Ludmila Sakhno

Higher-Order Spectral Densities of Fractional Random Fields

This paper presents the second- and higher-order spectral densities of stationary (in space) random fields arising as approximations of rescaled solutions of the heat and fractional heat equations with singular initial conditions. The development is based on the diagram formalism and the Riesz composition formula. Our results are the first step to full parametrization of higher-order spectra of some classes of fractional random fields.

On the Kaplan–Meier Estimator of Long-Range Dependent Sequences

The limiting distributions are obtained for the Kaplan–Meier estimator of unknown distribution function of stationary time series of the form G(Xj), where Xj is stationary Gaussian process with long-range dependence and G(·) is non-random function.

Motions with finite velocity analyzed with order statistics and differential equations

The aim of this paper is to derive the explicit distribution of the position of randomly moving particles on the line and in the plane (with different velocities taken cyclically) by means of order statistics and by studying suitable problems of differential equations. The two approaches are compared when both are applicable (case of the telegraph process). In some specific cases (alternating motions with skipping) it is possible to use the order statistics approach also to solve the equations governing the distribution. Finally, the approach based on order statistics is also applied in order to obtain the distribution of the position in the case of planar motion with three velocities conditioned on the number of changes of directions recorded.

Parameter estimation for reciprocal gamma Ornstein–Uhlenbeck type processes

We consider parameter estimation for a process of Ornstein–Uhlenbeck type with reciprocal gamma marginal distribution, to be called reciprocal gamma Ornstein–Uhlenbeck (RGOU) process. We derive minimum contrast estimators of unknown parameters based on both the discrete and the continuous observations from the process as well as moments based estimators based on discrete observations. We prove that proposed estimators are consistent and asymptotically normal. The explicit forms of the asymptotic covariance matrices are determined by using the higher order spectral densities and cumulants of the RGOU process.

Estimates for Functionals of Solutions to Higher-Order Heat-Type Equations with Random Initial Conditions

In the present paper we continue the investigation of solutions to higher-order heat-type equations with random initial conditions, which play the important role in many applied areas. We consider the random initial conditions given by harmonizable

Harmonic analysis tools for statistical inference in the spectral domain

\varphi

On a Szegö type limit theorem, the Hölder-Young-Brascamp-Lieb inequality, and the asymptotic theory of integrals and quadratic forms of stationary fields

φ-sub-Gaussian processes. The main results are the bounds for the distributions of the suprema over bounded and unbounded domains for solutions of such equations. The results obtained in the paper hold, in particular, for the case of Gaussian initial condition.

Fractional Non-Linear, Linear and Sublinear Death Processes

In this paper we consider fractional higher-order stochastic differential equations of the form \begin{align*} \left( \mu + c_\alpha \frac{d^\alpha}{d(-t)^\alpha} \right)^\beta X(t) = \mathcal{E}(t) , \quad t\geq 0,\; \mu>0,\; \beta>0,\; \alpha \in (0,1) \cup \mathbb{N} \end{align*} where