I. FOREWORD (PP. 3–5) (translated by Alexey Narykov)

The problem of extracting signals on the background of noise or similarly structured other signals is not new. The author started his work in this area in 1961, the same year when the articles by Stratonovich [100] and Nilsson [84] were published. It became clear that through merging of these two approaches: on the one hand, the general description of signal flows and the methods of constructing their a posteriori characteristics, and, on the other hand, of the abplication of statistical decision theory to "multi-signal" problems; it may become possible to obtain a variety of results that are interesting both in theoretical and in applied contexts.

It is also compelling to observe the book of Bogolubov [14] and the results produced by the Anglo-Indian school $[1,4$, $5,88]$ that developed the mathematical device for describing the ensemble of particles, which subsequently served as the foundation of the theory of random points of StratonovichKuznetsov [66]. Of interest is also a relatively scanty part of the literature on queueing theory, which is directly related to the description of the input flows $[22,33,35,54,56$, $62,90,91,109,111$, etc.]. Despite the applied research was hindered by the inconsistency in the developed knowledge on the flows of various kinds and clearly insufficient attention of mathematicians to this area, the available body of knowledge has been sufficient to obtain the first results $[20,21]$ that establish the connection between the problem formulation of Nilsson and known (or slightly generalized) methods of describing stochstic flows.

If the circumstances were different and the number of novel developments obtained in the "multi-signal" theory would gradually grow, one might have hoped for the moment to come when the results produced by the specialists working in the applied area would be rigorously reformulated by mathematicians to allow the theory to take the final shape. However, due to the lack of coordination in the efforts by the specialists operating in different domains, and whose works are somehow related to the announced theory, the process of emergence of this theory would hardly be rapid. Indeed, when it comes to radio engineers, the theory of flows is most familiar for those involved with the issues of reliability, but not of signal processing. Mathematicians that are interested in the theory of flows are mostly concerned with its application to the queueing theory, which is yet only loosely connected to the theory of statistical decisions. Researchers in the area of theoretical astronomy, who use the theory of flows to describe sets of galaxies, entrust the problem of extraction of lowcontrast star clusters to the experimenters [83]. Physicists who are interested in particle ensembles, which exist in conditions of physical interaction of special kinds, explicitly or implicitly treat the issue of registering particles in noise as empirical problem [36, 121]. The present book is written in order to jumpstart the establishment of the new direction in the theory of signal extraction.

The main results of this book are obtained from three research areas: the theory of stocahstic flows, the theory of posterior analysis, and the theory of statistical decisions. To some degree it uses special areas of mathematical program-ming - its nonlinear, integer and dynamical methods. The majority of results are obtained by the author (individually, or in collaboration) and partially published in recent times (in 1964-1968).

While preparing the manuscript, it turned out to be possible to significantly generalize and develop previously published results, as well as discover a number of unnecessary assumptions and even shortcomings in some of the author's publications. Therefore, it is appropriate to treat the later results as more correct ones.

The author anticipates the criticism to arrive from a number of directions. Perhaps, the book will be perceived by engineers as too mathematically complicated. Mathematicians, after having observed the author's attempts to systematically present the theory of flows, will probably criticise the lack of rigour. Let us, however, hope that this criticism will streamline the development of this area of science, which appears to the author so promising.

II. A BRIEF OVERVIEW OF THE THEORY OF STOCHASTIC FLOWS (PP. 48-49)

Without a claim to completeness, let us provide a short summary on the development of the theory of stochstic flows. A more detailed description can be found in $[4,35,54,86$, $109]$

Mathematics, physics and various adjacent sciences are well familiar with a Poisson flow, which is used to describe a number of interesting phenomena. This is the only kind of flows that is sometimes introduced in handbooks and monographs on the theory of stochastic processes, of which the theory of stochastic flows is a special case [50].

In 1910-20s, the new user of the theory of flows became the queueing theory [or the theory of service systems] with an application to telephone service, and later to many other domains $[33-35,54-56,62,90,91,109]$. A variety of stochastic flows has been slightly extended by introducing of recurrent flows, Bernoulli flows, flows with a group service requests, etc., yet the majority of results has been concerned with stationary Poisson flows. The central problem of the queueing theory was computation of the properties of the service systems, while little effort has been paid to determine convenient and universal methods of describing the flows, such as in [86].

Recurrent flows (renewal processes) have been studied by mathematicians outside the context of service systems. This resulted in a number of important developments [57].

Starting from 1930s stochastic flows have received the attention of physicists. Specifically, this was concerned with the developments of methods to describe ensembles of particles (in gasses, liquids, etc.). It appears that the groundbreaking work was reported in 1946 in a monograph by N.N. Bogolyubov [14]. He has used the systems of densities and a generating functional, which are convenient to describe the flows of various classes.

In 1949-1951 the theory of flows became an interest of the Anglo-Indian school of physicists and mathematics (H. J. Bhabha, A. Ramakrishnan, M. S. Bartlett, D. G. Kendall [1, $4,5,88]$ ) who used approaches that are close to the method of Bogolyubov. Their the main application was the description of showers of reproducing and mutually-transforming particles.

An important class of processes of particle reproduction and mutual transformation that take place for each individual particle independently from its origins and presence of its neighbours, and was first considered by A.N. Kolmogorov and N.A. Dmitriev [58], is the foundation of the theory of branching stochastic processes. At first, however, the particle coordinates were not taken into account and the focus was on their number exclusively [93], hence no direct connection to the theory of flows. Later the theory of branching processes was developed to include the dependency on age, energy, and any other particle coordinate [4, 94, 95, 98, 107], what extended the class of considered stochastic flows.

In 1956, P.I. Kuznetsov and R.L. Stratonovich published an article [66], which is based on the work of N.N. Bogolyubov, but offers a more convenient device of stochastic flows, referred by the authors as systems of random correlated points. Some special cases developed in this article exactly correspond to the results originating from the Anglo-Indian school.

Among early applications of the theory of stochastic flows to problems in radio engineering, there is a certain interest for the works of N.M. Sedyakin, generalized in the monograph [96], as well the article by I.S. Itskhoki [52].

In the recent times (1961-1967), the article by R.L. Stratonovich [100] has revealed the applicability of the theory of flows to the problem of extraction of signal flows from noise. This work is further developed in the publications by the author of the present books and of his employees [15-18, 20, 21, 39-41, 72]. Dealing with various applied problems has resulted into the development of a number of generalizations; however, there was no attempt to systematically introduce the theory of flows and expose the relation to other approaches to this theory as listed above.

Overall, the theory of stochastic flows can hardly be described as acquired its final shape. Despite its obvious merit for applied problems, stochastic flows are rarely covered in textbooks on stochastic processes. Because the theory of flows is used in significantly different applications, there is also no unified terminology. These factors make it more difficult for the specialists, who engaged with the practical problems, to learn and apply the theory of flows.