


Vol 214, No 1 (2016)
- Year: 2016
- Articles: 13
- URL: https://journal-vniispk.ru/1072-3374/issue/view/14726
Article
Preface



An Approach to the Inverse Problem of Brain Functional Mapping Under the Assumption of Gamma Distributed Myogram Noise Within Rest Intervals Using the Independent Component Analysis*
Abstract
This paper is devoted to the challenging task of brain functional mapping, which is posed by current techniques of noninvasive human brain investigation. One of them is magnetoencephalography, which is potentially a very powerful tool for scientific and practical research. Big data, retrieved from magnetoencephalography procedure, comprise information about brain processes. MEG data processing sets a highly ill-posed problem of the spatial reconstruction of MEG signal sources with a given accuracy. At the moment there are no universal tools for solving this problem accurately enough. A similar distribution of measured potentials on the human head surface may reflect the magnetic activity of different areas within the cerebral cortex. Nevertheless, under certain assumptions this task can be solved unambiguously. These assumptions are: the discreteness of signal sources, originating from distinct functional brain areas, and the superficial location of the signal sources. The obtained MEG signals are assumed to be the superposition of multi-dipole signals. In this case the solution of the inverse problem is a multi-dipole approximation. A more precise model can be constructed under the assumption of the gamma distribution of myogram responses within rest intervals by developing relative associative filter. The proposed algorithm of inverse problem solution consists of two main steps. The first step includes the application of Independent component analysis to primary/basic MEG signals for the determination of independent components. At the second step these independent components are treated separately as monodipole models to get isolated signal source locations for each component.



Portfolio Analysis with Transaction Costs Under Uncertainty*
Abstract
We obtain explicit formulas for the expected portfolio return and portfolio variance for portfolios with commission, which are in the general case unsmooth rational functions of the absolute value of portfolio weights. We prove that the function of expected portfolio return and portfolio variance function with commission are bounded. Two-asset portfolios with commission are investigated in detail.



Stability of Constant Retrial Rate Systems with NBU Input*
Abstract
We study the stability of a single-server retrial queueing system with constant retrial rate, general input and service processes. First, we present a review of some relevant recent results related to the stability criteria of similar systems. Sufficient stability conditions were obtained by Avrachenkov and Morozov (2014), which hold for a rather general retrial system. However, only in the case of Poisson input is an explicit expression provided; otherwise one has to rely on simulation. On the other hand, the stability criteria derived by Lillo (1996) can be easily computed but only hold for the case of exponential service times. We present new sufficient stability conditions, which are less tight than the ones obtained by Avrachenkov and Morozov (2010), but have an analytical expression under rather general assumptions. A key assumption is that interarrival times belongs to the class of new better than used (NBU) distributions. We illustrate the accuracy of the condition based on this assumption (in comparison with known conditions when possible) for a number of non-exponential distributions.



Limit Theorems for Queuing Systems with Regenerative Doubly Stochastic Input Flow*
Abstract
This article focuses on queuing systems with doubly stochastic Poisson regenerative input flow and an infinite number of servers. Service times have the heavy-tailed distribution. The analogs of the law of large numbers and the central limit theorem for the number of occupied servers are obtained. These theorems follow from results for systems with general doubly stochastic Poisson processes [1]. As examples, we consider systems in which the input flow is controlled by a semi-Markov modulated and Markov modulated processes.



Modeling High-Frequency Non-Homogeneous Order Flows by Compound Cox Processes*
Abstract
A micro-scale model is proposed for the evolution of a limit order book in modern high-frequency trading applications. Within this model, order flows are described by doubly stochastic Poisson processes (also called Cox processes) taking account of the stochastic character of the intensities of order flows. The models for the number of order imbalance (NOI) process and order flow imbalance (OFI) process are introduced as two-sided risk processes that are special compound Cox processes. These processes are sensitive indicators of the current state of the limit order book since time intervals between events in a limit order book are usually so short that price changes are relatively infrequent events. Therefore price changes provide a very coarse and limited description of market dynamics at time micro-scales. NOI and OFI processes track best bid and ask queues and change much faster than prices. They incorporate information about build-ups and depletions of order queues and they can be used to interpolate market dynamics between price changes and to track the toxicity of order flows. The proposed multiplicative model of stochastic intensities makes it possible to analyze the characteristics of the order flows as well as the instantaneous proportion of the forces of buyers and sellers without modeling the external information background. The proposed model gives the opportunity to link the micro-scale high-frequency dynamics of the limit order book with the macroscale models of stock price processes of the form of subordinated Wiener processes by means of limit theorems for special random sums and hence, to give a deeper insight in the nature of popular models of statistical regularities of the evolution of characteristics of financial markets such as generalized hyperbolic distributions and other normal variance-mean mixtures.



Scale-Free Property for Degrees and Weights in an N-Interactions Random Graph Model*
Abstract
A general random graph evolution mechanism is defined. The evolution is based on the interactions of N vertices. Besides the interactions of the new vertex and the old ones, interactions among old vertices are also allowed. Moreover, both preferential attachment and uniform choice are possible. A vertex in the graph is characterized by its degree and its weight. The weight of a given vertex is the number of interactions of the vertex. The asymptotic behavior of the graph is studied. Scale-free properties both for the degrees and the weights are proved. It turns out that any exponent in (2,∞) can be achieved. The proofs are based on discrete time martingale theory.



A Non-Uniform Bound of the Remainder Term in the Central Limit Theorem for Bernoulli Random Variables
Abstract
A bound for the remainder in the Esseen expansion is obtained in the case of Bernoulli random variables. The bound consists of two parts, uniform and non-uniform. The uniform part depends only on n and p, and the non-uniform part depends also on x. This bound is compared with other known bounds. It is shown how this result can be applied to the problem of the absolute constant in the Berry–Esseen inequality.



Estimation of the Parameters of Fractional-Stable Laws by the Method of Minimum Distance*
Abstract
Algorithm for statistical estimation of the parameters of fractional-stable distributions is described in this article. This algorithm is constructed on the basis of the method of distance minimization between the empirical and theoretical distributions. As the distance between the two distributions, the χ distance is considered. The main difficulty in dealing with fractional-stable distributions is the absence of explicit expressions for the probability density function. That is why the theoretical density is estimated by the histogram method. The results of the test calculations and the results of the estimation of the quadratic deviation are presented. The results obtained by this estimator are compared with the results obtained by the method of moments. An example of the use of the estimator for the approximation of the experimental data obtained in the investigation of gene expression by RNA sequencing technology is given as well. It is shown that the probability density function of the gene expression in a wide-enough domain can be described by fractional-stable distributions.






Moment-Type Estimates for Characteristic Functions with Application to Von Mises Inequality*
Abstract
We obtaine exact moment-type estimates for the real part and for the absolute value of a characteristic function with the first three moments being fixed. As a corollary, the Mises inequality for lattice distributions is improved and generalized to the case of fractional-order moments.



A Note on Characterizations of the Exponential Distribution*
Abstract
The following classical characterization of the exponential distribution is well known. Let X1,X2, . . . Xn be independent and identically distributed random variables. Their common distribution is exponential if and only if random variables X1 and n min(X1, . . .,Xn) have the same distribution. In this note we show that the characterization can be substantially simplified if the exponentiality is characterized within a broad family of distributions that includes, in particular, gamma, Weibull and generalized exponential distributions. Then the necessary and sufficient condition is the equality only expectations of these variables. A similar characterization holds for the maximum.





