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Durrett R. — Probability: Theory and Examples

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Название: Probability: Theory and Examples

Автор: Durrett R.

Аннотация:

Modern and measure-theory based, this text is intended primarily for the first-year graduate course in probability theory.

Язык:

Рубрика: Математика/Вероятность/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1996

Количество страниц: 525

Добавлена в каталог: 28.05.2005

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Предметный указатель

$\pi$ — $\lambda$ theorem      25 447
$\sigma$ -algebra = $\sigma$ -field      1
$\sigma$ -field generated by collection of sets      3
$\sigma$ -field generated by random variable      11
$\sigma$ -finite measure      441
Absolutely continuous      8 220 480
Adapted sequence      231
Age dependent branching process      370
Algebra      440
Algebra, generated by semi-algebra      441
Almost sure convergence      12
Alternating renewal process      218
Anosov map      354
Arcsine laws for Brownian motion      396
Arcsine laws for random walks      199 200
Arithmetic distribution      205
Asymptotic density of subset of Z      440
Asymptotic equipartition property for i.i.d. sequences      61
Asymptotic equipartition property for stationary sequences      360
Backgammon      399
Backwards martingale      265
Ballot Theorem      198 267
Banach — Tarski theorem      455
Bayes' formula      222
Benford's law      345
Bernstein polynomials      37
Berry — Esseen theorem      126
Birth and death chains      283 295 301 308
Birthday problem      83
Blackwell's renewal theorem      206
Blumenthal's 0-1 law      384
Bonferroni inequalities      22
Borel sets      2
Borel — Cantelli Lemmas      47 50 240 256
Borel's paradox      224
Bounded convergence theorem      17 466
Branching process      254
Brother sister mating      284
Brownian bridge      431
Brownian motion      375
Brownian motion, continuity of paths      377
Brownian motion, Markov property      381
Brownian motion, modulus of continuity      396
Brownian motion, nondifferentiability      380
Brownian motion, quadratic variation      380
Brownian motion, reflection principle      394
Brownian motion, scaling relation      375
Brownian motion, strong Markov property      390
Brownian motion, tail $\sigma$ -field      386
Brownian motion, zeros of      392
Cantor set      8
Caratheodary's extension theorem      441
Card trick      316
Carleman's condition      110
Cauchy distribution      44
Cauchy — Schwarz inequality      15 465
Central limit theorem for i.i.d. sequences      112 405
Central limit theorem for i.i.d. vectors      170
Central Limit Theorem for Martingales      412 417
Central limit theorem for prime divisors      124
Central limit theorem for random indices      116
Central limit theorem for renewals      116
Central limit theorem for stationary sequences      419
Central order statistic      84
Chain dependent r.v.'s      417
Change of variables formula      18
Chapman — Kolmogorov equation      287
Characteristic function      92
Characteristic function in $R^{d}$       169
Characteristic function, inversion formula      95
Chebyshev's inequality      15 225
Chung — Fuchs theorem      189
Closed set in Markov chain      293
Completion      453
Conditional expectation      219
Conditional variance formula      253
Continued fractions      340
Continuity theorem for ch.f.      99 170
Continuous mapping theorem      87
Convergence in probability      35
Convergence of types      159
Convergence, almost surely      12
Convergence, weak      82 166
Converging together lemma      91
Convolution      30
Countably generated $\sigma$ -field      8
Coupon collector's problem      38 144
Cramer — Wold device      170
Cramer's estimates of ruin      213
Cube, high dimensional      37
Cyclic decomposition      318
De Finetti's theorem      269
de Moivre — Laplace theorem      81
Decomposition theorem      295
Delayed renewal process      206
Density function      7
Directly Riemann integrable      215
distribution      4
Distribution function      4
Dominated Convergence Theorem      16 49 264 468
Donsker's theorem      406
Doob's inequality      250
Double exponential distribution      85
Doubly stochastic      300
Dual transition probability      302
Dubin's inequality      238
Ehrenfest chain      283 301
Empirical distribution      59 428
entropy      61 357
Erdoes — Kac central limit theorem      124
Ergodic sequence      338
Ergodic theorem      341
Exchangeable $\sigma$ -field      174
Extended real line      12
Extreme value distribution      85
Fatou's lemma      16 48 86 467
Filtration      231
First entrance decomposition      290
First passage percolation      371
Friedman's urn      257
Fubini's theorem      470
Galton — Watson process      245
Gaussian processes      376
Germ $\sigma$ -field      384
GI/G/1 queue      327 332
Glivenko — Cantelli theorem      59
Gumbel distribution      85
Hahn decomposition      477
Hamburger moment problem      111
Harris chain      325
Head runs      54
Helly's selection theorem      88
Hewitt — Savage 0-1 law      174 268
Histogram correction      114
Hitting time      176
Hoelder's inequality      15 465
Holtsmark distribution      160
I.i.d.      36
Inclusion-exclusion formula      22
Independence, defined      23
Index of a stable law      157
Indicator function      4
Infinitely divisible distribution      161
Invariant set      337
Inversion formula for ch.f.      95 169
Investment problem      61
Irreducible set in Markov chain      293
Jensen's inequality      14 225 464
Jordan decomposition      478
Kac's recurrence theorem      348

