Search

We present some product representations for random variables with the Linnik, Mittag-Leffler, and Weibull distributions and establish the relationship between the mixing distributions in these representations. The main result is the representation of the Linnik distribution as a normal scale mixture with the Mittag-Leffler mixing distribution. As a corollary, we obtain the known representation of the Linnik distribution as a scale mixture of Laplace distributions. Another corollary of the main representation is the theorem establishing that the distributions of random sums of independent identically distributed random variables with finite variances converge to the Linnik distribution under an appropriate normalization if and only if the distribution of the random number of summands under the same normalization converges to the Mittag-Leffler distribution., 1. Introduction Usually the Mittag-Leffler and Linnik distributions are mentioned in the literature together as examples of geometric stable distributions. Since these distributions are very often reffered to as weak [...]

In this paper, product representations are obtained for random variables with the Weibull distribution in terms of random variables with normal, exponential and stable distributions yielding scale mixture representations for the corresponding distributions. Main attention is paid to the case where the shape parameter [gamma] of the Weibull distribution belongs to the interval (0, 1]. The case of small values of [gamma] is of special interest, since the Weibull distributions with such parameters occupy an intermediate position between distributions with exponentially decreasing tails (e.g., exponential and gamma-distributions) and heavy-tailed distributions with Zipf-Pareto power-type decrease of tails. As a by-product result of the representation of the Weibull distribution with [gamma] [member of] (0, 1) in the form of a mixed exponential distribution, the explicit representation of the moments of symmetric or one-sided strictly stable distributions are obtained. It is demonstrated that if [gamma] [member of] (0, 1], then the Weibull distribution is a mixed half-normal law, and hence, it can be limiting for maximal random sums of independent random variables with finite variances. It is also demonstrated that the symmetric two-sided Weibull distribution with [gamma] [member of] (0, 1] is a scale mixture of normal laws. Necessary and sufficient conditions are proved for the convergence of the distributions of extremal random sums of independent random variables with finite variances and of the distributions of the absolute values of these random sums to the Weibull distribution as well as of those of random sums themselves to the symmetric two-sided Weibull distribution. These results can serve as theoretical grounds for the application of the Weibull distribution as an asymptotic approximation for statistical regularities observed in the scheme of stopped random walks used, say, to describe the evolution of stock prices and financial indexes. Also, necessary and sufficient conditions are proved for the convergence of the distributions of more general regular statistics constructed from samples with random sizes to the symmetric two-sided Weibull distribution., 1. Introduction. The Weibull distribution In probability theory and mathematical statistics it is conventional to use the term Weibull distribution for a special absolutely continuous probability distribution concentrated on the [...]

One of the most popular experimental techniques for investigation of brain activity is the so-called method of evoked potentials: the subject repeatedly makes some movements (by his/her finger), whereas brain activity and some auxiliary signals are recorded for further analysis. The key problem is the detection of points in the myogram that correspond to the beginning of the movements. The more precisely the points are detected, the more successfully the magnetoencephalogram is processed aiming at the identification of sensors that are closest to the activity areas. This paper proposes a statistical approach to this problem based on mixtures models that uses a specially modified method of moving separation of mixtures of probability distributions (MSM-method) to detect the start points of the finger's movements. We demonstrate the correctness of the new procedure and its advantages as compared with the method based on the notion of the myogram window variance., Introduction The human brain is the center of the nervous system; the cerebral cortex processes all information. Therefore, research on brain activity is one of the most important problems of [...]

We consider a multidimensional inhomogeneous birth-death process and obtain bounds for the probabilities of the corresponding one-dimensional processes., 1. Introduction and preliminaries Multidimensional birth-death processes are objects of a number of studies in queueing theory. Firstly, the problem of the product form solutions for such models was considered; [...]

The paper presents an empirical approach to the analysis of broadband Fourier-spectra of the low-frequency structural plasma turbulence based on a priori assumptions concerning the number of processes and the Gaussian form of the spectral components. This type of turbulence in toroidal plasma devices is described by the mathematical model of non-stationary continuous-time random walk, namely compound doubly stochastic Poisson process or compound Cox process. Measurements of low-frequency (in ranges of frequencies to 5 MHz) long-wave (3..6 cm) fluctuations spectra were obtained at the plasma edge in the L-2M stellarator by the technique of Doppler reflectometry. Under some experimental conditions, the efficiency of the methodology was demonstrated. The spectra were successfully decomposed into a few components. There are more than two harmonics in broadband Fourier-spectra besides a harmonic connected with poloidal rotation of the plasma. Those additional harmonics correspond to fluctuations rotating in the directions of electron and ion drift in toroidal plasma devices. In all modes it was possible to reveal the components corresponding to the poloidal rotation of the plasma (which could be defined by a radial electric field) and phase velocity of the two types of structural turbulence (which could be defined by electron gradient and ion gradient instabilities). The suggested description of probabilistic and spectral characteristics of low-frequency plasma turbulence allow us to formulate the correct problem of characterization of structural turbulence by a system of stochastic differential equations. The equations of the system should involve stochastic processes with densities in the form of mixtures of probability distributions. Such a comprehensive approach assumes a correct comparison of different structural turbulence models (caused by drift dissipative, ion-sound, gradient instabilities, etc.) with characteristics of the obtained stochastic processes., 1. Introduction The investigation of low-frequency fluctuations in the L-2M stellarator plasma proves that the processes may also be characterized as structural turbulence [1,24]. It obeys the mathematical model of [...]

An improved and corrected version of the functional limit theorem is proved establishing weak convergence of random walks generated by compound doubly stochastic Poisson processes (compound Cox processes) to Levy processes in the Skorokhod space under more realistic moment conditions. As corollaries, theorems are proved on convergence of random walks with jumps having finite variances to Levy processes with variance-mean mixed normal distributions, in particular, to stable Levy processes, generalized hyperbolic and generalized variance-gamma Levy processes., 1. Introduction In financial mathematics the evolution of (the logarithms of) stock prices and financial indexes on small time horizons is often modeled by random walks. A simplest example of [...]

Context. We provide the first results from the complete SNAD adaptive learning pipeline in the context of a broad scope of data from large-scale astronomical surveys. Aims. The main goal of this work is to explore the potential of adaptive learning techniques in application to big data sets. Methods. Our SNAD team used Active Anomaly Discovery (AAD) as a tool to search for new supernova (SN) candidates in the photometric data from the first 9.4 months of the Zwicky Transient Facility (ZTF) survey, namely, between March 17 and December 31, 2018 (58 194 = MJD= 58 483). We analysed 70 ZTF fields at a high galactic latitude and visually inspected 2100 outliers. Results. This resulted in 104 SN-like objects being found, 57 of which were reported to the Transient Name Server for the first time and with 47 having previously been mentioned in other catalogues, either as SNe with known types or as SN candidates. We visually inspected the multi-colour light curves of the non-catalogued transients and performed fittings with different supernova models to assign it to a probable photometric class: Ia, Ib/c, IIP, IIL, or IIn. Moreover, we also identified unreported slow-evolving transients that are good superluminous SN candidates, along with a few other non-catalogued objects, such as red dwarf flares and active galactic nuclei. Conclusions. Beyond confirming the effectiveness of human-machine integration underlying the AAD strategy, our results shed light on potential leaks in currently available pipelines. These findings can help avoid similar losses in future large-scale astronomical surveys. Furthermore, the algorithm enables direct searches of any type of data and based on any definition of an anomaly set by the expert. [ABSTRACT FROM AUTHOR]