Interpolation, correlation identities, and inequalities for infinitely divisible variables (Q1282021)

The authors derive formulas for expressions of the form \(\mathbb{E} G(X_1)- \mathbb{E} G(X_0)\) as a (kind of) convex combination of \(\mathbb{E} G(X_\alpha)\), where the random vector \(X_\alpha\) is a (kind of) convex combination of random vectors \(X_1\) and \(X_0\). A frequently used example concerns \(X_\alpha\) with characteristic function \(\varphi_1^\alpha \varphi_0^{1-\alpha}\), where the characteristic functions \(\varphi_1\) and \(\varphi_0\) of \(X_1\) and \(X_0\) are infinitely divisible. The simplest case concerns a known formula for Gaussian random vectors. Special cases yield results for variances and covariances; the latter leading to results on association. Repeated application of these formulas gives rise to expansions for variances and inequalities for covariances. The paper concludes with some `evidence' in favour of the so-called Gaussian correlation conjecture: \(\text{Cov(\textbf{1}}(X\in A_1), \mathbf{1}(X\in A_2))\geq 0\), where \(X\) is a \(d\)-dimensional Gaussian random vector and \(A_1\) and \(A_2\) are symmetric convex sets in \(\mathbb{R}^d\), and \(d\geq 3\).