Probability with martingales

Описание

vi Contents
First Borel-Cantelli Lemma (BCl). 2.8. Definitions, liminf En,(En,ev).
2.9. Exercise.
Chapter 3: Random Variables 29
3.1. Definitions. S-measurable function, mS, (mS)+,bS. 3.2. Elementary
Propositions on measurability. 3.3. Lemma. Sums and products of mea-
measurable functions are measurable. 3.4. Composition Lemma. 3.5. Lemma
on measurability of infs, liminfs of functions. 3.6. Definition. Random
variable. 3.7. Example. Coin tossing. 3.8. Definition, cr-algebra generated
by a collection of functions on Q. 3.9. Definitions. Law, Distribution Func-
Function. 3.10. Properties of distribution functions. 3.11. Existence of random
variable with given distribution function. 3.12. Skorokod representation of
a random variable with prescribed distribution function. 3.13. Generated
a-algebras - a discussion. 3.14. The Monotone-Class Theorem.
Chapter 4: Independence 38
4.1. Definitions of independence. 4.2. The 7r-system Lemma; and the
more familiar definitions. 4.3. Second Borel-Cantelli Lemma (BC2). 4.4.
Example. 4.5. A fundamental question for modelling. 4.6. A coin-tossing
model with applications. 4.7. Notation: IID RVs. 4.8. Stochastic processes;
Markov chains. 4.9. Monkey typing Shakespeare. 4.10. Definition. Tail c-
algebras. 4.11. Theorem. Kolmogorov's 0-1 law. 4.12. Exercise/Warning.
Chapter 5: Integration 49
5.0. Notation, etc. /i(/) :=: J fdfi, fi(f\A). 5.1. Integrals of non-negative
simple functions, SF+. 5.2. Definition of//(/), / £ (mS)+. 5.3. Monotone-
Convergence Theorem (MON). 5.4. The Fatou Lemmas for functions (FA-
TOU). 5.5. 'Linearity'. 5.6. Positive and negative parts of /. 5.7. Inte-
grable function, C1 (S, S,//). 5.8. Linearity. 5.9. Dominated Convergence
Theorem (DOM). 5.10. Scheffe's Lemma (SCHEFFE). 5.11. Remark on
uniform integrability. 5.12. The standard machine. 5.13. Integrals over
subsets. 5.14. The measure ///, / £ (mS)+.
Chapter 6: Expectation 58
Introductory remarks. 6.1. Definition of expectation. 6.2. Convergence
theorems. 6.3. The notation E(X;F). 6.4. Markov's inequality. 6.5.
Sums of non-negative RVs. 6.6. Jensen's inequality for convex functions.
6.7. Monotonicity of Cp norms. 6.S. The Schwarz inequality. 6.9. C2:
Pythagoras, covariance, etc. 6.10. Completeness of Cp A < p < oo). 6.11.
Orthogonal projection. 6.12. The 'elementary formula' for expectation.
6.13. Holder from Jensen.

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