V. E. Bening

Generalized poisson models and their applications in insurance and finance

Basic notions of probability theory random variables, their distributions and moments generating and characteristic functions random vectors stochastic independence weak convergence of random variables and distribution functions Poisson theorem law of large numbers central limit theorem stable laws the Berry-Esseen inequality asymptotic expansions in the central limit theorem elementary properties of random sums stochastic processes Poisson process the definition and elementary properties of a Poisson process Poisson process as a model of chaotic displacement of points in time the asymptotic normality of a Poisson process elementary rarefaction of renewal processes convergence of superpositions of independent stochastic processes characteristic features of the problem approximation of distributions of randomly indexed random sequences by special mixtures the transfer theorem relations between the limit laws for random sequences with random and non-random indices necessary and sufficient conditions for the convergence of distributions of random sequences with independent random indices convergence of distributions of randomly indexed sequences to identifiable location or scale mixtures the asymptotic behaviour of extremal random sums convergence of distributions of random sums the central limit theorem and the law of large numbers for random sums a general theorem on the asymptotic behaviour of superpositions of independent stochastic processes the transfer theorem for random sums of independent identically distributed random variables in the double array limit scheme compound Poisson distribution mixed and compound Poisson distributions discrete compound Poisson distributions the asymptotic normality of compound Poisson distributions the Berry-Esseen inequality for Poisson random sums non-central Lyapunov fractions asymptotic expansions for compound Poisson distributions the asymptotic expansions for the quantiles of compound Poisson distributions exponential inequalities for the probabilities of large derivations of Poisson random sums an analog of Bernshtein-Kolmogorov inequality the application of Esscher transforms to the approximation of the tails of compound Poisson distributions estimates of convergence rate in local limit theorems for Poisson random sums classical risk processes the definition of the classical risk process - its asymptotic normality the Pollaczek-Khinchin-Beekman formula for the ruin probability in the classical risk process approximations for the ruin probability with small safety loading asymptotic expansions for the ruin probability with small safety loading approximations for the ruin probability asymptotic approximations for the distributions of the surplus in general risk processes a problem of inventory control a non-classical problem of optimization of the initial capital doubly stochastic Poisson processes (Cox processes) the asymptotic behaviour of random sums

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.

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.

Low-Frequency Structural Plasma Turbulence in the L-2M Stellarator

Experiments on the L-2M stellarator have shown the occurrence of steady-state low-frequency strong structural (LFSS) turbulence throughout the entire plasma column. A key feature of LFSS turbulence is the presence of stochastic plasma structures. It is shown that different types of LFSS turbulence are correlated throughout the entire plasma volume. Stable non-Gaussian probability density distributions of all of the fluctuating plasma parameters are measured. The characteristic spatial and time scales of LFSS turbulence, which is responsible for non-Brownian diffusion in plasma, are determined.

New approach to the probabilistic-statistical analysis of turbulent transport processes in plasma

It is proposed to apply the statistical analysis of the increments of fluctuating particle fluxes to examine the probability characteristics of turbulent transport processes in plasma. Such an approach makes it possible to pass over to the analysis of the dynamical probability characteristics of the process under study. It is shown that, in the plasmas of the L-2M stellarator and the TAU-1 linear device, the increments of local fluctuating particle fluxes are stochastic in character and their distributions are scale mixtures of Gaussians. In particular, in TAU-1, the increments obey a Laplacian distribution (which is a scale mixture of Gaussians with an exponential mixing distribution). A mathematical model is proposed to explain such distributions. Possible physical mechanisms responsible for the random character of the increments of fluctuating particle fluxes are discussed.

On M n (t)/M n (t)/S queues with catastrophes

We consider the Mn(t)/Mn(t)/S queue with catastrophes. The bounds on the rate of convergence to the limit regime and the estimates of the limit probabilities are obtained.

Some bounds for M(t)/M(t)/S queue with catastrophes

We consider the M(t)/M(t)/S queue with catastrophes. The bounds of the rate of convergence to the limit regime and the estimates of the limit probabilities are obtained. We also study the bounds for the mean of the queue and consider an example.

Turbulent Transport Processes in a Plasma As a Diffusion Process with Random Time

It is proposed to apply the statistical analysis of the increments of fluctuating particle fluxes to analyzing the probability characteristics of turbulent transport processes in plasma. Such an approach makes it possible to analyze the dynamic probability characteristics of the process under study. It is shown that, in the plasmas of the L-2M stellarator and the TAU-1 linear device, the increments of local fluctuating particle fluxes are of stochastic character and that their distributions are scale mixtures of Gaussians. In particular, for TAU-1, the increments have the Laplacian distribution (which is a scale mixture of Gaussians with an exponential mixing distribution). This implies that the rate of flux variations is a diffusion process with random time. It is shown that the characteristic growth (damping) time of fluctuations is one order of magnitude shorter than their characteristic correlation time. Physical mechanisms that may be responsible for the random character of the growth (damping) of fluctuations are discussed.

ON THE RATE OF CONVERGENCE OF L- AND R-STATISTICS UNDER ALTERNATIVES*

Asymptotic distributions of test statistics under alternatives are important from the point of view of their power properties. When the limiting distributions of test statistics are specified under the hypothesis in a certain sense, LeCams third lemma ([4], Chapter 6) enables one to obtain their limiting distributions under close alternatives. In this paper we generalize LeCams third lemma by using the rate of convergence in the case of asymptotically efficient test statistics. A general lemma is proved which is specified to linear combinations of order statistics (L-statistics) and linear rank statistics (R-statistics). Edgeworth-type asymptotic expansions for these statistics under alternatives are considered in [3].