Vol 214, No 1 (2016)

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.

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.

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.

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.

In this paper we consider the problem of estimating a function from the noised observations via thresholding its wavelet coefficients and prove the strong consistency of the risk estimator with the adaptive threshold.

The following classical characterization of the exponential distribution is well known. Let X₁,X₂, . . . X_n be independent and identically distributed random variables. Their common distribution is exponential if and only if random variables X₁ and n min(X₁, . . .,X_n) 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.