
V. Yu. KorolevLomonosov Moscow State University | MSU · Department of Mathematical Statistics
V. Yu. Korolev
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Publications (291)
In this paper, the representability of the generalized Student’s distribution as uniform and normal-scale mixtures is considered. It is also shown that the generalized Burr and the Snedecor–Fisher distributions can be represented as the scale mixtures of uniform, folded normal, exponential, Weibull or Fréchet distributions. New multiplication-type...
In the paper, quasi-exponentiated normal distributions are introduced for any real power (exponent) no less than two. With natural exponents, the quasi-exponentiated normal distributions coincide with the distributions of the corresponding powers of normal random variables with zero mean. Their representability as scale mixtures of normal and expon...
We consider the time-inhomogeneous Prendiville model with failures and repairs. The property of weak ergodicity is considered, and estimates of the rate of convergence for the main probabilistic characteristics of the model are obtained. Several examples are considered showing how such estimates are obtained and how the limiting characteristics the...
A generalized multivariate problem due to V. M. Zolotarev is considered. Some related results on geometric random sums and (multivariate) geometric stable distributions are extended to a more general case of “anisotropic” random summation where sums of independent random vectors with multivariate random index having a special multivariate geometric...
In the paper, upper bounds for the rate of convergence in laws of large numbers for mixed Poisson random sums are constructed. As a measure of the distance between the limit and pre-limit laws, the Zolotarev ζ-metric is used. The obtained results extend the known convergence rate estimates for geometric random sums (in the famous Rényi theorem) to...
The problem considered is the computation of the (limiting) time-dependent performance characteristics of one-dimensional continuous-time Markov chains with discrete state space and time varying intensities. Numerical solution techniques can benefit from methods providing ergodicity bounds because the latter can indicate how to choose the position...
In the paper, a survey of the main results concerning univariate and multivariate exponential power (EP) distributions is given, with main attention paid to mixture representations of these laws. The properties of mixing distributions are considered and some asymptotic results based on mixture representations for EP and related distributions are pr...
In applied probability, the normal approximation is often used for the distribution of data with assumed additive structure. This tradition is based on the central limit theorem for sums of (independent) random variables. However, it is practically impossible to check the conditions providing the validity of the central limit theorem when the obser...
A version of the central limit theorem is proved for sums with a random number of independent and not necessarily identically distributed random variables in the double array limit scheme. It is demonstrated that arbitrary normal mixtures appear as the limit distribution. This result is used to substantiate the log-normal finite mixture approximati...
In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate stable distributions. Multivariate analogs of the Mittag–Leffler distribution are introduced. Some properties of these distributions are discussed. The main focus is on the representations of the corresponding random vectors...
Mathematical models are proposed for statistical regularities of maximum daily precipitation within a wet period and total precipitation volume per wet period. The proposed models are based on the generalized negative binomial (GNB) distribution of the duration of a wet period. The GNB distribution is a mixed Poisson distribution, the mixing distri...
Extreme values are considered in samples with random size that have a mixed Poisson distribution generated by a doubly stochastic Poisson process. Limit theorems are proved for the distributions of max-compound Cox processes establishing necessary and sufficient conditions for the convergence of these distributions.
We present new mixture representations for the generalized Linnik distribution in terms of normal, Laplace, and generalized Mittag–Leffler laws. In particular, we prove that the generalized Linnik distribution is a normal scale mixture with the generalized Mittag–Leffler mixing distribution. Based on these representations, we prove some limit theor...
In the paper, upper bounds for the rate of convergence in laws of large numbers for mixed Poisson random sums are constructed. As a measure of the distance between the limit and pre-limit laws, the Zolotarev $\zeta$-metric is used. The obtained results extend the known convergence rate estimates for geometric random sums (in the famous R{\'e}nyi th...
This paper is largely a review. It considers two main methods used to study stability and to obtain appropriate quantitative estimates of perturbations of (inhomogeneous) Markov chains with continuous time and a finite or countable state space. An approach is described to the construction of perturbation estimates for the main five classes of such...
In this paper, we display a method for the computation of convergence bounds for a non-stationary two-processor heterogeneous system with catastrophes, server failures and repairs when all parameters varying with time. Based on the logarithmic norm of linear operators, the bounds on the rate of convergence and the main limiting characteristics of t...
Extreme values are considered in samples with random size that has a mixed Poisson distribution being generated by a doubly stochastic Poisson process. We prove some inequalities providing bounds on the rate of convergence in limit theorems for the distributions of max-compound Cox processes.
