A note on the product correlation rule
Suppose \(R_ 1(\cdot,\cdot)\) and \(R_ 2 (\cdot, \cdot)\) are two covariance functions of stochastic processes on \([0,1]\). Define on \({\underset \sim s}\), \({\underset \sim t} \in [0,1]^ 2\), \(R({\underset \sim s},{\underset \sim t}) = R_ 1 (s_ 1,t_ 1) R_ 2 (s_ 2, t_ 2)\); then for \({\underset \sim t}^ 1,{\underset \sim t}^ 2,\dots,{\underset \sim t}^ n \in [0,1]^ 2\), the matrix \(C=\{R({\underset \sim t}^ i,{\underset \sim t}^ j)\}\), \(i,j=1,\dots,n\) is semipositive definite. Write \({\underset \sim t}^ i = (t^ i_ 1, t^ i_ 2)\); then \(C=C_ 1 \circ C_ 2\) is the Hadamard product of \(C_ 1\), \(C_ 2\) where \(C_ k = R_ k(t^ i_ 1,t^ j_ 2)\), \(i,j=1, \dots, n;k = 1,2\). The author gives a condition ensuring the positive definiteness of \(C\) even though \(C_ 1\), \(C_ 2\) are only semipositive definite.
covariance functions of stochastic processes
scientific article; zbMATH DE number 566097