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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 Edition
Авторы: Gnedenko B.V., Kolmogorov A.N., Chung K.L.
Язык: 
Рубрика: Математика/
Статус предметного указателя: Готов указатель с номерами страниц
ed2k:
Год издания: 1968
Количество страниц: 298
Добавлена в каталог: 09.12.2012
Операции: 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 ( ) distribution 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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