantity is equal to 𝔼f(X_{1}, … X_{m}) 𝔼g(X_{m+1}, …, X_{k}) which finishes the proof. □

Examples. (i) Let f(x_{1}, …, x_{m}) = (1−|x_{1}|)_{+} ⋯ (1−|x_{m}|)_{+}, and g(x_{m+1}, …, x_{k}) = (1 − |x_{m+1}|)_{+} ⋯ (1 − |x_{k}|)_{+}, where the function (1 − |t|)_{+} is equal to 1 − |t| if t ∈ [−1, 1], and is equal to zero otherwise. It is well known that the function (1 − |t|)_{+} is positive definite, and hence f and g are positive definite. Thus, by Theorem 1, for every m < k and every jointly stable random variables X_{1}, …, X_{k},

𝔼((1 − |X_{1}|)_{+} · . . . (1 − |X_{k}|)_{+}) ≥
𝔼((1 − |X_{1}|)_{+} · . . . (1 − |X_{m}|)_{+}) 𝔼((1 − |X_{m+1}|)_{+} · . . . (1 − |X_{k}|)_{+}).

The latter inequality can be generalized by taking any functions f and g of the form f(x_{1}, …, x_{m}) = f_{1}(x_{1}) … f_{m}(x_{m}), g(x_{m+1}, …, x_{k}) = f_{m+1}(x_{m+1}) … f_{k}(x_{k}), where f_{1}, …, f_{k} are even functions on ℝ which are convex and decreasing on [0, ∞). Such functions f_{i} are positive definite by a well-known result of Polya.

(ii) Let q_{1}, …, q_{k} ∈ (0, 2], f(x_{1}, …, x_{m}) = exp(−|x_{1}|^{q_{1}} − ⋯ − |x_{m}|^{q_{m}}), and g(x_{m+1}, …, x_{k}) = exp(−|x_{m+1}|^{q_{m+1}} − ⋯ − |x_{k}|^{q_{k}}). Since for any q ∈ (0, 2] the function exp(−|t|^{q}) is positive definite, it follows that f and g are positive definite. Therefore, for every m < k

𝔼( exp(−|X_{1}|^{q_{1}} − ⋯ − |X_{k}|^{q_{k}})) ≥
𝔼( exp(−|X_{1}|^{q_{1}} − ⋯ − |X_{m}|^{q_{m}}))𝔼( exp(−|X_{m+1}|^{q_{m+1}} − ⋯ − |X_{k}|^{q_{k}})).

Remarks. (i) In the case of jointly Gaussian random variables the result of Theorem 1 can be extended to some classes of continuous functions f and g with power growth at infinity and such that their Fourier transforms (in the sense of distributions) are non-negative locally integrable functions with power growth at infinity. To do that, consider the convolutions of the functions f and g with Gaussian densities e_{n} approaching the δ-function as n → ∞, and slightly modify the proof of Theorem 1.

(ii) Y. Hu has recently proved that, for any even convex functions f and g on ℝ^{n} and jointly Gaussian random variables X_{1}, …, X_{n}, the random variables f(X_{1}, …, X_{n}) and g(X_{1}, …, X_{n}) are non-negatively correlated (private communication from T. Schlumprecht; compare the result of Hu with our Example 1).

3. On the local minimum in the correlation for Gaussian measures of symmetric convex sets

Let ν be the standard symmetric Gaussian measure on ℝ^{n}. Is it true that

(6)  ν(F ∩ G) ≥ ν(F)ν(G)

for all symmetric convex sets F and G in ℝ^{n} ? In 1977, L. Pitt proved that the answer is positive in the case n = 2. However, the question of whether the answer is positive for every dimension n is still open.

It can be seen that it suffices to consider the sets F = {x ∈ ℝ^{n} : |(x, ξ_{1})| ≤ 1, …, |(x, ξ_{k})| ≤ 1} and G = {x ∈ ℝ^{n} : |(x, ξ_{k+1})| ≤ 1, …, |(x, ξ_{2k})| ≤ 1}, where k is an integer, and ξ_{1}, …, ξ_{k}, ξ_{k+1}, …, ξ_{2k} ∈ ℝ^{n}. For these sets F and G, inequality (6) can be written in the form

(7)  P ( max_{1≤i≤2k} |X_{i}| < 1) ≥ P ( max_{1≤i≤k} |X_{i}| < 1) P ( max_{k+1≤i≤2k} |X_{i}| < 1),

where X_{1}, …, X_{2k} are the jointly Gaussian random variables generated by the vectors ξ_{1}, …, ξ_{k}, ξ_{k+1}, …, ξ_{2k} ∈ ℝ^{n} and a standard Gaussian random vector Y in ℝ^{n}, so that X_{i} = (Y, ξ_{i}) for each i.

