Ash R.B., Doléans-Dade C.A. — Probability and Measure Theory :: Электронная библиотека попечительского совета мехмата МГУ

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Ash R.B., Doléans-Dade C.A. — Probability and Measure Theory

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Название: Probability and Measure Theory

Авторы: Ash R.B., Doléans-Dade C.A.

Аннотация:

Probability and Measure Theory, Second Edition, is a text for a graduate-level course in probability that includes essential background topics in analysis. It provides extensive coverage of conditional probability and expectation, strong laws of large numbers, martingale theory, the central limit theorem, ergodic theory, and Brownian motion.

- Clear, readable style
- Solutions to many problems presented in text
- Solutions manual for instructors
- Material new to the second edition on ergodic theory, Brownian motion, and convergence theorems used in statistics
- No knowledge of general topology required, just basic analysis and metric spaces
- Efficient organization

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: Second edition

Год издания: 2000

Количество страниц: 516

Добавлена в каталог: 20.03.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

$L^{p}$ spaces      84ff.
$\mu$ -integrable function      39
$\sigma$ -algebra      4
$\sigma$ -field      4
$\sigma$ -field, generated by a class of sets      4
$\sigma$ -field, induced by a random object      216
$\sigma$ -finite set fiinction      9
Absolute homogeneity      127
Absolute moments and central moments      191
Absolutely continuous, function      72
Absolutely continuous, measure (or signed measure)      60 65
Absolutely continuous, random variable      175
Absolutely continuous, random vector      177
Abstract Lebesgue integral      37ff.
Adapted process      426
Additivity theorem      46
Adjoint      164
Algebra of sets      3
Almost everywhere (a.e.)      48
Almost invariant set      350
Almost surely (a.s.)      176
Annihilator      164
Approximation theorem      19 32 40 92 94
atom      225
Baire category theorem      158
Banach space      128
Bernoulli shift      394
Bernoulli trials      170 199 309
Bessel's inequality      133
beta function      202
Bochner's theorem      296
Borel - Cantelli Lemma      68 285
Borel - Cantelli lemma, second      238
Borel measurable function      36ff.
Borel sets      6 121
Bounded linear operator      142
Bounded sequence of distribution functions      300
Bounded variation      74
Branching processes      251 282
Brennan's gambling model      284
Brownian bridge      406
Brownian motion      401
Cantor function      79 175
Cantor set      35 79
Caratheodory extension theorem      19
Cauchy - Schwarz inequality      85 95 130
Cauchy - Schwarz inequality, for sums      91
Cauchy density      176
Cauchy sequence      89
Cauchy sequence, in measure      96
Central limit theorem      290ff. 343
Central moments      191
Chain rule      71
Characteristic function      291 341
Chebyshev's inequality      88 192
Closed graph theorem      163
Closed linear operator      163
Closed subspace      133
Complete measure space      17
Complete orthonormal set      136
Completeness of $L^{p}$       89 94
Completion of a measure space      18
Complex measure      71
Complex-valued Borel measurable function      83
Complex-valued random variable      193
Conditional entropy      376 387
Conditional expectation      201 ff. 210
Conditional expectation, given a $\sigma$ -field      217
Conditional probability      171 201 ff. 210
Conditional probability, given a $\sigma$ -field      219
Conjugate isometry      145
Conjugate space      156
Conservative transformation      355
Continuity from below and above      10
Continuity point of a distribution function      124
Continuous functions dense in $L^{p}$       92
Continuous linear functionals      144-147
Continuous linear operator      142
Continuous paths      399
Continuous process      399
Continuous random variable      175
Convergence almost everywhere      48 96
Convergence almost uniformly      97
Convergence in distribution      290 297
Convergence in measure      96
Convergence in probability      96 297
Convergence of sequences of measurable functions      96ff.
Convergence of transformed sequences      335
Convergence of types      314 447
Convergence theorems for submartmgales      259ff.
Convergence to a normal distribution      307ff.
Convex function      253
Convex set      133
Convolution      328
Correlation coefficient      194
Countably additive set function      5 62
Counting measure      6 10 20 89 94 112
Covariance      194 343
Cram $\acute{e}$ r - Wold device      343
Cylinder      113 117
De Morgan laws      1
Degenerate random variable      298
Density (density function) of a random variable      175
Density (density function) of a random vector      177
Density (density function) of a signed measure      68
Difference operator      27
Differentiating under the integral sign      53-54
Differentiation of measures      76
Differentiation of real-valued functions      78
Discrete probability space      167
Discrete random variable      174
Discrete random vector      177
Distribution function      22 24 28 30 337
Distribution function of a random variable      173
Distribution function of a random vector      176
Distribution of a random object      185
Dominated Convergence Theorem      50 100 223
Dot product      130
Dyadic rationale      404
Egoroff's theorem      98
Empirical distribution function      331
Ensemble average      349
entropy      376 378 386 391
Ergodic theorems      355-357 361
Ergodic transformation      350
Essential supremum      93
Events      166
Events, prior to a stopping time      270 414
Expectation      188
Exponential density      175
Extended monotone convergence theorem      49 223
Extended random variable      173
Extension of a measure      17 19 21 24 30
Fatou's lemma      223
Feller's theorem      314 443
Field of sets      3
Finite set function      8
Finitely additive set function      5
Flow      345
For sums      91
Fourier series      141
Fourier transform      292 296
Fubini's Differentiation Theorem      83
Fubini's theorem      105 108 109 111 112
Functions of independent random objects are independent      179
Gambler's ruin      200
Gamma distribution      324
Gamma function      202
Gaussian random vector      449
Generalized Bernoulli trials      170

