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Subject and purpose. The task of designing various radio engineering devices, such as filters, delay lines, antennas with a given radiation pattern, requires the development of methods for generating random sequences (the values of the system parameters) with given correlation properties, since the spectral characteristics of the listed and similar to them systems are expressed in terms of the Fourier transforms of correlators. The purpose of this paper is to represent the function of the transition probability of random sequences with long-range correlations in a form convenient for numerical generation of sequences, and to study the statistical properties of the latter. Method and methodology. An adequate mathematical tool for solving such problems is the higher order Markov chains. The statistical characteristics of these objects are determined by their transition probability function, which in the general case can have a very complex form. In this paper, the transition probability function is assumed to be additive and linear with respect to the values of the random variable. It is assumed that the state space of the sequence belongs to the set of real numbers. Results. The equations that relate the correlation functions of the sequence to the weight coefficients of the memory function, determined in their turn by the transition probability function, are derived and analytically solved. Conclusion. It is shown that the correlation functions of the additive Markov chain are completely determined by the variance of the random variable and the weight coefficients of the memory function. The agreement of the obtained analytical results with the results of numerical realization of the additive Markov sequence is demonstrated. Examples of possible correlation scenarios in higher order additive linear chains are given.

A Markov sequence is a basic statistical model representing uncertain sequential data, and it is used within a plethora of applications, including speech recognition, image processing, computational biology, radio-frequency identification (RFID), and information extraction. The problem of querying a Markov sequence is studied under the conventional semantics of querying a probabilistic database, where queries are formulated as finite-state transducers. Specifically, the complexity of two main problems is analyzed. The first problem is that of computing the confidence (probability) of an answer. The second is the enumeration of the answers in the order of decreasing confidence (with the generation of the top-k answers as a special case), or in an approximate order thereof. In particular, it is shown that enumeration in any subexponential- approximate order is generally intractable (even for some fixed transducers), and a matching upper bound is obtained through a proposed heuristic. Due to this hardness, a special consideration is given to restricted (yet common) classes of transducers that extract matches of a regular expression (subject to prefix and suffix constraints), and it is shown that these classes are, indeed, significantly more tractable. [ABSTRACT FROM AUTHOR]

We consider the problem of the construction of the estimator-process of the unknown finite-dimensional parameter in the case of the observations of nonlinear autoregressive process. The estimation is done in two or three steps. First we estimate the unknown parameter by a learning relatively short part of observations and then we use the one-step MLE idea to construct an-estimator process which is asymptotically equivalent to the MLE. To have the learning interval shorter we introduce the two-step procedure which leads to the asymptotically efficient estimator-process too. The presented results are illustrated with the help of two numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: We give a simple and entirely elementary proof of Gasper''s Theorem on the Markov sequence problem for Jacobi polynomials. It is based on the spectral analysis of an operator that arises in the study of a probabilistic model of colliding molecules introduced by Marc Kac, and the methods developed here yield new estimates relevant to the collision model. [Copyright &y& Elsevier]

Two approaches to studying the correlation functions of the binary Markov sequences are considered. The first of them is based on the study of probability of occurring different "words" in the sequence. The other one uses recurrence relations for correlation functions. These methods are applied for two important particular classes of the Markov chains. These classes include the Markov chains with permutative conditional probability functions and the additive Markov chains with the small memory functions. The exciting property of the self-similarity (discovered in Phys. Rev. Lett.90, 110601 (2003) for the additive Markov chain with the step-wise memory function) is proved to be the intrinsic property of any permutative Markov chain. Applicability of the correlation functions of the additive Markov chains with the small memory functions to calculating the thermodynamic characteristics of the classical Ising spin chain with long-range interaction is discussed. [ABSTRACT FROM AUTHOR]

The paper consists of three parts. The first part is dedicated to a Markov monotonous random search on a general optimization space. Under certain restrictions, an upper bound for the complexity of search is presented in an integral form, suitable for further analysis. This estimate is applied to the case of a torus, where several specific results on the rate of convergence are obtained with the help of a supplementary optimization problem, discussed in Appendix. [Copyright &y& Elsevier]

A problem of an optimal stopping of a Markov sequence is considered. The existence conditions and the structure of optimal and ∊-optimal (∊ > 0) multiple stopping rules are obtained. [ABSTRACT FROM AUTHOR]