It is easy to see that, to prove inequality (6), it suffices to consider the case where the vectors ξ_{i}, i = 1, …, 2k are linearly independent. For example, if n < 2k and the system of vectors ξ_{i} has rank n, we can transfer everything to the space ℝ^{2k}, and consider the vectors η_{i} = ξ_{i} + ϵe_{i} ∈ ℝ^{2k}, i = 1, …, 2k where, for each i, either e_{i} = 0 or ∥e_{i}∥ = 1 and e_{i} is orthogonal to each of the vectors ξ_{j}, j = 1, …, 2k and e_{j}, j ≠ i, so that the vectors η_{i} are linearly independent in ℝ^{2k}. Then inequality (7) for the random variables generated by the vectors η_{i} would imply inequality (7) for the random variables generated by ξ_{i}’s by taking the limit as ϵ → 0 and applying the Lebesgue dominated convergence theorem.

Assume that the vectors ξ_{i} ∈ ℝ^{2k}, i = 1, …, 2k are linearly independent. Then the joint distribution μ of random variables X_{1}, …, X_{2k} is a non-singular Gaussian measure in ℝ^{2k}, and the left-hand side of (7) is equal to

P ( max_{1≤i≤2k} |X_{i}| < 1) = μ([−1, 1]^{2k}).

We fix the scalar products (ξ_{i}, ξ_{j}) for all choices of i, j with either 1 ≤ i, j ≤ k or k+1 ≤ i, j ≤ 2k, and consider the quantity μ([−1, 1]^{2k}) as a function of k^{2} variables b_{i,j} = Cov(X_{i}, X_{j}), i = 1, …, k, j = k + 1, …, 2k. To prove Pitt’s inequality, one has to show that this function has a global minimum at zero. Being unable to do that we show instead that the function has a local minimum at zero. This fact is a simple consequence of Theorem 2 below.

In the proof of Theorem 2 we use one result about log-concave functions. A non-negative function f on ℝ^{k} is called log-concave if, for every choice of x, y ∈ ℝ^{k}, and 0 ≤ t ≤ 1,

f(tx + (1 − t)y) ≥ f(x)^{t}f(y)^{1−t}.

This means that the function log(f) is concave. Prekopa (1973) and Leindler (1972) have proved that if f is a log-concave function on ℝ^{k} and 0 < m < k, then the function

g(x_{1}, …, x_{m}) = ∫_{ℝ^{k−m}} f(x_{1}, …, x_{m}, z_{1}, …, z_{k−m}) dz

is also log-concave.

Theorem 2. Let F and G be symmetric convex sets in ℝ^{k}, and μ_{B} be a nonsingular probability Gaussian measure in ℝ^{2k} with the covariance matrix A = [ \begin{matrix}A & B \\ B^{T} & C \end{matrix} ] . Fix the k × k matrices A and C, and consider B = (b_{i,j})_{i,j=1}^{k} as a

variable from the space ℝ^{k^{2}}. Then the function B ↦ μ_{B}(F × G) has a local minimum at the point B = 0.

Proof. Without loss of generality, we may suppose that F and G have compact closure. Let χ_{F} , χ_{G} be the indicator functions of the sets F and G. Taking Fourier transforms, we obtain

μ_{B}(F × G) = ∫_{ℝ^{k}} ∫_{ℝ^{k}} χ_{F} (x)χ_{G}(y) dμ_{B}(x, y)

= ∫_{ℝ^{k}} ∫_{ℝ^{k}} χ̂_{F} (x)χ̂_{G}(y) exp(−\frac{1}{2}(x^{T} Ax + y^{T} Cy + 2x^{T} By)) dx dy.

Taking the second partial derivative by b_{i,j} and b_{m,n}, we get

H_{i,j,m,n} = \frac{∂^{2}}{∂b_{i,j}∂b_{m,n}} μ_{B}(F × G)

= ∫_{ℝ^{k}} ∫_{ℝ^{k}} χ̂_{F} (x)χ̂_{G}(y)(x_{i}x_{m}y_{j}y_{n}) exp(−\frac{1}{2}(x^{T} Ax + y^{T} Cy + 2x^{T} By)) dx dy

= \frac{1}{(2π)^{k}|A|^{1/2}} ∫_{F} ∫_{G} \frac{∂^{4}}{∂x_{i}∂x_{m}∂y_{j}∂y_{n}} exp(−\frac{1}{2}(x, y)^{T} A^{−1}(x, y)) dy dx.

The fact that |A| ≠ 0, and the validity of using Parseval’s Equality in the latter equations, follow from the non-singularity of the measure μ_{B}.

Since the sets F and G are symmetric, the partial derivative of the function B ↦ μ_{B}(F × G) by each b_{i,j} is equal to zero at the point B = 0. In order to show that there is a local minimum at B = 0, we need to know that H is positive definite when B = 0. Furthermore, by a change of variables, we see that it is sufficient to consider the special case when A = C = I. Hence, we need to show the positive definiteness of

H_{i,j,m,n} = \frac{1}{(2π)^{2k}} L_{i,m}K_{j,n}

where

L_{i,m} = ∫_{F} (x_{i}x_{m} − δ_{i,m}) exp(−\frac{1}{2}x^{T} x) dx

and

K_{j,n} = ∫_{G}(y_{j}y_{n} − δ_{j,n}) exp(−\frac{1}{2}y^{T} y) dy.

