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Gnedenko B.V., Kolmogorov A.N., Chung K.L. - Limit Distributions for Sums of Independent Random Variables. Revised Edition
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Название: Limit Distributions for Sums of Independent Random Variables. Revised EditionАвторы: Gnedenko B.V., Kolmogorov A.N., Chung K.L.Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1968Количество страниц: 298Добавлена в каталог: 09.12.2012Операции: Положить на полку |
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Предметный указатель
Accompanying laws 98 Additive characteristic 44 Asymptotic expansion 10 Asymptotic expansion for continuous distributions 220 Asymptotic expansion for densities 228 Asymptotic expansion for lattice distributions 241 Asymptotic expansion for nonlattice distributions 210 Asymptotic expansion, failure of 222 Asymptotically constant summands 95 Axiomatics 20 Bernoulli distribution 217 Berry - Eseen theorem 201 Borel closure 16 Borel field 16 Canonical decomposition of set functions 30 Canonical representation, of infinitely divisible laws 76 Canonical representation, of laws of the class L 149 Canonical representation, of stable laws 164 Canonical representation, uniqueness of 76 80 Cauchy's law 72 Cauchy's law, local limit theorem for 236 Central limit theorem 3 4 see Characteristic exponent of a stable law 171 Characteristic function(s) 44 Characteristic function(s), convergence of a sequence of 53 Characteristic function(s), derivatives of 63 Characteristic function(s), inequalities on 53 55 Characteristic function(s), inversion formula 48 Characteristic function(s), logarithm of 64 Characteristic function(s), uniqueness of 50 Chebyshev - Hermite polynomials 191 Chebyshev's expansion 191 Chebyshev's two problems 4 Chi-square ( 72 Class L 145 Class L, canonical representation 149 Class L, criterion for 147 Class L, example with finite variance 152 Complete inverse image 17 composition 28 Condition ( 232 Conditional mathematical expectation 21 248 250 Conditional probability 21 248 250 Conditionally compact set of distributions 38 Convergence for normalized sums 152-157 Convergence in probability 105 Convergence with probability one 137 Convergence, failure of 223 Convergence, of densities 222 Convergence, of distributions and measures see "Weak convergence" Convergence, of series 137 Convergence, of sums of independent summands, most general form 98-101 112 116-124 see "Domain Convergence, of variances 97 Cramer's condition (C) 209 Cramer's theorem, on asymptotic expansion 220 Cramer's theorem, on normal distribution 51 DeMoivre - Laplace theorem 5 density 23 Distance (of Levy) 33 Distribution function 14 24 see "Probability Domain, of a normal law 172 Domain, of a stable law 175 Domain, of attraction 172 Domain, of normal attraction 181 Domain, of partial attraction 184 Domain, theorems on 189-190 Elementary event 20 Exponential distribution 9 Feller's form of the central limit theorem 130 Field of sets 16 Field of sets, Borel 16 Field of sets, unit of 16 Gnedenko's theorem, on convergence of infinitely divisible laws 87 Gnedenko's theorem, on convergence of sums 112 Identically distributed random variables 162 Improper distribution 24 Improper distribution, theorems on 56 Incomplete gamma function 9 Independence 26 250 Infinitely divisible distributions 71 Infinitely divisible distributions as limit distributions in partial attraction 184 Infinitely divisible distributions as limit distributions of sums of infinitesimal summands 115 Infinitely divisible distributions, canonical representation 76 Infinitely divisible distributions, convergence of a sequence of 87 Infinitely divisible distributions, examples 71-76 Infinitely divisible distributions, examples with striking properties 81-83 Infinitely divisible distributions, generation by Poisson laws of 74 Infinitely divisible random variable 71 Infinitesimal summands 95 Infinitesimal summands, criterion for 96 Infinitesimal summands, uniformly 126 Inversion formula 48 Khintchine's theorem, on characterization of infinitely divisible laws 115-116 Khintchine's theorem, on partial attraction 184 Kolmogorov - Khintchine criterion for convergence of series 137 Kolmogorov's formula 85 Laplace's theorem 2 Lattice distribution 58 212 231 Lattice distribution, (maximum) span of 58 60 232 Law of Large Numbers 3 4 105 133-139 see Law of large numbers, Khintchine's form of 138 Law of large numbers, Kolmogorov - Feller form of 135 Lebesgue integral 19 Levy - Khintchine formula, for infinitely divisible laws 70 Levy - Khintchine formula, for stable laws 62 Levy's formula 84 Lindeberg - Feller theorem 103 Lindeberg's condition 5 Local limit theorems 231 Lyapunov's condition 5 103 Lyapunov's theorem 201 Markov's condition 4 Mathematical expectation 14 Measurable, function 19 Measurable, mapping 17 Measurable, set 19 Measure 17 Measure in product space 27 Measure, carrier of 17 Measure, complete 250 Measure, generated 17 Measure, normalization of 20 Measure, perfect 18 Median 95 Method of moments 4 Metric space, of characteristic functions 52 Metric space, of distributions 37 moment 62 Moment, absolute, central, absolute central 62 Normal distribution 3 Normal distribution as infinitely divisible 71 Normal distribution, convergence to 102 126-132 143 172 181 Normal distribution, convergence to density of 224 228 Normal distribution, local limit theorems for 233 241 Normal distribution, special role of 126 Normalized sum 145 Pearson curves 72 Pearson curves as infinitely divisible 86 Poisson's law 47 Poisson's law as infinitely divisible 72 247 Poisson's law, convergence to 104 132 Poisson's law, generating infinitely divisible laws 74 Poisson's law, special position in classical probability 8 145 Poisson's limit theorem for rare events 3 8 Poisson's limit theorem for rare events, generalized 8 Possible value 23 Principal branch 64 probability 20 Probability, density 23 Probability, distribution 22 Probability, distribution, continuous, discrete 23 Probability, distribution, joint 22 Probability, distribution, n-dimensional 23 Probability, distribution, proper, improper 24 Raikov's theorem, on Poisson's law 51 Raikov's theorem, on relative stability 143 Random, event 21 Random, function 68 Random, variable 21 Random, variable, critique of 13 Random, variable, existence of 246 Random, vector 22 Random, walk 6 Relative stability 139 Remainder term 196 Remainder term, estimation of 201 Remainder term, extremal case 217 Remainder term, lattice case 212 Remainder term, nonlattice case 208 Riemann zeta-function 75 Semi-invariant 65 Stable laws 162 Stable laws, canonical representation of 164 Stable laws, convergence to 175 181 Stable laws, convergence to density of 227 Stable laws, examples of 171 Stable laws, local limit theorem for 236 Stable laws, properties of 182-183 Stable sequence 105 Stable type 162 Stieltjes integral 29 Stieltjes integral, discussion of 15 Stieltjes sums 26 Stochastic process with independent increments 127 Stochastically (strongly) continuous process 128 Symmetrical distribution function 51 Theory of errors 3 Type of distribution functions 40 Type of distribution functions, improper, proper 40 Unimodal distribution function 157 252 Unimodal distribution function, criteria for 157 160 Unimodal distribution function, examples of 253 Unimodal distribution function, vertex of 157 Uniqueness theorem, for canonical representation 76 80 Uniqueness theorem, for characteristic function 50 Unitary law 7 Unitary law, convergence to see "Law of large numbers" Unitary law, theorems on 57-58 Universal law (of Doeblin) 189 Variance 44 Variation, negative 30 Variation, positive 30 Variation, total 30 Weak convergence, characterizations of 33 Weak convergence, in terms of characteristic functions 53 Weak convergence, of distributions 32 Weak convergence, of measures 39
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