In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate elliptically contoured stable distributions. It is demonstrated that these distributions form a special subclass of scale mixtures of multivariate elliptically contoured normal distributions. Some properties of these distribu...
The paper is largely of a review nature. It considers two main methods used to study stability and obtain appropriate quantitative estimates of perturbations of (inhomogeneous) Markov chains with continuous time and a finite or countable state space. An approach is described to the construction of perturbation estimates for the main five classes of...
In this paper, we study a wide and flexible family of discrete distributions, the so-called generalized negative binomial (GNB) distributions, which are mixed Poisson distributions with the mixing laws belonging to the class of generalized gamma (GG) distributions. This family was introduced by E.W. Stacy as a particular family of lifetime distribu...
The analysis of the real observations of precipitation based on the novel statistical approach using the negative binomial distribution as a model for describing the random duration of a wet period is considered and discussed. The study shows that this distribution fits very well to the real observations and generalized standard methods used in met...
The formalism of compound Cox processes was used for statistical analysis of turbulent density fluctuations of plasma during a transient process at electron cyclotron resonance heating of L-2M stellarator plasmas. Short-wavelength (k = 20–30 cm⁻¹) density fluctuations in the central region of the plasma column were measured by the collective backsc...
We study a non-stationary Markovian queueing model of a two-processor heterogeneous system and obtain basic limiting characteristics for this model. Some specific examples are considered illustrated by the corresponding plots.
We consider a multidimensional inhomogeneous birth-death process. In this paper, a general situation is studied in which the intensity of birth and death for each coordinate (“each type of particle”) depends on the state vector of the whole process. A one-dimensional projection of this process on one of the coordinate axes is considered. In this ca...
Some results of the statistical analysis of the observed regularities in some characteristics of the precipitation process are presented in the paper. The importance of this problem is emphasized by that the information concerning the regularities in duration of wet and dry periods plays a significant role in predicting floods and preventing them....
Extreme values are considered in samples with random size that have a mixed Poisson distribution that is generated by a doubly stochastic Poisson process. Some inequalities are proved relating the distributions and moments of extrema with those of the leading process (the mixing distribution). Limit theorems are proved for the distributions of max-...
The paper presents improved mathematical models and methods for statistical regularities in the behavior of some important characteristics of precipitation: duration of a wet period, maximum daily and total precipitation volumes within a such period. The asymptotic approximations are deduced using limit theorems for statistics constructed from samp...
Nonuniform estimates are presented for the rate of convergence in the central limit theorem for sums of a random number of independent identically distributed random variables. Two cases are studied in which the summation index (the number of summands in the sum) has the binomial or Poisson distribution. The index is assumed to be independent of th...
The article provides new mixture represenations for the generalized Mittag-Leffler distribution. In particular, it is shown that for values of the “generalizing” parameter not exceeding one, the generalized Mittag-Leffler distribution is a scale mixture of the half-normal distribution laws, classic Mittag-Leffler distributions, or generalized Mitta...
This paper is a further development of the results of [20] where, based on the negative binomial model for the duration of wet periods measured in days [16], an asymptotic approximation was proposed for the distribution of the maximum daily precipitation volume within a wet period. This approximation has the form of a scale mixture of the Fr´echet...
We consider a multidimensional inhomogeneous birth-death process (BDP) and obtain bounds on the rate of convergence for the corresponding one-dimensional processes.
We present new mixture representations for the generalized Linnik distribution in terms of normal, Laplace and generalized Mittag-Leffler laws. In particular, we prove that the generalized Linnik distribution is a normal scale mixture with the generalized Mittag-Leffler mixing distribution. Based on these representations, we prove some limit theore...
An approach to user identification based on deviations of their topic trends in operation with text information is presented. An approach is proposed to solve this problem; the approach implies topic analysis of the user’s past trends (behavior) in operation with text content of various (including confidential) categories and forecast of their futu...
The generalized negative binomial distribution (GNB) is a new flexible family of discrete distributions that are mixed Poisson laws with the mixing generalized gamma (GG) distributions. This family of discrete distributions is very wide and embraces Poisson distributions, negative binomial distributions, Sichel distributions, Weibull–Poisson distri...
The generalized negative binomial distribution (GNB) is a new flexible family of discrete distributions that are mixed Poisson laws with the mixing generalized gamma (GG) distributions. This family of discrete distributions is very wide and embraces Poisson distributions, negative binomial distributions, Sichel distributions, Weibull--Poisson distr...
In this paper we study a non-stationary Markovian queueing model of a two-processor heterogeneous system with time-varying arrival and service rates. We obtain the bounds on the rate of convergence and find the main limiting characteristics of the queue-length process.
