V. Yu. Korolev

On the Upper Bound for the Absolute Constant in the Berry–Esseen Inequality

This paper describes the history of the search for unconditional and conditional upper bounds of the absolute constant in the Berry–Esseen inequality for sums of independent identically distributed random variables. Computational procedures are described. New estimates are presented from which it follows that the absolute constant in the classical Berry–Esseen inequality does not exceed 0.5129.

Convergence of Random Sequences with Independent Random Indices II

Necessary and sufficient conditions are obtained for the convergence of some statistics constructed from samples with random sizes.

Nonparametric estimation of the ruin probability for generalized risk processes

In this paper we construct a statistical estimator of the ruin probability for a generalized risk process characterized by the stochastic character of the premium rate and of the intensity of insurance payments. The asymptotic properties of the proposed estimator are considered. Algorithms are proposed for the construction of approximate nonparametric confidence intervals for the ruin probability.

On approximations to generalized Poisson distributions

In this paper three methods of the construction of approximations to generalized Poisson distributions are considered: approximation by a normal law, approximation by asymptotic distributions, the so-called Robbins mixtures, and approximation with the help of asymptotic expansions. Uniform and (for the first two methods) nonuniform estimates of the accuracy of the corresponding approximations are given. Some estimates for the concentration functions of generalized Poisson distributions are also presented.

Generalized Hyperbolic Laws as Limit Distributions for Random Sums

A general theorem is proved stating necessary and sufficient conditions for the convergence of the distributions of sums of a random number of independent identically distributed random variables to one-parameter variance-mean mixtures of normal laws. As a corollary, necessary and sufficient conditions for convergence of the distributions of sums of a random number of independent identically distributed random variables to generalized hyperbolic laws are obtained. Convergence rate estimates are presented for a particular case of special continuous time random walks generated by compound doubly stochastic Poisson processes.

Asymptotic Properties of Extrema of Compound Cox Processes and Their Applications to Some Problems of Financial Mathematics

Necessary and sufficient conditions are presented for the weak convergence of one-dimensional distributions of extrema of compound doubly stochastic Poisson processes whose jumps have zero expectation and finite variance. Convergence rate estimates are given. The obtained results are applied to the problem of prediction of stock prices.

Modeling high-frequency order flow imbalance by functional limit theorems for two-sided risk processes

A micro-scale model is proposed for the evolution of the limit order book. Within this model, the flows of orders (claims) are described by doubly stochastic Poisson processes taking account of the stochastic character of intensities of bid and ask orders that determine the price discovery mechanism in financial markets. The process of order flow imbalance (OFI) is studied. This process is a sensitive indicator 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. The OFI process tracks best bid and ask queues and change much faster than prices. It incorporates information about build-ups and depletions of order queues so that it can be used to interpolate market dynamics between price changes and to track the toxicity of order flows. The two-sided risk processes are suggested as mathematical models of the OFI process. The multiplicative model is proposed for the stochastic intensities making it possible to analyze the characteristics of order flows as well as the instantaneous proportion of the forces of buyers and sellers, that is, the intensity imbalance (II) process, 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 functional limit theorems of probability theory and hence, to give a deeper insight in the nature of popular subordinated Wiener processes such as generalized hyperbolic Levy processes as models of the evolution of characteristics of financial markets. In the proposed models, the subordinator is determined by the evolution of the stochastic intensity of the external information flow.

A new moment-type estimate of convergence rate in the Lyapunov theorem

The inequality

Two-sided bounds on the rate of convergence for continuous-time finite inhomogeneous Markov chains

\Delta_n\le 0.3197\cdot \sum_{i=1}^n(\beta_i+\sigma_i^3) (\sum_{i=1}^n\sigma_i^2)^{-3/2}

Statistical analysis and modelling of turbulent fluxes in the plasma of the L-2M stellarator and the FT-2 tokamak

is proved for the uniform distance