Preface xiii 1 Basic notions of probability theory 1 1.1 Random variables, their distributions and moments 1 1.2 Generating and characteristic functions 11 1.3 Random vectors. Stochastic independence 21 1.4 Weak convergence of random variables and distribution functions ... 24 1.5 Poisson theorem 30 1.6 Law of large numbers. Central limit theorem. Stable laws 35 1.7 The Berry-Esseen inequality 45 1.8 Asymptotic expansions in the central limit theorem 47 1.9 Elementary properties of random sums 56 1.10 Stochastic processes 62 2 Poisson process 69 2.1 The definition and elementary properties of a Poisson process 69 2.2 Poisson process as a model of chaotic displacement of points in time. . 72 2.3 The asymptotic normality of a Poisson process 74 2.4 Elementary rarefaction of renewal processes 76 3 Convergence of superpositions of independent stochastic processes 83 3.1 Characteristic features of the problem 83 3.2 Approximation of distributions of randomly indexed random sequences by special mixtures 85 3.3 The transfer theorem. Relations between the limit laws for random sequences with random and non-random indices 89 3.4 Necessary and sufficient conditions for the convergence of distributions of random sequences with independent random indices 91 3.5 Convergence of distributions of randomly indexed sequences to identifiable location or scale mixtures. The asymptotic behavior of extremal random sums 98 3.6 Convergence of distributions of random sums. The central limit theorem and the law of large numbers for random sums 105 3.7 A general theorem on the asymptotic behavior of superpositions of independent stochastic processes 115

No abstract

No abstract

No abstract