Association of infinitely divisible random vectors (Q1346143)

The (components of a) random vector \(X\) in \(\mathbb{R}^ d\), \(d \geq 2\), are called associated if \(\text{cov} (f(X)\), \(g(X)) \geq 0\) for any two nondecreasing, real functions \(f\) and \(g\). It is known that for an infinitely divisible (inf div) random vector to be associated it is sufficient that its Lévy measure to be concentrated on the two orthants \(\mathbb{R}^ d_ +\) and \(\mathbb{R}^ d_ -\). The author shows that this condition, which I call C here, is not necessary, by constructing an example of an associated inf div random vector in \(\mathbb{R}^ 2\) that does not satisfy condition C. Further two examples are given: one showing that C is equivalent to a stronger notion than association, another presenting a class of inf div random vectors for which C is equivalent to association.