Skorokhod A.V., Prokhorov Y.V. (Ed) — Basic Principles and Applications of Probability Theory :: Электронная библиотека попечительского совета мехмата МГУ

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Skorokhod A.V., Prokhorov Y.V. (Ed) — Basic Principles and Applications of Probability Theory

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Название: Basic Principles and Applications of Probability Theory

Авторы: Skorokhod A.V., Prokhorov Y.V. (Ed)

Аннотация:

The book is an introduction to probability written by one of the famous experts in this area. Readers will learn about the basic concepts of probability and its applications, preparing them for more advanced and specialized works.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 260

Добавлена в каталог: 26.02.2008

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Предметный указатель

$a \vee b, \ a\wedge b$       75
      12 88
$\mathcal A \vee \mathcal B$       54
$\mu_n \Rightarrow \mu$       119
$\nu \ll \mu$       88
$\sigma$ -algebra (field)      14 27
$\sigma$ -algebra (field), Borel      29
$\sigma$ -algebra (field), cylinder      47
$\sigma$ -algebra (field), predictable      107
$\sigma$ -algebra (field), tail      56
$\varepsilon$ -entropy      240
$\varepsilon$ -optimum      218
Absolute continuity      15
Adapted processes      104ff
Adaptedness      106
Additive control cost      223
Algebra(s)      24ff 46 47 53ff 100ff 140ff 210
Algebra(s), independent      54
Algebra(s), of events      11 24
Algebra(s), product      24
Almost surely      35
Amount of information      235ff
Analytic set      221 222
Arithmetic distribution      69
Arzela, C.      136
atom      21 28
Axioms      21 28
Axioms, Kolmogorov      14
Axioms, von Mises      14
Backward (first) equation      156ff
Bayes theorem      22
Bayes, T.      22ff 209ff
Bayesian decision      209
Bellman's equation      223 230
Bellman, R.      223 230 234 273
Bernoulli's scheme      24
Bernoulli's theorem      16 25
Bernoulli, J.      16 24 25 61 67
Bernstein, S.N.      23
Binomial distribution      25 26 39
Binomial probabilities      22
Birkhoff's theorem      126
Birkhoff, G.D.      126
Bochner's theorem      45
Bochner, S.      45ff 118
Boltzmann, L.      20
Borel $\sigma$ -algebra      29
Borel — Cantelli lemma      56
Borel, E.      29ff 56ff 82 85 93ff 139 141 161 180 202ff 267
Bose — Einstein statistics      20
Bose, S.      20
Brownian motion      8 84
Buffon's problem      29
Buffon, G.      29
Cantelli, F.P.      56 65 66 82 97 180
Cauchy problem      145
Cauchy, A.      64 90 145 166 167 182 184
Causes (hypotheses)      22
Central limit theorem      132 202
Central moment      31
Change-point problem      266ff 273
Channel capacity      246ff
Channel capacity, ergodic      254
Chaos      5
Chapman — Kolmogorov equation      99 149ff
Chapman, D.G.      99 149ff
Characteristic function      44ff 59 69 74 75 85—87 132
Characteristic function, general      86
