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N. N. Leonenko

Publications and source records attributed to N. N. Leonenko.

10 recordsLinked to original sources
arXiv2025

High-level moving excursions for spatiotemporal Gaussian random fields with long range dependence

The asymptotic behavior of an extended family of integral geometric random functionals, including spatiotemporal Minkowski functionals under moving levels, is analyzed in this paper. Specifically, sojourn measures of spatiotemporal long-range dependence (LRD) Gaussian random fields are considered in this analysis. The limit results derived provide general reduction principles under increasing domain asymptotics in space and time. The case of time-varying thresholds is also studied. Thus, the family of morphological measures considered allows the statistical and geometrical analysis of random physical systems displaying structural changes over time. Motivated by cosmological applications, the derived results are applied to the context of sojourn measures of spatiotemporal spherical Gaussian random fields. The results are illustrated for some families of spatiotemporal Gaussian random fields displaying complex spatiotemporal dependence structures.

math.PR↗
arXiv2022

Sojourn functionals for spatiotemporal random fields with long-memory

This paper addresses the asymptotic analysis of sojourn functionals of spatiotemporal Gaussian random fields with long-range dependence (LRD) in time also known as long memory. Specifically, reduction theorems are derived for local functionals of nonlinear transformation of such fields, with Hermite rank m larger than or equal to 1, under general covariance structures. These results are proven to hold, in particular, for a family of non--separable covariance structures belonging to Gneiting class. For m=2, under separability of the spatiotemporal covariance function in space and time, the properly normalized Minkowski functional, involving the modulus of a Gaussian random field, converges in distribution to the Rosenblatt type limiting distribution for a suitable range of the long memory parameter.

math.PR↗
arXiv2017

Space-time fractional diffusion of Riesz-Bessel type on regular bounded open domains

Fractional (in time and in space) evolution equations defined on Dirichlet regular bounded open domains, driven by fractional integrated in time Gaussian spatiotemporal white noise, are considered here. Sufficient conditions for the definition of a weak-sense Gaussian solution, in the mean-square sense, are derived. The temporal, spatial and spatiotemporal Hölder continuity, in the mean-square sense, of the formulated solution is obtained, under suitable conditions, from the asymptotic properties of the Mittag-Leffler function, and the asymptotic order of the eigenvalues of a fractional polynomial of the Dirichlet negative Laplacian operator on such bounded open domains.

math.DS↗
arXiv2016

Rosenblatt distribution subordinated to gaussian random fields with long-range dependence

The Karhunen-Loève expansion and the Fredholm determinant formula are used to derive an asymptotic Rosenblatt-type distribution of a sequence of integrals of quadratic functions of Gaussian stationary random fields on R^d displaying long-range dependence. This distribution reduces to the usual Rosenblatt distribution when d=1. Several properties of this new distribution are obtained. Specifically, its series representation in terms of independent chi-squared random variables is given, the asymptotic behavior of the eigenvalues, its Lèvy-Khintchine representation, as well as its membership to the Thorin subclass of self-decomposable distributions. The existence and boundedness of its probability density is then a direct consequence.

math.ST↗
arXiv2015

Non-central limit theorems for random fields subordinated to gamma-correlated random fields

A reduction theorem is proved for functionals of Gamma-correlated random fields with long-range dependence in d-dimensional space. In the particular case of a non-linear function of a chi-squared random field with Laguerre rank equal to one, we apply the Karhunen-Loéve expansion and the Fredholm determinant formula to obtain the characteristic function of its Rosenblatt-type limit distribution. When the Laguerre rank equals one and two, we obtain the multiple Wiener-Itô stochastic integral representation of the limit distribution. In both cases, an infinite series representation in terms of independent random variables is constructed for the limit random variables.

math.SP↗
arXiv2015

On a class of minimum contrast estimators for Gegenbauer random fields

The article introduces spatial long-range dependent models based on the fractional difference operators associated with the Gegenbauer polynomials. The results on consistency and asymptotic normality of a class of minimum contrast estimators of long-range dependence parameters of the models are obtained. A methodology to verify assumptions for consistency and asymptotic normality of minimum contrast estimators is developed. Numerical results are presented to confirm the theoretical findings.

math.ST↗
arXiv2013

Ergodicity and mixing bounds for the Fisher-Snedecor diffusion

We consider the Fisher-Snedecor diffusion; that is, the Kolmogorov-Pearson diffusion with the Fisher-Snedecor invariant distribution. In the nonstationary setting, we give explicit quantitative rates for the convergence rate of respective finite-dimensional distributions to that of the stationary Fisher-Snedecor diffusion, and for the $β$-mixing coefficient of this diffusion. As an application, we prove the law of large numbers and the central limit theorem for additive functionals of the Fisher-Snedecor diffusion and construct $P$-consistent and asymptotically normal estimators for the parameters of this diffusion given its nonstationary observation.

math.ST↗
arXiv2010

On spectral properties and statistical analysis of Fisher-Snedecor diffusion

We consider the problem of parameter estimation for an ergodic diffusion with Fisher-Snedecor invariant distribution, to be called Fisher-Snedecor diffusion. We compute the spectral representation of its transition density, which involves a finite number of discrete eigenfunctions (Fisher-Snedecor polynomials) as well as a continuous part. We propose moments based estimators (related to the Fisher-Snedecor polynomials) and prove their consistency and asymptotic normality. Furthermore, we propose a statistical test for the distributional assumptions on the marginal distribution of the Fisher-Snedecor diffusion, based on the moment condition derived from the corresponding Stein's equation.

math.ST↗