An approach is proposed to the construction of general lower bounds for the rate of convergence of probability characteristics of continuous-time inhomogeneous Markov chains with a finite state space in terms of special “weighted” norms related to total variation. We study the sharpness of these bounds for finite birth–death–catastrophes process an...
In this paper we present a method for the computation of convergence bounds for four classes of multiserver queueing systems, described by inhomogeneous Markov chains. Specifically, we consider an inhomogeneous M/M/S queueing system with possible state-dependent arrival and service intensities, and additionally possible batch arrivals and batch ser...
In this paper, two approaches are proposed to the definition of abnormally extremal precipitation. These approaches are based on the negative binomial model for the distribution of duration of wet periods measured in days. This model demonstrates excellent fit with real data and provides a theoretical base for the determination of asymptotic approx...
Finite inhomogeneous continuous-time Markov chains are studied. For a wide class of such processes an approach is proposed for obtaining sharp bounds on the rate of convergence to the limiting characteristics. Queueing examples are considered.
The paper contains an overview of some properties of the Mittag-Leffler distribution. Main attention is paid to its representability as a mixed exponential law. The possibility to represent the Mittag-Leffler distribution as a scale mixture of half-normal and uniform distributions is discussed as well. It is shown that the Mittag-Leffler distributi...
A limit theorem is proved for doubly stochastically rarefied renewal processes. It is shown that under rather general conditions, as limit laws in limit theorems for mixed geometric random sums, there appear mixed exponential and mixed Laplace distributions. Some known and new properties of these distributions are reviewed. Also, some nonobvious pr...
On the basis of the negative binomial distribution of the duration of wet periods calculated per day, an asymptotic model is proposed for distributing the maximum daily rainfall volume during the wet period, having the form of a mixture of Frechet distributions and coinciding with the distribution of the positive degree of a random variable having...
In this paper we propose the median modifications of the EM-algorithm and demonstrate their advantages in comparison with conventional methods by the example dealing with the numerical solution of the problem of decomposition of the volatility of financial indexes. We provide examples of volatility decompositions for AMEX, CAC 40, NIKKEI, and NASDA...
It is proved that the negative binomial distributions with the shape parameter less than one are mixed geometric distributions. The mixing distribution is written out explicitly. Thus, the similar result of L. Gleser, stating that the gamma distributions with the shape parameter less than one are mixed exponential distributions, is transferred to t...
In the paper general theorems concerning the asymptotic deficiencies of sample median based on the sample of random size are presented. These results make it possible to compare the quality of the sample median constructed from samples with both random and non-random sizes in terms of additional observations. The cases of the binomial distribution...
In the paper, the concepts of π-mixed geometric and π-mixed binomial distributions are introduced within the setting of Bernoulli trials with a random probability of success. A generalization of the Rényi theorem concerning the asymptotic behavior of rarefied renewal processes is proved for doubly stochastic rarefaction resulting in that the limit...
In the present paper we present the results of a statistical analysis of some characteristics of precipitation events and propose a kind of a theoretical explanation of the proposed models in terms of mixed Poisson and mixed exponential distributions based on the information-theoretical entropy reasoning. The proposed models can be also treated as...
An inhomogeneous retrial queueing model is studied. Bounds for the rate of convergence in null ergocic case are obtained.
Based on the negative binomial model for the duration of wet periods measured in days, an asymptotic approximation is proposed for the distribution of the maximum daily precipitation volume within a wet period. This approximation has the form of a scale mixture of the Frechet distribution with the gamma mixing distribution and coincides with the di...
A new class of discrete GG-mixed Poisson distributions is considered as the family of mixed Poisson distributions in which the mixing laws belong to the class of generalized gamma (GG) distributions. The latter was introduced by E. W. Stacy as a special family of lifetime distributions containing gamma, exponential power and Weibull distributions....
In this paper we propose a new approach to evaluating and analyzing the volatility of financial indices, in particular, stock prices. This approach is based on a multidimensional interpretation of the volatility of one-dimensional processes. The foundation of this approach is a model based on the limit theorems for compound doubly stochastic Poisso...
New estimates are obtained for the rate of convergence of the negative binomial distribution with parameters (r, p) to the gamma-distribution with parameters (r, r) when p → 0. The main result is a considerable improvement of convergence rate estimates for regular statistics constructed from samples with negatively binomially distributed random siz...
Построены оценки отклонений функций концентрации сумм независимых случайных величин с конечными дисперсиями от функции распределения полунормального закона без предположения о существовании у слагаемых моментов высших порядков. Полученные результаты перенесены на пуассон-биномиальные, биномиальные и пуассоновские случайные суммы. При тех же предпол...
