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Ludmila Sakhno

Publications and source records attributed to Ludmila Sakhno.

5 recordsLinked to original sources
arXiv2015

Spectral functions related to some fractional stochastic differential equations

In this paper we consider fractional higher-order stochastic differential equations of the form \begin{align*} \left( μ+ c_α\frac{d^α}{d(-t)^α} \right)^βX(t) = \mathcal{E}(t) , \quad t\geq 0,\; μ>0,\; β>0,\; α\in (0,1) \cup \mathbb{N} \end{align*} where $\mathcal{E}(t)$ is a Gaussian white noise. We derive stochastic processes satisfying the above equations of which we obtain explicitly the covariance functions and the spectral functions.

math.PR↗
arXiv2013

Fractional Non-Linear, Linear and Sublinear Death Processes

This paper is devoted to the study of a fractional version of non-linear $\mathpzc{M}^ν(t)$, $t>0$, linear $M^ν(t)$, $t>0$ and sublinear $\mathfrak{M}^ν(t)$, $t>0$ death processes. Fractionality is introduced by replacing the usual integer-order derivative in the difference-differential equations governing the state probabilities, with the fractional derivative understood in the sense of Dzhrbashyan--Caputo. We derive explicitly the state probabilities of the three death processes and examine the related probability generating functions and mean values. A useful subordination relation is also proved, allowing us to express the death processes as compositions of their classical counterparts with the random time process $T_{2 ν} (t)$, $t>0$. This random time has one-dimensional distribution which is the folded solution to a Cauchy problem of the fractional diffusion equation.

math.PR↗
arXiv2009

On spectral representations of tensor random fields on the sphere

We study the representations of tensor random fields on the sphere basing on the theory of representations of the rotation group. Introducing specific components of a tensor field and imposing the conditions of weak isotropy and mean square continuity, we derive their spectral decompositions in terms of generalized spherical functions. The properties of random coefficients of the decompositions are characterized, including such an important question as conditions of Gaussianity.

math.PR↗
arXiv2008

On a Szego Type Limit Theorem, the Holder-Young-Brascamp-Lieb Inequality, and the Asymptotic Theory of Integrals and Quadratic Forms of Stationary Fields

Many statistical applications require establishing central limit theorems for sums, integrals, or for quadratic forms of functions of a stationary process. A particularly important case is that of Appell polynomials, since the Appell expansion rank" determines typically the type of central limit theorem satisfied by these functionals. We review and extend here to multidimensional indices a functional analysis approach to this problem proposed by Avram and Brown (1989), based on the method of cumulants and on integrability assumptions in the spectral domain; several applications are presented as well.

math.PR↗