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Williams D. Probability with Martingales
Williams D.  Probability with Martingales









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: Probability with Martingales

: Williams D.

:

The author adopts the martingale theory as his main theme in this introduction to the modern theory of probability, which is, perhaps, at a practical level, one of the most useful mathematical theories ever devised.


: en

: /

:

ed2k: ed2k stats

: 1991

: 251

: 03.03.2006

: | | ID
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$\pi$-system      (1.6)
$\pi$-system, Uniqueness Lemmas      (1.6 4.2)
$\sigma$-algebra      (1.1)
'extension of measures', existence      (1.7)
'extension of measures', uniqueness      (1.6)
A.N. KOLMOGOROVs Definition of Conditional Expectation      (9.2)
A.N. KOLMOGOROVs Inequality      (14.6)
A.N. KOLMOGOROVs Law of the Iterated Logarithm      (A4.1 14.7)
A.N. KOLMOGOROVs Strong Law of Large Numbers      (12.10 14.5)
A.N. KOLMOGOROVs Three-Series Theorem      (12.5)
A.N. KOLMOGOROVs Truncation Lemma      (12.9)
A.N. KOLMOGOROVs Zero-One (01) Law      (4.11 14.3)
Abracadabra      (4.9 E10.6)
Adapted process      (10.2)
Adapted process, Doob decomposition      (12.11)
Algebra of sets      (1.1)
Almost everywhere = a.e.      (1.5)
Almost surely = a.s.      (2.4)
Atoms of $\sigma$-algebra      (9.1 14.13)
Atoms of distribution function      (16.5)
Azuma Hoeffding inequality      (E14.2)
Baire category theorem      (A1.12)
Banach Tarski paradox      (1.0)
Bayes' formula      (15.715.9)
Bellman optimality principle      (E10.2 15.3)
Black Scholes option-pricing formula      (15.2)
Blackwells Markov chain      (E4.8)
Bochners theorem      (E16.5)
Borel Cantelli Lemmas, First = BC1      (2.7)
Borel Cantelli Lemmas, Levys extension of      (12.15)
Borel Cantelli Lemmas, Second = BC2      (4.3)
Bounded Convergence Theorem = BDD      (6.2 13.6)
Branching process      (Chapter 0 E12.1)
Burkholder Davis Gundy inequality      (14.18)
Caratheodorys Lemma      (A1.7)
Caratheodorys Theorem, proof      (A1.8)
Caratheodorys Theorem, statement      (1.7)
Central limit theorem      (18.4)
Cesaros Lemma      (12.6)
Characteristic functions, convergence theorem      (18.1)
Characteristic functions, definition      (16.1)
Characteristic functions, inversion formula      (16.6)
Chebyshev's inequality      (7.3)
coin tossing      (3.7)
Conditional expectation      (Chapter 9)
Conditional expectation, properties      (9.7)
Conditional probability      (9.9)
Consistency of Likelihood-Ratio Test      (14.17)
Contraction property of conditional expectation      (9.7
Convergence in probability      (13.5 A13.2)
Convergence theorems for integrals, BDD      (6.2 13.6)
Convergence theorems for integrals, DOM      (5.9)
Convergence theorems for integrals, Fatou      (5.4)
Convergence theorems for integrals, for UI RVs      (13.7)
Convergence theorems for integrals, MON      (5.3)
Convergence theorems for martingales, Downward      (14.4)
Convergence theorems for martingales, for UI case      (14.1)
Convergence theorems for martingales, Main      (11.5)
Convergence theorems for martingales, Upward      (14.2)
d-system      (A1.2)
Differentiation under integral sign      (A16.1)
Distribution function for RV      (3.103.11)
Dominated Convergence Theorem = DOM      (5.9)
Dominated Convergence Theorem = DOM, conditional      (9.7
Downward Theorem      (14.4)
Dynkins Lemma      (A1.3)
Events      (Chapter 2)
Events, independent      (4.1)
Expectation      (6.1)
Expectation, conditional      (Chapter 9)
Extinction probability      (0.4)
Fair game      (10.5)
Fair game, unfavourable      (E4.7)
Fatou Lemmas for functions      (5.4)
Fatou Lemmas for sets      (2.6 2.7
Fatou Lemmas, conditional version      (9.7
Filtered space, filtration      (10.1)
Filtering      (15.615.9)
Finite and $\sigma$-finite measures      (1.5)
Forward Convergence Theorem for supermartingales      (11.5)
Fubinis Theorem      (8.2)
Gamblers ruin      (E10.7)
Gambling strategy      (10.6)
Hardy space $\mathcal{H}_{0}^{1}$      (14.18)
harnesses      (15.1015.12)
Hedging strategy      (15.2)
Helly Bray lemma      (17.4)
Hitting times      (10.12)
Hoeffdings Inequality      (E14.2)
Hoelders inequality      (6.13)
Hunts Lemma      (E14.1)
Independence and product measure      (8.4)
Independence, $\pi$-system criterion      (4.2)
Independence, and conditioning      (9.7 9.10)
Independence, definitions      (4.1)
Inequalities, Azuma Hoeffding      (E14.2)
Inequalities, Burkholder Davis Gundy      (14.18)
Inequalities, Chebyshev      (7.3)
Inequalities, Doobs $\mathcal{L}^{p}$      (14.11)
Inequalities, Hoelder      (6.13)
