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Quasi-associatedness of a Gaussian system of random vectors. (English. Russian original) Zbl 1218.62059
Russ. Math. Surv. 57, No. 6, 1243-1244 (2002); translation from Usp. Mat. Nauk 57, No. 6, 199-200 (2002).
From the paper: L.D. Pitt [Ann. Probab. 10, 496–499 (1982; Zbl 0482.62046)] proved that a Gaussian system \(\{\xi_t, t \in T\}\) of real random variables is associated if and only if the covariances \(\text{cov}(\xi_s, \xi_t)\) are non-negative for all \(s, t \in T\). According to K. Joag-Dey and F. Proschan [Ann. Stat. 11, 286–295 (1983; Zbl 0508.62041)] negative associatedness of a Gaussian system is equivalent to the condition that \(\text{cov}(\xi_s, \xi_t)\leq 0\) for \(s\neq t\). The goal of this note is to prove that Gaussian systems of random vectors \(\{\xi_t, t \in T\}\) with values in \(\mathbb R^s\) are quasi-associated.

MSC:
62H20 Measures of association (correlation, canonical correlation, etc.)
62H05 Characterization and structure theory for multivariate probability distributions; copulas
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