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