Inequalities, Jensen      (6.6)
Inequalities, Jensen, and in conditional form      (9.7
Inequalities, Khinchine - see      (14.8)
Inequalities, Kolmogorov      (14.6)
Inequalities, Markov      (6.4)
Inequalities, Minkowski      (6.14)
Inequalities, Schwarz      (6.8)
Infinite products of probability measures      (8.7 Chapter
Infinite products of probability measures, Kakutanis Theorem on      (14.12 14.17)
Integration      (Chapter 5)
J.L. DOOBs Convergence Theorem      (11.5)
J.L. DOOBs Convergence Theorem, $\mathcal{L}^{p}$ inequality      (14.11)
J.L. DOOBs Convergence Theorem, Decomposition      (12.11)
J.L. DOOBs Convergence Theorem, Optional Sampling Theorem      (A14.314.4)
J.L. DOOBs Convergence Theorem, Optional Stopping Theorem      (10.10 A14.3)
J.L. DOOBs Convergence Theorem, Submartingale Inequality      (14.6)
J.L. DOOBs Convergence Theorem, Upcrossing Lemma      (11.2)
Jensens Inequality      (6.6)
Jensens inequality, conditional form      (9.7
Kakutanis Theorem on likelihood ratios      (14.12 14.17)
Kalman Bucy filter      (15.615.9)
Kroneckers lemma      (12.7)
Laplace transforms, and weak convergence      (E18.7)
Laplace transforms, inversion      (E7.1)
Law of random variable      (3.9)
Law of random variable, joint laws      (8.3)
Least-squares-best predictor      (9.4)
Lebesgue integral      (Chapter 5)
Lebesgue measure = Leb      (1.8 A1.9)
Lebesgue spaces $\mathcal{L}^{p}$, $L^{p}$      (6.10)
Likelihood-Ratio Test, consistency of      (14.17)
Mabinogion sheep      (15.315.5)
Markov chain      (4.8 10.13)
Markovs inequality      (6.4)
Martingale      (Chapters 1015!)
Martingale transform      (10.6)
Martingale, convergence theorem      (11.5)
Martingale, definition      (10.3)
Martingale, Optional-Sampling Theorem      (Chapter A14)
Martingale, Optional-Stopping Theorem      (10.910.10 A14.3)
Measurable function      (3.1)
Measurable space      (1.1)
Measure space      (1.4)
Minkowskis inequality      (6.14)
Moment problem      (E18.6)
Monkey typing Shakespeare      (4.9)
Monotone-Class Theorem      (3.14 A1.3)
Monotone-Convergence Theorem for functions      (5.3 Chapter
Monotone-Convergence Theorem for sets      (1.10)
Monotone-Convergence Theorem, conditional version      (9.7
Narrow convergence      see "Weak convergence"
Option pricing      (15.2)
Optional time      (10.8) see
Optional-Sampling Theorem      (Chapter A14)
Optional-Stopping Theorems      (10.910.10 A14.3)
Orthogonal projection      (6.11)
Orthogonal projection and conditional expectation      (9.49.5)
Outer measures      (A1.6)
P.LEVYs Convergence Theorem for CFs      (18.1)
P.LEVYs Downward Theorem for martingales      (14.4)
P.LEVYs Extension of Borel Cantelli Lemmas      (12.15)
P.LEVYs Inversion formula for CFs      (16.6)
P.LEVYs Upward Theorem for martingales      (14.2)
Polyas urn      (E10.1 E10.8)
Previsible (= predictable) process      (10.6)
Probability density function = pdf      (6.12)
Probability density function = pdf, joint      (8.3)
Probability measure      (1.5)
Probability triple      (2.1)
Product measures      (Chapter 8)
Pythagorass theorem      (6.9)
Radon Nikodym theorem      (5.14 14.1314.14)
Random signs      (12.3)
Random walk, hitting times      (10.12 E10.7)
Random walk, on free group      (EG.3EG.4)
Record Problem      (E4.3 E12.2 18.5)
Regular conditional probability      (9.9)
Riemann integral      (5.3)
Sample path      (4.8)
Sample point      (2.1)
Sample space      (2.1)
Schwarz inequality      (6.8)
Star Trek problems      (E10.10 E10.11 E12.3)
Stopped process      (10.9)
Stopping times      (10.8)
Stopping times, associated $\sigma$-algebras      (A14.1)
Strassens Law of the Iterated Logarithm      (A4.2)
Strong Laws      (7.2 12.10 12.14 14.5)
Submartingales and supermartingales, convergence theorem      (11.5)
Submartingales and supermartingales, definitions      (10.3)
Submartingales and supermartingales, optional sampling      (A14.4)
Submartingales and supermartingales, optional stopping      (10.910.10)
Superharmonic functions for Markov chains      (10.13)
Symmetrization technique      (12.4)
Tail $\sigma$-algebra      (4.104.12 14.3)
Tchebycheff = Chebyshev      (7.3)
Three-series theorem      (12.5)
Tightness      (17.5)
Tower property      (9.7
Truncation Lemma      (12.9)
Uniform integrability      (Chapter 13)
Uniqueness lemma      (1.6)
Upcrossing lemma      (11.111.2)
Weak convergence      (Chapter 17)
Weak convergence and characteristic functions      (18.1)
Weak convergence and Laplace transforms      (E18.7)
Weak convergence and moments      (E18.318.4)
Weierstrass approximation theorem      (7.4)
Zero-one law = 01 law      (4.11 14.3)
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