Geometric distribution      328
Glivenko - Cantelli theorem      331
Good sets principle      5 11 34 36
Gram - Schmidt process      139
Gramian      139
H $\ddot{o}$ lder inequality      85 94 95
Hahn - Banach theorem      153 155
Helly's theorem      300 339
Hermite polynomials      139
Hewitt - Savage zero-one law      246
Hilbert space      128
Hitting time      270
Idempotent linear operator      144
Improper integrals      57-58
Incompressible transformation      355
Increasing function      22 28 337
Indefinite integral      61 75
Independent and identically distributed (iid) random variables      241 308
Independent classes of sets      181
Independent events      168
Independent random variables (extended random variables, random objects)      177
Indicator      37
Infinite sequences of random variables      196
Infinitely divisible distributions      322-329
Infinitely recurrent transformation      355
Initial distribution      196
Inner measure      233
Inner product      128
Inner product space      128
Integrable function      39 83
integral      37ff.
Integrating an infinite series term by term      53-54
Invariant set      350
Inversion formula      292 341
Irreducible Maikov chain      285
Isometric isomorphism      137
Isomorphic transformations      391
It $\hat{o}$ integrals      426
It $\hat{o}$ 's differentiation formula      433
Jensen's inequality      254
Joint distribution function      177
Joint entropy      376
Jointly Gaussian random variables      449
Jordan - Hahn decomposition theorem      62
Kolmogorov - Sinai theorem      393
Kolmogorov Extension Theorem      118
Kolmogorov strong law of large numbers      240
Kolmogorov three series theorem      282
Kolmogorov zero-one law      245
Kolmogorov's inequality      237
Kronecker lemma      236
last element      266 270 274
Law of the iterated logarithm      410-413
Lebesgue - Stieltjes measure      22 24 27 30
Lebesgue decomposition theorem      70
Lebesgue integrable fucntion      53
Lebesgue integral      37ff.
Lebesgue measurable function      41
Lebesgue measurable sets      26 31 34
Lebesgue measure      26 31
Lebesgue set      81
Legendre polynomials      139
Length      6
Levy - Khintchine representation      327
Levy's extension of the Borel - Cantelli lemma      285
Levy's theorem      304 305 342
lim inf      2
lim sup      2
Limit of a sequence of sets      2
Limit under the integral sign      53-54
Lindeberg's theorem      307
Line of support theorem      253
Linear functional      144
Linear manifold      133
Linear operator      142
Lipschitz condition      81
Lower limit      2
Lower semicontinuous (LSC) functions      122 441
Lower variation      62
Lyapunov's condition      309
Markov chains      196 198 251 261 262 285 368-374
Markov property      369 414
Martingale      248ff. 420
Martingale, convergence theorems      257ff.
Martingale, differences (orthogonality of)      280
Maximal ergodic theorem      357
Mean      191 343
Mean ergodic theorem      365-366
Measurable cylinder      113 117
Measurable function      36
Measurable process      400
Measurable rectangle      102 113 117
Measurable set      36
Measurable space      36
Measurable transformation      345
Measure      5 6
Measure, concentrated on a set      6 26 60
Measure, space      5
Measure-preserving transformation      52 345
Measures on infinite product spaces      113ff.
Minimal $\sigma$ -field over a class of sets      4
Minkowski inequality      86 94
Minkowski inequality, for sums      91
Mixing transformation      352
Moment-generating property of characteristic functions      299
Moments      191
Monotone class theorem      18 21
Monotone Convergence Theorem      46 222
Multivariate normal distribution      449
Mutually singular measures      68
Negative part      38 62
Non-anticipating process      426
Nondegenerate random variable      314
Nonnegative definite funtion      296
Norm      87 127
Norm of a linear operator      142
Normal density      176 192 297
Normed linear space      127
Normed sums      317
Nowhere differentiability of Brownian motion paths      408
Null space      145
One-sided shifts      143 346 353
Open mapping theorem      161
Optional sampling theorem      273
Optional skipping theorem      257
Optional stopping theorem      279
Ornstein's theorem      397-398
Orthogonal (perpendicular) complement      135
Orthogonal (perpendicular) direct sum      135
Orthogonal (perpendicular) elements      132
Orthonormal basis      135-137
Orthonormal set      132
Outer measure      16 21 233
Parallelogram law      131
Parseval relation      136
Path      399
permutations      346 353
Pointwise converence of linear operators      148
Pointwise ergodic theorem      361
Poisson distribution      321
Poisson process      399
Poisson type      324
Polya urn scheme      262
Positive contraction operator      356
Positive part      38 62
Positive-homogeneity      153
Pre - Hilbert space      128
Principle of uniform boundedness      158
Probability function      174 177
Probability measure      5 6

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