We consider the asymptotic behavior of the empirical availability function, which is an important reliability characteristic of technical, communication, information, or transport systems. Our main focus is the case of violation of the classical assumptions of homogeneity of failure flow or the existence of the expectation of failure-free performan...
Weakly ergodic continuous-time countable Markov chains are studied. We obtain uniform in time bounds for approximations via truncations by analogous smaller chains under some natural assumptions.
Linnik distributions (symmetric geometrically stable distributions) are widely applied in financial mathematics, telecommunication systems modeling, astrophysics, and genetics. These distributions are limiting for geometric sums of independent identically distributed random variables whose distribution belongs to the domain of normal attraction of...
A problem related to the Bernoulli trials with a random probability of success is considered. First, as a result of the preliminary experiment, the value of the random variable π ∈ (0, 1) is determined that is taken as the probability of success in the Bernoulli trials. Then, the random variable N is determined as the number of successes in k ∈ N B...
The paper demonstrates a way for application of a methodology for the stochastic analysis of random processes based on the method of moving separation of finite normal mixtures to analyze the non-negative time series. We suggest to noise the initial data by adding i.i.d. normal random variables with known parameters. Then the one-dimensional distri...
Рассматриваются слабо эргодичные счетные марковские цепи с непрерывным временем. При некоторых естественных дополнительных условиях получены равномерные по времени оценки аппроксимации для таких цепей с помощью усечений аналогичными цепями с меньшим числом состояний.
We prove some new product representations for random variables with the
Linnik, Mittag-Leffler and Weibull distributions. 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 mix...
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-st...
One of 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 which correspond...
We consider a multidimensional inhomogeneous birth-death process and obtain bounds for the probabilities of the corresponding one-dimensional processes.
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 Lévy processes in the Skorokhod space under more realistic moment conditions. As corollaries, theorems are proved on convergence of random wal...
In this paper, product representations are obtained for random variables with theWeibull 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 γ of theWeibull distribution bel...
In this paper one presents method for the computation of convergence bounds for four classes of multiserver queueing systems, described by inhomogeneous Markov chains. Specifically one considers inhomogeneous $M/M/S$ queueing system with possibly state-dependent arrival and service intensities and additionally possible batch arrivals and batch serv...
The paper presents two techniques for the identification of extremal loading via determination of special thresholds in order to distinguish between “normal” and “extreme” values in information data flows. Both algorithms are based on the Rényi limit theorem on rarefaction of renewal processes flows and the Pickands–Balkema–de Haan theorem on the a...
A class of inhomogeneous Markovian queuing systems with possible catastrophic failures and group arrival of customers in the case of empty queue is considered; basic estimates of the rate of convergence and stability for this class are obtained.
Рассматривается асимптотическое поведение выборочного коэффициента готовности - важного показателя надежности технических, коммуникационных, информационных и транспортных систем - при нарушении классических предположений однородности потока отказов или восстановлений, существования математического ожидания времени безотказной работы или ремонта. Ра...
Estimates are constructed for the deviation of the concentration functions of sums of independent random variables with finite variances from the folded normal distribution function without any assumptions concerning the existence of the moments of summands of higher orders. The obtained results are extended to Poisson-binomial, binomial and Poisso...
In the paper, we discuss the transformation of the asymptotic distribution of asymptotically normal statistics if the sample size is replaced by a random variable. We also discuss the asymptotic expansions of the distribution function and concentration functions of statistics constructed from samples with random sizes.
The main goal of an efficient profiling of software is to minimize the runtime overhead under certain constraints and requirements. The traces built by a profiler during the work, affect the performance of the system itself. One of important aspect of an overhead arises from the randomness of variability in the context in which the application is e...
We introduce an inhomogeneous birth-death process with birth rates λ
k
(t), death rates µ
k
(t), and possible transitions to/from zero with rates β
k
(t), r
k
(t) respectively, and obtain ergodicity bounds for this process.
In this paper one presents the extension of the transient analysis of the class of continuous-time birth and death processes defined on non-negative integers with special transitions from and to the origin. From the origin transitions can occur to any state. But being in any other state, besides ordinary transitions to neighbouring states, a transi...
In this paper one presents the extension of the transient analysis of the class of continuous-time birth and death processes defined on non-negative integers with special transitions from and to the origin. From the origin transitions can occur to any state. But being in any other state, besides ordinary transitions to neighbouring states, a transi...
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 imbala...
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. Based on these representations, we prove some limit theorems for a wide class of rather simple statistics constructed from samples with rando...