arXiv subjects

Mikhail Moklyachuk

Publications and source records attributed to Mikhail Moklyachuk.

24 records · Page 2Linked to original sources

arXiv2020

On Distributions of One Class of Random Sums and their Applications

We propose results of the investigation of properties of the random sums of random variables. We consider the case, where the number of summands is the first moment of an event occurrence. An integral equation is presented that determines distributions of random sums. With the help of the obtained results we analyse the distribution function of the time during which the Geiger-Muller counter will not lose any particles, the distribution function of the busy period of a redundant system with renewal, and the distribution function of the sojourn times of a single-server queueing system.

math.PR↗

arXiv2018

Filtering of Multidimensional Stationary Sequences with Missing Observations

The problem of mean-square optimal linear estimation of linear functionals which depend on the unknown values of a multidimensional stationary stochastic sequence from observations of the sequence with a noise and missing observations is considered. Formulas for calculating the mean-square errors and the spectral characteristics of the optimal linear estimates of the functionals are proposed under the condition of spectral certainty, where spectral densities of the sequences are exactly known. The minimax (robust) method of estimation is applied in the case where spectral densities are not known exactly while some sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and minimax spectral characteristics are proposed for some special sets of admissible densities.

math.ST↗

arXiv2016

Filtering Problem for Functionals of Stationary Processes with Missing Observations

The problem of the mean-square optimal linear estimation of the functional $Aξ=\ \int\limits_{R^s}a(t)ξ(-t)dt,$ which depends on the unknown values of stochastic stationary process $ξ(t)$ from observations of the process $ξ(t)+η(t)$ at points $t\in\mathbb{R} ^{-} \backslash S $, $S=\bigcup\limits_{l=1}^{s}[-M_{l}-N_{l}, \, \ldots, \, -M_{l} ],$ $R^s=[0,\infty) \backslash S^{+},$ $S^{+}=\bigcup\limits_{l=1}^{s}[ M_{l}, \, \ldots, \, M_{l}+N_{l}]$ is considered. Formulas for calculating the mean-square error and the spectral characteristic of the optimal linear estimate of the functional are proposed under the condition of spectral certainty, where spectral densities of the processes $ξ(t)$ and $η(t)$ are exactly known. The minimax (robust) method of estimation is applied in the case where spectral densities are not known exactly, but sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and minimax spectral characteristics are proposed for some special sets of admissible densities.

math.ST↗

arXiv2016

Minimax Extrapolation Problem For Harmonizable Stable Sequences With Noise Observations

We consider the problem of optimal linear estimation of the functional $A ξ~=~\sum_{j = 0}^{\infty} a_j ξ_j$ that depends on the unknown values $ξ_j,j=0,1,\dots, $ of a random sequence $\{ξ_j,j\in\mathbb Z\}$ from observations of the sequence $\{ξ_k+η_k,k\in\mathbb Z\}$ at points $k = -1, -2, \dots$, where $\{ξ_k,k\in\mathbb Z\}$ and $\{η_k,k\in\mathbb Z\}$ are mutually independent harmonizable symmetric $α$-stable random sequences which have the spectral densities $f(θ)>0$ and $g(θ)>0$ satisfying the minimality condition. The problem is investigated under the condition of spectral certainty as well as under the condition of spectral uncertainty. Formulas for calculating the value of the error and the spectral characteristic of the optimal linear estimate of the functional are derived under the condition of spectral certainty where spectral densities of the sequences are exactly known. In the case of spectral uncertainty where spectral densities of the sequences are not exactly known, while a class of admissible spectral densities is given, relations that determine the least favorable spectral densities and the minimax spectral characteristic are derived.

math.PR↗

arXiv2016

Filtering of Continuous Time Periodically Correlated Isotropic Random Fields

The problem of optimal linear estimation of functionals depending on the unknown values of a random field $ζ(t,x)$, which is mean-square continuous periodically correlated with respect to time argument $t\in\mathbb R$ and isotropic on the unit sphere ${S_n}$ with respect to spatial argument $x\in{S_n}$. Estimates are based on observations of the field $ζ(t,x)+θ(t,x)$ at points $(t,x):t\leq 0,x\in S_{n}$, where $θ(t,x)$ is an uncorrelated with $ζ(t,x)$ random field, which is mean-square continuous periodically correlated with respect to time argument $t\in\mathbb R$ and isotropic on the sphere ${S_n}$ with respect to spatial argument $x\in{S_n}$. Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of functionals are derived in the case of spectral certainty where the spectral densities of the fields are exactly known. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics are proposed in the case where the spectral densities are not exactly known while a class of admissible spectral densities is given.

math.ST↗

arXiv2016

Minimax interpolation of sequences with stationary increments and cointegrated sequences

We consider the problem of optimal estimation of the linear functional $A_Nξ=\sum_{k=0}^Na(k)ξ(k)$ depending on the unknown values of a stochastic sequence $ξ(m)$ with stationary increments from observations of the sequence $ξ(m)+η(m)$ at points of the set $\mathbb{Z}\setminus\{0,1,2,\ldots,N\}$, where $η(m)$ is a stationary sequence uncorrelated with $ξ(m)$. We propose formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional in the case of spectral certainty, where spectral densities of the sequences are exactly known. We also consider the problem for a class of cointegrated sequences. We propose relations that determine the least favorable spectral densities and the minimax spectral characteristics in the case of spectral uncertainty, where spectral densities are not exactly known while a set of admissible spectral densities is specified.

math.PR↗