A translation of Problemy Peredachi Informatsii

Volume 33, Number 2, April–June, 1997
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Sensitivity of the $\varepsilon$-Entropy of Stationary Continuous-Time Gaussian Processes
M. S. Pinsker, V. V. Prelov, and S. Verdú
pp. 95–113

Abstract—Let $N=N(t)$ and $Z=Z(t)$ be independent continuous-time stationary random processes, and let $N$ be Gaussian. Denote by $\bar H_{\varepsilon}(N+\theta Z)$ the $\varepsilon$-entropy (relative to the mean-square-error criterion) of the process $N+\theta Z$. We prove that for any entropy-regular process $Z$, the limit $$ S_{\bar H_{\varepsilon}}(N,Z)=\lim_{\theta\to 0}\frac{1}{\theta^2}\left[\bar H_{\varepsilon} (N+\theta Z)-\bar H_{\varepsilon}(N)\right], $$ called the sensitivity of the $\varepsilon$-entropy, exists. Moreover, in this case, the equality $S_{\bar H_{\varepsilon}}(N,Z)=S_{\bar H_{\varepsilon}}(N,\bar Z)$ holds, where $\bar Z=\bar Z(t)$ is a stationary Gaussian process with the same autocorrelation function as $Z$. An explicit expression for $S_{\bar H_{\varepsilon}}(N,Z)$ in terms of the spectral densities of $N$ and $Z$ is also derived. Similar results for discrete-time processes have been obtained in [1, 2].


Spectrum of Codes Associated with the Grassmannian G(3,6)
D. Yu. Nogin
pp. 114–123

Abstract—In the code associated with the Grassmann variety $G(3,6)$ over an arbitrary finite field $\mathbb{F}_q$, words of five different weights exist. Their weight distribution is found.


Risk-Efficient Estimation of the Parameter of an Autoregressive Process
A. A. Veksler
pp. 124–138

Abstract—The problem of the constructing procedures for estimation of the parameter of first-order autoregression is considered. The estimates obtained are efficient relative to the standard risk (i.e., the sum of the observation cost and estimation cost). The proposed procedures are based on the least-squares method (LSM) with a random number of observations (sequential estimates). In studying properties of the sequential modification of the LSM estimate, we obtained an expansion of the mean duration of the estimation procedure, which is an analog of the Smith theorem in the renewal theory.


Asymptotic Behavior of the Conditional Distributions of Diffusion Processes with Rapidly Oscillating Contamination
M. L. Kleptsina and A. P. Serebrovski
pp. 139–148

Abstract—The asymptotic behavior of the conditional distribution density of the slow component of a diffusion process is considered for the case where the coefficients of drift and diffusion depend on the rapidly oscillating component. Theorems of the type of the law of large numbers and central limit theorem are proved.


Estimation of a Limiting Distribution Density and Its Derivatives from Observations with Weakening Dependence
V. A. Vasil'ev and G. M. Koshkin
pp. 149–162

Abstract—We study properties of nonparametric kernel estimators for the derivatives of a multivariate distribution density. The distribution is such that the sequence of conditional distributions of dependent random variables $\varepsilon_n$ conforming with a nondecreasing $\sigma$-algebra flow $\{{\cal F}_n\}$ converges to this distribution. The principal part of the asymptotic mean-square error of the studied estimator with an improved rate of convergence is found. For asymptotically weakening dependence of the variables $\varepsilon_n$, the expression obtained coincides with a similar expression for the case of independent observations. The convergence with probability one and uniform asymptotic normality of the density derivative estimator under consideration is ascertained.


Fault Detection in Network Realizations of Systems of Monotone Boolean Functions
Yu. L. Sagalovich and V. Yu. Solomennikov
pp. 163–173

Abstract—The minimal disjunctive normal form of a monotone Boolean function does not contain variables with negation and, therefore, permits a network realization without inverters, which is attractive in itself. On the other hand, a set of conjunctions without negations does not possess properties of a separating system, which creates an obstacle to fault detection in a network. Nevertheless, in this paper, we prove that, under some conditions, a network remains testable, and a considerable reduction of the volume of computations while constructing diagnosis facilities is achieved.


On Some Asymptotical Properties of Design Matrices for Screening Experiments
V. V. Illarionov
pp. 174–179

Abstract—We perform asymptotic estimation of existence bounds for design matrices of screening experiments in a linear model and also of the fraction of matrices with strong separation in the general ensemble of $q$-ary matrices.


Stable Operation of a Nonstable Communication Network with a Protocol of Random Multiple Access
A. A. Nazarov
pp. 180–189

Abstract—The domain of stable operation for a communication network modeled as a queueing system without a stationary regime is determined. The duration of stay in this domain and the corresponding probability distribution are found. Time-probabilistic characteristics of such a network with a protocol of random multiple access are obtained.