Kakutani dichotomy      244
Kochen — Stone lemma      55
Kolmogorov's continuity criterion      378
Kolmogorov's cycle condition      303
Kolmogorov's extension theorem      32 474
Kolmogorov's maximal inequality      62
Kolmogorov's zero-one law      62
Kronecker's lemma      64
Ky Fan metric      91
Ladder variables      179
Large deviations      70-78
Lattice distribution      131
Law of the iterated logarithm      434
Lazy janitor      59
Lebesgue decomposition      479
Lebesgue measure      2
Levy measure      164
Levy metric      91
Levy — Khintchine Theorem      163
Levy's 0-1 law      263
Levy's modulus of continuity      396
Lindeberg — Feller theorem      116
Lindeberg — Feller theorem for martingales      414
Littlewood's Principles      461
Lognormal distribution      107
Longest common subsequence      363
Lyapunov's theorem      121
m-dependent      421
M/G/l queue      282 298 309 323
M/M/ $\infty$ queue      299 310
Marginal distributions      166
Markov chain      277
Markov chain, additive functional      323
Markov chain, convergence theorem      314
Markov property      285
Markov property of Brownian motion      381
Markov's inequality      15
Martingale      231
Martingale, $L^{p}$ convergence theorem      252
Martingale, $L^{p}$ maximum inequality      251
Martingale, convergence theorem      235
Martingale, orthogonality of increments      253
Matching      142
Maxima, limit theorems for      85
Maximal ergodic lemma      341
Mean      13 19
Measurable map      9
Measurable space      1
Measure      1
Measure, on an algebra      440
Measure, preserving      336
Method of moments      107
Minkowski's inequality      466
Mixing      350 423
moment      19
Moment problem      107
Monotone class theorem      280
Monotone Convergence Theorem      16 225 467
Monte Carlo integration      416
Moving average processes      422
Multivariate normal distribution      171
Mutually singular measures      478
Negative set      476
Nice measurable space      33
Nonarithmetic distribution      205
Nonmeasurable set      453
Normal number      346
Null recurrent      307
Null set      478
Occupancy problem      40 143 148
Optional random variable      176
Optional stopping theorem      273
Ornstein's coupling      322
Outer measure      449
Pairwise independent      34
Parseval relation      191
Pedestrian delay      212
Period of a state      313
Permutable event      173
Poisson process      145 149 151
Pollaczek — Khintchine formula      334
Polya's criterion      104
Polya's distribution      98
Polya's urn scheme      241
Positive recurrent      307
Positive set      476
Predictable sequence      233
Probability measure      2
Probability space      1
Product spaces      3
Products of random matrices      367
Radon — Nikodym derivative      220 241
Radon — Nikodym theorem      220 480
Random index c.l.t.      116
Random matrices      367
Random permutations      38 118
Random permutations, increasing sequences in      369
Random permutations, longest common subequences      363
Random variable      3
Random vector      9
Random walk      173 see
Random walk, on $Z^{d}$       185 321 346
Random walk, on graphs      302
Random walk, on hypercube      318
Random walk, on tree      322
Random walk, range of      346 362
Random walk, symmetric      175
Ratio limit theorems      324
Record values      52 118
Record values, from improving populations      362
Recurrent random walk      185
Recurrent state      292
Reflection principle      197 288 394
Regular conditional probability      229
Renewal equation      210
Renewal measure      205
Renewal process, alternating      218
Renewal process, delayed      206
Renewal process, stationary      207
Renewal process, terminating      212
Renewal Theorem      206 214
Renewal theorem, central limit theorem      116
Renewal theorem, strong law      58
Residual waiting time      217
Riemann integration      462
Riemann — Lebesgue lemma      462
Scheffe's theorem      83
Self-normalized sums      116
Semi-algebra      441
Semi-continuous functions      12
Sequence space      279
Shannon — McMillan — Breiman th.      357
Shannon's Theorem      60
Shift on sequence space      177
shuffling cards      317
Signed measure      476
Simple function      455
Simple random walk      171 176 180
Simple random walk, asymmetric      181 274 297 300
Singular measure      8
Skorokhod's representation      402
Slowly varying      153
Solovay's theorem      455
Span of a lattice distribution      131
St. Petersburg paradox      44
Stable laws      157 159
Stationary distribution      300 306
Stationary measure      300
Stationary renewal process      207
Stationary sequence      335

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