Update: This conjecture has now been solved! See A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions by T. Royen, and Royen’s proof of the Gaussian correlation inequality by Rafał Latała, and Dariusz Matlak.
We continue investigating the Gaussian correlation conjecture in this post. This states that if is the standard Gaussian measure on then
for all symmetric and convex sets . In this entry, we consider a stronger `local’ version of the conjecture, which has the advantage that it can be approached using differential calculus. Inequality (1) can alternatively be stated in terms of integrals,
This is clearly equivalent to (2) when are indicator functions of convex symmetric sets. More generally, using linearity, it extends to all nonnegative functions such that and are symmetric and convex subsets of for positive . A class of functions lying between these two extremes, which I consider here, are the log-concave functions. The reason for concentrating on this class of functions is that they satisfy the following stability result, due to Prékopa (available here) and Leindler.
Theorem 1 If is log-concave then so is .
The approach used in this post is as follows; Given an nxn matrix , the idea is to consider a probability measure with respect to which we have joint normal random variables and , all with zero mean and unit variance, and covariances given by
The random variables then have covariance matrix
This must be positive semidefinite or, equivalently, in the operator norm. Writing for expectation with respect to the probability measure , the Gaussian correlation conjecture is equivalent to the statement
Furthermore, considering as a function of , its partial derivatives can be calculated as follows,
Using this, the Taylor expansion to second order about Q=0 can be written as,
(we use the summation convention where repeated indices in a product are summed over). From Theorem 1, the function is log-concave and, therefore the matrix is positive semidefinite. So, to second order in Q,
If are strictly log-concave, so that are positive definite, then this shows that has a local minimum at . This suggests the following conjecture, which is stronger than the Gaussian correlation inequality.
Conjecture: For log-concave functions on , then over all matrices , the function is minimized at . In fact, we conjecture that it never has a (strict) local minimum at any .
Certainly, this conjecture would imply (3). Being a stronger version of the Gaussian correlation inequality, we can be less sure that this statement holds than the original conjecture. However, if it is true then it has one advantage. That is, it can be approached locally with respect to the covariances, only considering infinitesimal changes in Q. In fact, it can be shown to be equivalent to the following (for any fixed log-concave and differentiable ). I do not give the proof here, to save space, but it is just an application of equation (4).
Conjecture: For all log-concave functions , the matrix is never symmetric and negative definite.
If this can be proven, then the correlation conjecture follows. As demonstrated in 1977 by Pitt, the Gaussian correlation conjecture will follow if this matrix always has nonnegative trace (this was used to prove the two dimensional case of the conjecture), which would certainly imply that it can’t be negative definite. It is easily shown that the matrix will indeed have nonnegative trace if the Gaussian measure is replaced by the standard (Lebesgue) measure. This is because is log-concave and symmetric in (by Theorem 1), and is therefore maximized at . Expanding to second order in y shows that is symmetric and positive semidefinite. In fact, in two dimensions it can be shown that
has nonnegative trace (see Pitt). From this it follows that has nonnegative trace for all decreasing functions . In particular, letting be the Gaussian density, the above conjecture follows. I don’t know whether a similar argument can be carried across to an arbitrary number of dimensions.
Consider, for example, the case where is a function of for a positive definite matrix . This includes the cases where is a centered ellipsoid (proven in 1999 by Hargé) and, as a limit of centered ellipsoids, where it is a symmetric slab (i.e., the region between two parallel hyperplanes, independently proven in 1967 by both Khatri and Šidák). Say, . Then,
The final inequality follows because and are necessarily decreasing along lines going radially out from the origin, implying that and are both nonpositive. Consequently, is not positive definite in this case and the Gaussian correlation inequality follows for centered ellipsoids.