On long-range dependence of random measures. (English) Zbl 1384.60083

Vašata, Daniel

Adv. Appl. Probab. 48, No. 4, 1235-1255 (2016).

Summary: This paper deals with long-range dependence of random measures on \(\mathbb{R}^{d}\). By examples, it is demonstrated that one must be careful in order to define it consistently. Therefore, we define long-range dependence by a rather specific second-order condition and provide an equivalent formulation involving the asymptotic behaviour of the Bartlett spectrum near the origin. Then it is shown that the defining condition may be formulated less strictly when the additional isotropy assumption holds. Finally, we present an example of a long-range dependent random measure based on the 0-level excursion set of a Gaussian random field for which the corresponding spectral density and its asymptotics are explicitly derived.

MSC:

60G57	Random measures
60D05	Geometric probability and stochastic geometry
52A40	Inequalities and extremum problems involving convexity in convex geometry

Keywords:

long-range dependence; random measure; Bartlett spectrum