PROBLEMS OF INFORMATION TRANSMISSION

A translation of *Problemy Peredachi Informatsii*

Volume 18, Number 4, October–December, 1982

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** Lower Bound for Error Probability in a Discrete Memoryless Channel with Feedback
**

A.Yu. Sheverdyaev

pp. 227–234

**Abstract**—It is shown that in the case of block transmission of length
$N$ with feedback over a discrete memoryless channel, the error probability satisfies
the lower spherical-packing bound at rates less than the channel capacity, while the
probability of correct reception satisfies the Arimoto upper bound at rates greater
than the capacity. Thus, the logarithmic asymptotic behavior (as $N\to\infty$) of the
error probability and probability of correct reception is known at least for all
rates greater than the critical one.

** Capacity in the Case of Transmission by Smooth Surfaces in Multiparametric
Gaussian White Noise
**

M. Nusbaum

pp. 235–244

**Abstract**—The author computes the asymptotic behavior of the capacity
of a channel with multiparametric Gaussian white noise when the signal is a smooth
periodic function of many variables. Constraints are imposed on the average power of
the signal and its partial derivatives. The derivatives may be of different order
$\beta_T$ in different directions. It is shown that when the noise intensity
$\varepsilon^2$ tends to zero, the capacity increases as
$\varepsilon^{-2/(\gamma+1)}$, where $\gamma=(\sum\beta_i^{-1})^{-1}$.

** On Methods of Code Lengthening
**

V. A. Zinov'ev and S. N. Litsyn

pp. 244–254

**Abstract**—The available methods of increasing code lengths are
presented in a more general form, leading to a variety of optimal codes. The
initial codes are provided by embedded systems of generalized concatenated
codes, as well as BCH, Hamming, Preparata, Kerdock, Goethals, and
Delsarte–Goethals codes.

** Multivariate Asymptotically Optimal Stochastic Approximation Procedure
**

A. M. Benderskii and M. B. Nevel'son

pp. 255–262

**Abstract**—The authors consider the problem of estimating the
root of the regression equation in the multivariate case. It is shown that if
there exists an estimate that converges to the unknown value of the root at
some rate, then it can be used, practically without any additional
conditions, to construct a recursive procedure of stochastic-approximation
type which is optimal in the sense of the minimum covariance matrix of the
limiting normal law.

** Sequential Parameter Estimation and Filtering of Stochastic Processes
**

A. G. Tartakovskii

pp. 263–272

**Abstract**—The article considers problems of truncated
sequential estimation of constant and variable random parameters in a
Bayesian formulation. Conditions are given under which the
(nonasymptotically) optimal estimation procedure is a nonsequential
procedure. An optimal sequential-estimation procedure is obtained for the
parameter of an exponential distribution.

** On Minimax Filtering of Functions in $L_2$
**

G. K. Golubev

pp. 272–278

**Abstract**—The article considers the problem of filtering of nonrandom
functions $s(t)\in\Sigma_T$ in white noise. It is assumed that class $\Sigma_T$ is
made up of all possible responses of a linear time-optimal filter. The minimax
quadratic risk and minimax estimate for $s(t)$ are obtained in this case.

** Upper Bound for the Delay in a Multiple-Random-Access System with a Splitting
Algorithm
**

B. S. Tsybakov and N. B. Likhanov

pp. 279–285

**Abstract**—Papers [R.G. Gallager, *Proc. AFOSR Workshop
in Commun. Theory Appl.*, 1978, pp. 74–76; B.S. Tsybakov and V.A. Mikhailov,
*Probl. Peredachi Inf.*, 1980, vol. 16, no. 4,
pp. 65–79] propose a multiple-random-access
algorithm which, in a sense, has been a record-setter in terms of
transmission rate up to now. In this paper we obtain an upper bound for the
mean-session packet-transmission delay for this algorithm.

** Unidirectional Loop Communication Networks with Packet Switching
**

L. V. Andreev

pp. 285–298

**Abstract**—The article describes a model of a unidirectional
loop communications network with packet switching, in which all stations can
exchange messages; the model is a generalization of a number of models
investigated earlier. Formulas are obtained for the basic characteristics of
the network for an extensive class of priority distributions between flows
entering the network.

** Some Inequalities for a Multiphase System
**

A. A. Pukhal'skii

pp. 299–307

**Abstract**—The author considers a multiphase queuing system
with intermediate storage of finite capacity and blocking. Properties of
external monotonicity for the capacity of the system are established. A bound
is obtained for the capacity of the three-phase system.

** INDEX
**

pp. 309–314

BRIEF COMMUNICATIONS

** Decay of the Dependence on an Initial State for Some Multicomponent Markov Systems
**

L. G. Mityushin

pp. 117–123 (Russian issue)

**Abstract**—The notion of dependence between two Markov-related random
variables is introduced. For two classes of multicomponent Markov systems, we
estimate the dependence of the state at time $t$ from an initial state.

** On the Asymptotic Expansion of the Mean Square Risk for a Bayesian Estimate
**

E. K. Trishchenko

pp. 123–125 (Russian issue)

**Abstract**—We consider the problem of estimation of an unknown parameter
affected by unit Gaussian noise. We show that continuity of the a priori density
$p(x)$ of this parameter and mean-square differentiability of $\sqrt{p(x)}$ are
necessary and sufficient conditions for the second term of the asymptotic expansion
of the mean square risk of a Bayesian estimate to be of order $n^{-2}$.