Samorodnitsky, Gennady; Taqqu, Murad S. Lévy measures of infinitely divisible random vectors and Slepian inequalities. (English) Zbl 0843.60019 Ann. Probab. 22, No. 4, 1930-1956 (1994). Two \(\mathbb{R}^d\)-valued random variables \(X\), \(Y\) satisfy the right Slepian inequality if \(P(X > \lambda) \geq P(Y>\lambda)\) for each \(\lambda \in \mathbb{R}^d\). This inequality has the interpretation that the components of \(X\) are more positively dependent than those of \(Y\). For centered Gaussian vectors this is by Slepian’s results just the ordering given by covariances. The authors study Slepian inequalities for the whole family of measures given by the convolution semigroup of infinitely divisible random vectors \(X\) and \(Y\) without Gaussian component. Roughly speaking the Slepian inequalities for the convolution semigroup are determined by the corresponding inequalities for their Lévy measures and the shift vectors provided the Lévy measure integrates the identity locally at zero. But also the general case can be treated similarly. The paper contains further interesting application, for instance for the Ornstein-Uhlenbeck process. Reviewer: A.Janssen (Düsseldorf) Cited in 4 Documents MSC: 60E07 Infinitely divisible distributions; stable distributions 60E15 Inequalities; stochastic orderings Keywords:Slepian inequality; infinitely divisible random vectors; stable random vectors; convolution semigroup × Cite Format Result Cite Review PDF Full Text: DOI