PROBLEMS OF INFORMATION TRANSMISSION

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.