Characteristic function, joint (n-dimensional)      42ff
Characteristic functional      44ff
Charge      150
Chebyshev's inequality      35
Chebyshev's theorem      60
Chebyshev, P.L.      35 60 61 197
Clapeyron, B.      9
Coding      248
Communication channel      244
Communication channel, determinate (noiseless)      245
Communication channel, noisy      245
Complete, group of events      22ff
Complete, measurable process      108
Complete, set of functions      45 57 88 122
Completeness      35 105
Conditional expectation      31—33 129 136
Conditional probability      21ff 32ff 213ff 238
Conditional probability, regular      34
Confidence interval      197
Consistency      42 49
Continuous random variable      29ff
Control cost      216ff
Control cost, additive      223
Control strategy      217
Controlled diffusion process      233
Controlled stochastic process      215
Controlled stochastic process, discrete      215
Controlled stochastic process, Markov      223
Convergence, almost surely      35
Convergence, in probablity      35
Convergence, of sequences      82
Convergence, weak      119
Convergence, with probability 1      13 16 35
Convolution equation      74
Covariance (function)      42ff 101ff 162 176 178 202 205 259
Critical region      203 204
Cylinder $\sigma$ -algebra      47
Cylinder set      47ff
Cylinder set base      50
Decision function      208ff
Decision function, minimax      209
Decoding      248ff
Delay time      40
DeMoivre — Laplace theorem      26
density      30 32 39—41 44 88 155 162 176 204 205 212 240ff 260ff
Denumerable homogeneous Markov process      161
Determinism      5
Diffusion coefficients      161ff 171 178 233 234
Diffusion operator      162 165
Diffusion process      161ff 171 178 233
Dirac, P.      20
Dirichiet, P.G.      185
Dirichlet problem      173
Discrete random variable      30
Disorder problem      See change-point problem 266
distribution      17 19 26ff 58ff 64ff 74ff 85ff 93 99ff 104 120ff 132ff 147ff 155 162 167 177 189 196ff 216ff 223 226 240 245 251 267ff
Distribution function(s)      30 38 39 49 59 198
Distribution function(s), finite-dimensional      42 43 48 124 150
Distribution function(s), joint      41 42 58
Distribution function(s), sample (empirical)      198 199
Distribution span      69 71
Distribution, continuous      155 240
Distribution, discrete      39
Distribution, exponential      39 40
Distribution, geometric      39
Distribution, n-dimensional      41
Distribution, normal (Gaussian)      40 41
Distribution, Poisson      26 39 83
Distribution, symmetric      62
Distribution, uniform      39 41
Distribution, weak      50 52
Donsker — Prokhorov theorem      135
Donsker, M.      135
Doob, J.L.      94 139ff
Einstein, A.      20
Elementary events, llff      19ff 236
Elliptic equations      185
Empirical distribution function      198ff
Encounter problem      29
entropy      235ff 249ff
Entropy, conditional      238 241
Entropy, properties      237