Since H = L ⊗ K, it is sufficient to show that L and K are negative definite, and clearly it is enough just to prove it for L.

Thus we desire to show that

∑_{i,m} L_{i,m}α_{i}α_{m} = ∫_{F} ((∑_{i} α_{i}x_{i})^{2} − ∥α∥_{2}^{2}) exp(−\frac{1}{2}x^{T} x) dx < 0

for all α ≠ 0. But by a change of variables, it is sufficient to show

∫_{F} (x_{1}^{2} − 1) exp(−\frac{1}{2}x^{T} x) dx < 0

for every convex symmetric set F with compact closure.

To show this, we see this as

∫_{−∞}^{∞}(x_{1}^{2} − 1) exp(−\frac{1}{2}x_{1}^{2})ϕ(x_{1}) dx_{1},

where

ϕ(x_{1}) = ∫_{ℝ^{k−1}} χ_{F} (x_{1}, …, x_{k}) exp(−\frac{1}{2}(x_{2}^{2} + ⋯ + x_{k}^{2})) dx_{2} … dx_{k}.

Since χ_{F} (x) exp(−\frac{1}{2}(x_{2}^{2} + ⋯ + x_{k}^{2})) is log-concave in ℝ^{k}, the result of Prekopa and Leindler mentioned before the formulation of Theorem 2 implies that ϕ is also logconcave. Since ϕ is also symmetric, it follows that ϕ(x_{1}) = ϕ_{1}(|x|), where ϕ_{1} is a decreasing function. Furthermore, since F has compact closure, ϕ_{1} is non-constant. Hence in order to show that

∫_{−∞}^{∞}(x_{1}^{2} − 1) exp(−\frac{1}{2}x_{1}^{2})ϕ(x_{1}) dx_{1} < 0,

it is sufficient to show that for all 0 < a < ∞

θ(a) = ∫_{−a}^{a}(x_{1}^{2} − 1) exp(−\frac{1}{2}x_{1}^{2}) dx_{1} < 0.

The function under the latter integral has antiderivative −x_{1} exp(−\frac{1}{2}x_{1}^{2}), so the result follows. □

Finally, we present one more argument showing that inequality (6) would be proved if one showed that the function from Theorem 2 had global minimum at zero.

Let A = C = I. Since the sets F and G are convex, their topological boundaries have zero Lebesgue measure. Let ν be standard Gaussian measure on ℝ^{k}. Then μ_{0}(F × G) = ν(F)ν(G), whereas lim_{λ→1} μ_{λI}(F × G) = ν(F ∩ G). To see this last assertion, note that

μ_{λI}(F × G) = \frac{1}{((2π(1 − λ^{2}))^{k}} ∫_{F} ∫_{G} exp(−\frac{1}{2(1 − λ^{2})}(x^{T} x − 2λx^{T} y + y^{T} y)) dy dx

which, making the substitution x = u + v, y = u − v

= \frac{1}{(π(1 − λ^{2}))^{k}} ∫_{ℝ^{k}} ∫_{(F −u)∩(u−G)} exp(−\frac{u^{2}}{1 + λ} − \frac{v^{2}}{1 − λ}) dv du.

Now, if u is not in the boundary of F or the boundary of G, then it is easily seen that

lim_{λ→1} \frac{1}{(\sqrt{π}(1 − λ))^{k}} ∫_{(F −u)∩(u−G)} exp(−\frac{v^{2}}{1 − λ}) dv = χ_{F ∩G}(u).

Hence the last assertion follows by Lebesgue’s law of dominated convergence. It is clear now that, if the function μ_{B} has global minimum at zero then μ_{λI}(F ×

G) ≥ μ_{0}(F × G), and, hence, ν(F ∩ G) ≥ ν(F)ν(G). However, the question of whether the function from Theorem 2 has global minimum at zero remains open.

Acknowledgements. We would like to thank T. Schlumprecht, G. Schechtman and J. Zinn for bringing the problem to our attention and providing us with updated information including their unpublished results.

References

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3. M.-L. T. Lee, S. Rachev and G. Samorodnitsky (1990), Association of stable random variables, Ann. Probab. 18, 1759–1764.

4. L. Leindler (1972), On a certain converse of Hölder’s inequality II, Acta. Sci. Math. Szeged 33, 217–223.

5. W. Orlicz (1933), Uber unbedingte Konvergenz in Funktionen Raumen I and II, Studia Math. 4, 33–37 and 41–47.

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7. L. Pitt (1982), Positively correlated normal variables are associated, Ann. Probab. 10, 496– 499.

8. A. Prekopa (1973), On logarithmic concave measures and functions, Acta Sci. Math. (Szeged) 34, 335–343.

9. T. Schlumprecht, G. Schechtman and J. Zinn (1994), On the Gaussian measure of the intersection of symmetric convex sets, preprint.

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Division of Mathematics and Statistics, University of Texas at San Antonio, San Antonio, TX 78249, U.S.A.

  E-mail address: koldobsk@ringer.cs.utsa.edu

Department of Mathematics, University of Missouri–Columbia, Columbia, MO 65211

  E-mail address: stephen@mont.cs.missouri.edu