On the rate of convergence to Rosenblatt-type distribution

Abstract

The main result of the article is the rate of convergence to the Rosenblatt-type distributions in non-central limit theorems. Specifications of the main theorem are discussed for several scenarios. In particular, special attention is paid to the Cauchy, generalized Linnik's, and local-global distinguisher random processes and fields. Direct analytical methods are used to investigate the rate of convergence in the uniform metric.

Weak convergence of weighted additive functionals of long-range dependent fields

“…Also, note that our result is a refined version of Lemma 1 in [13] as we consider long-range dependent random fields with general covariance functions (satisfying Assumption 1) while [13] only studied the case B(t) = (1 + t 2 ) −α/2 , 0 < α < 1. The obtained results can be applied to more general settings than integrals over homothetic regions considered in [1,2,12].…”

On asymptotics of discretized functionals of long-range dependent functional data

“…If α ∈ (0, n), then the covariance function B(x) satisfying Assumption 3.1 is not integrable, which corresponds to the long-range dependence case (Anh et al (2015)).…”

Reduction Principle for Functionals of Vector Random Fields

“…Remark 1 The random variable X κ,j (∆) is given in terms of the multiple Wiener-Itô stochastic integral. For κ = 2, the probability distribution of X 2,j (∆) is known as a Rosenblatt-type distribution which is a generalisation of the Rosenblatt distribution to an arbitrary set ∆(r) (see Anh et al 2015;Taqqu 1975).…”

“…They studied the multivariate limit theorems for functionals of stationary Gaussian series under long-range dependence, short-range dependence and a mixture of both. Excellent surveys of limit theorems for long-range dependent random fields can be found in Anh et al 2015;Ivanov and Leonenko 1989;Leonenko 1999;Spodarev 2014;Stoev and Taqqu 2007. Limit theorems for Minkowski functionals of stationary and isotropic Gaussian random fields have been studied under short and long-range dependence assumptions by Ivanov and Leonenko 1989. The CLT was proved for broad classes of random fields under different conditions in Bulinski et al 2012;Demichev 2015. Limit theorems for sojourn measures were discussed for geometric functionals of homogeneous isotropic Gaussian random fields that exhibit long-range dependence in Ivanov and Leonenko 1989.…”