Equal likelihood      10 28
Ergodic channel      253ff
Ergodic theorem      17
Ergodic theorem, maximal      127
Estimator      196ff 209 265
Estimator, unbiased      196ff
Events      7 10ff 39 53ff 61ff 71 80 96 105 113 147 149 195 225 236 238
Events, complete group      22ff
Events, mutually independent      23
Exit time (first)      154 170
Expectation      30ff 60 68 110 111 129 132 140ff 166 175ff 186 208 211 218 266
Expected value      See Expectation
Fatou, P.      90
Fermi — Dirac statistics      20
Fermi, E.      20
Filtering      257 266ff
Filtering, nonlinear      274
Filtering, nonpredictive      265
Finite-dimensional distributions      48 99 125 149 151 228
Finite-dimensional distributions, consistency      48
Flow of $\sigma$ -algebras      105 108 170
Forward (second) equation      156 159 271
Fourier, J.B.J.      44 75 76 145 206 262ff
Fubini, G.      88 100 102
Fundamental sequence      37
Gaussian distribution      40ff 87 202 205
Gaussian measure      87 91
Gaussian random function      44 104
Generating function      60
Geometric distribution      39
Geometrical probabilities      28
Gikhman, I.I.      139 145 273
Goodness-of-fit      200
Harmonic function      226
Hegel, G.W.F.      6
Homogeneous Markov chain      125 226
Homogeneous Markov chain, denumerable      161
Homogeneous Markov chain, temporally      151
Independence      15 16 23 53ff 137 139 177
Independent algebras      24 53
Independent increments      78ff 104 110 140 176ff 268
Independent increments, stationary      86 110
Independent random elements      57
Indicator function      12 28
Indistinguishable processes      109
Interval estimation      197
Invariance principle      132
Ito, K.      145
Jordan decomposition      88
Jordan, C      88
Jump component      85
Kac's formula      167
Kac, M.      145 167 168 187 189
Kakutani, S.      88
Karhunen theorem      116
Karhunen, K.      116 118
Khinchin, A. Ya.      145 187 189
Kolmogorov equation(s)      156 162 271
Kolmogorov — Smirnov test      200
Kolmogorov's inequality      61
Kolmogorov's theorems      48 96
Kolmogorov's three-series theorem      65
Kolmogorov's zero-one law      56
Kolmogorov, A.N.      14ff 48 56 61ff 82 87 94ff 139 141ff 165 187 189 200 201 271 274
Ladder functionals      74
Laplace transform      59 68
Laplace, P.      6 26 40 59 68 176
Law of Large Numbers      10 16 17 60 61 130
Law of Large Numbers, strong      65 199
Law of normal fluctuations      17
Law of rare events      26
Lebesgue, H.      29 37ff 93 102 121 179
Levy's decomposition      79
Levy's formula      83
Levy, P.      79 83 86
Lindeberg's condition      133
Lindeberg's theorem      133
Lindeberg, J.W.      133ff
Lyapunov's theorem      134
Lyapunov, A.A.      134
Markov chain      125
Markov chain, controlled      223ff
Markov chain, filtering      269
Markov process, controlled      234
Markov process, definition      98
Markov process, filtering      269
Markov process, homogeneous (with stationary transition probabilities)      151 225ff 242 271
Markov process, realization      225
Markov property      125
Markov, A.A.      98 99 125 139 145ff 165ff 177 178 187 189 223ff 240ff 269ff
Martingale      110ff 150ff 164ff 182ff 274
Martingale, uniformly integrable      114
Maximal ergodic theorem      127
Maxwell distribution      8
Maxwell — Boltzmann statistics      20
Maxwell, J.C.      8 20
Mean      5 30 41ff 69 83ff lOlff 132 162 179 199ff 248 258
Measurable mapping      38 42 87 124
Measurable random function      100
Measure(s)      15 24 38 45ff 57 86ff 102 116 119 121ff 136 140 145ff 156 159 165 177 202ff
Measure(s), absolute continuity of      87
Measure(s), continuity set of      120
Measure(s), derivative of      88
Measure(s), ergodic      131
Measure(s), invariant      125 140
Measure(s), orthogonal      88
Measure(s), product      88 91 206
Measure(s), Radon      121
Measure(s), singular      87 88
Measure(s), tight      121
Measure(s), weakly compact      122
Measure-preserving transformation      124
Message(s)      205 236 237 244ff
Metric transitivity      130
Minimax decision function      209
Minimax risk      210
Minlos — Sazonov theorem      51
Minlos, R.A.      5]
Mises, R. von      14 67
Modification      79 80 93 97ff 109 110 135 153 161 205
Modification, measurable      100ff 109 110
Modification, of martingales      114 115
Modification, regular      93
Moivre, A. de      26 40
moment      31
Moment function      43
Monotone collection      27 109
Monotone collection, theorem      27
Multinomial probabilities      25
Newton, I.      6
Neyman — Pearson theorem      204
Neyman, J.      203ff 273
Nikodym, D.      15 32 87
Noisy (noiseless) channel      245ff
Nonarithmetic distribution      70
Nonlinear filtering      274
Nonpredictive filtering      265
Nonrandomized control      225
Normal distribution      44 132 133 162 200 205 206
Null hypothesis      202
Optimum ( $\varepsilon$ -)control      216 225ff 234
Optimum stopping      273
Optimum strategy      218
Order statistics      198 199
Overshoot      74 77
Overshoot time      74
Parabolic equations      145
Pearson, E.S.      203ff
Petrovsky, I.G.      145 187 189
Phase space      38 147

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