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Gnedenko B.V., Kolmogorov A.N. — Limit Distributions for Sums of Independent Random Variables
Gnedenko B.V., Kolmogorov A.N. — Limit Distributions for Sums of Independent Random Variables

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Название: Limit Distributions for Sums of Independent Random Variables

Авторы: Gnedenko B.V., Kolmogorov A.N.


This is a translation of the Russian book ПРЕДЕЛЬНЫЕ РАСПРЕДЕЛЕНИЯ ДЛЯ СУММ НЕЗАВИСИМЫХ СЛУЧАЙНЫХ ВЕЛИЧИН (1949). There are various points of contact with the treatises by P. Levy [76] and by H. Cramer [21], but much of the material in the book has been hitherto available only in periodical articles, many of which are in Russian. The systematic account presented here combines generality with simplicity, making some of the most important and difficult parts of the theory of probability easily accessible to the reader. Beyond a knowledge of the calculus on the level of, say, Hardy's Pure Mathematics, the book is formally self-contained. However, a certain amount of mathematical maturity, perhaps a touch of single-minded perfectionism, is needed to penetrate the depth and appreciate the classic beauty of this definitive work.

Язык: en

Рубрика: Математика/

Серия: Посвящена 110-летию со дня рождения Колмогорова Андрея Николаевича

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1968

Количество страниц: 293

Добавлена в каталог: 19.05.2006

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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 — Esseen 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
Characteristic exponent of a stable law      171
Characteristic function      44
Characteristic function, derivatives of      63
Characteristic function, inversion formula      48
Characteristic function, logarithm of      64
Characteristic function, uniqueness of      50
Characteristic functions, convergence of a sequence of      53
Characteristic functions, inequalities on      53 55
Chebyshev — Hermite polynomials      191
Chebyshev’s expansion      191
Chebyshev’s two problems      4
Chi-square ($\chi^2$) 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 ($\omega$)      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 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
Convergence of variances      97
Convergence with probability one      137
Convergence, failure of      223
Cramer’s condition (C)      209
Cramer’s theorem on asymptotic expansion      220
Cramer’s theorem on normal distribution      51
de Moivre — Laplace theorem      55
density      23
Distance (of Levy)      33
Distribution function      14 24 “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
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      62
Moment, absolute central      62
Moment, 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      23
Probability distribution, discrete      23
Probability distribution, improper      24
Probability distribution, joint      22
Probability distribution, n-dimensional      23
Probability distribution, proper      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      40
Type of distribution functions, 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 in terms of characteristic functions      53
Weak convergence of distributions      32
Weak convergence of measures      39
Weak convergence, characterizations of      33
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