Victor Korolev

An improvement of the Berry–Esseen inequality with applications to Poisson and mixed Poisson random sums

By a modification of the method that was applied in study of Korolev & Shevtsova (2009), here the inequalities and are proved for the uniform distance ρ(F n ,Φ) between the standard normal distribution function Φ and the distribution function F n of the normalized sum of an arbitrary number n≥1 of independent identically distributed random variables with zero mean, unit variance, and finite third absolute moment β3. The first of these two inequalities is a structural improvement of the classical Berry–Esseen inequality and as well sharpens the best known upper estimate of the absolute constant in the classical Berry–Esseen inequality since 0.33477(β3+0.429)≤0.33477(1+0.429)β3<0.4784β3 by virtue of the condition β3≥1. The latter inequality is applied to lowering the upper estimate of the absolute constant in the analog of the Berry–Esseen inequality for Poisson random sums to 0.3041 which is strictly less than the least possible value 0.4097… of the absolute constant in the classical Berry–Esseen inequality. As corollaries, the estimates of the rate of convergence in limit theorems for compound mixed Poisson distributions are refined.

On truncations for weakly ergodic inhomogeneous birth and death processes

Abstract We investigate a class of exponentially weakly ergodic inhomogeneous birth and death processes. We consider special transformations of the reduced intensity matrix of the process and obtain uniform (in time) error bounds of truncations. Our approach also guarantees that we can find limiting characteristics approximately with an arbitrarily fixed error. As an example, we obtain the respective bounds of the truncation error for an Mt/Mt/S queue for any number of servers S. Arbitrary intensity functions instead of periodic ones can be considered in the same manner.

Perturbation bounds and truncations for a class of Markovian queues

We consider time-inhomogeneous Markovian queueing models with batch arrivals and group services. We study the mathematical expectation of the respective queue-length process and obtain the bounds on the rate of convergence and error of truncation of the process. Specific queueing models are shown as examples.

A general theorem on the limit behavior of superpositions of independent random processes with applications to Cox processes

A general theorem is presented which gives necessary and sufficient conditions of the convergence of one-dimensional distributions of appropriately centered and normalized superpositions of independent stochastic processes. This theorem is used as a tool for obtaining limit theorems for doubly stochastic Poisson processes (Cox processes).

Ergodicity And Perturbation Bounds For Inhomogeneous Birth And Death Processes With Additional Transitions From And To The Origin

Abstract Service life of many real-life systems cannot be considered infinite, and thus the systems will be eventually stopped or will break down. Some of them may be re-launched after possible maintenance under likely new initial conditions. In such systems, which are often modelled by birth and death processes, the assumption of stationarity may be too strong and performance characteristics obtained under this assumption may not make much sense. In such circumstances, time-dependent analysis is more meaningful. In this paper, transient analysis of one class of Markov processes defined on non-negative integers, specifically, inhomogeneous birth and death processes allowing special transitions from and to the origin, is carried out. Whenever the process is at the origin, transition can occur to any state, not necessarily a neighbouring one. Being in any other state, besides ordinary transitions to neighbouring states, a transition to the origin can occur. All possible transition intensities are assumed to be non-random functions of time and may depend (except for transition to the origin) on the process state. To the best of our knowledge, first ergodicity and perturbation bounds for this class of processes are obtained. Extensive numerical results are also provided.

Modelling Of Statistical Fluctuations Of Information Flows By Mixtures Of Gamma Distributions.

The paper describes statistical approach to the analysis of traffic of information flows. Stochastic structure of traffic process can be modelled by finite probability mixtures, e.g., mixtures of gamma distributions. The approach is demonstrated on real data from the official website of the Russian Academy of Sciences.

Bounds of the accuracy of the normal approximation to the distributions of random sums under relaxed moment conditions

We improve bounds of accuracy of the normal approximation to the distribution of a sum of independent random variables under relaxed moment conditions, in particular, under the absence of moments of orders higher than the second. We extend these results to Poisson binomial, binomial, and Poisson random sums. Under the same conditions, we obtain bounds for the accuracy of approximation of the distributions of mixed Poisson random sums by the corresponding limit law. In particular, we construct these bounds for the accuracy of approximation of the distributions of geometric, negative binomial, and Poisson-inverse gamma (Sichel) random sums by the Laplace, variance gamma, and Student distributions, respectively.

Statistical analysis of precipitation events

In the present paper we demonstrate 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 the result of following the popular Bayesian approach.

Modeling and analyzing licensed shared access operation for 5G network as an inhomogeneous queue with catastrophes

The framework of licensed shared access (LSA) to spectrum seems to become one of the trends of 5G wireless networks. The framework assumes the simultaneous access to spectrum by at least two parties - the primarily owner (incumbent), which has the highest priority, and several secondary users (licensees), which have lower priorities. The critical up-to-date problem is the development of the corresponding radio admission control and load balancing algorithms that form an essential part of the LSA agreement between the parties. The algorithm of binary use of spectrum gives an absolute priority to the incumbent, e.g. the airport using spectrum for aeronautical telemetry purposes. In the paper, capturing the inhomogeneous in time nature of rates of requests for access to spectrum and average times of spectrum use, we propose a queuing model of binary access to spectrum as seen from the licensees point of view. The queue is described by an inhomogeneous birth and death process with catastrophes and repairs. The main aim of the paper is to find the bounds on the rates of convergence to the limiting characteristics of the queue - average number of users, blocking probability, and probability of service interruption due to the incumbents need for spectrum. Not only the acceptable upper thresholds on the limiting characteristics are important for consideration but also the corresponding bounds showing the moment in time when the system becomes stable and the LSA licensee could really access to spectrum.

Modeling High-Frequency Non-Homogeneous Order Flows by Compound Cox Processes

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 orders imbalance (NOI) process and order flow imbalance (OFI) process are introduced as two-sided risk processes which 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 macro-scale 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.