Chapter 27: Probability theory
Terminology; distributions as Radon measures; distribution functions; densities; transformations of random variables; *distribution functions and convergence in measure.
Independent families of random variables; characterizations of independence; joint distributions of (finite) independent families, and product measures; the zero-one law; E(X×Y), Var(X+Y); distribution of a sum as convolution of distributions; Etemadi's inequality; *Hoeffding's inequality.
The strong law of large numbers
(1/(n+1))Σi=0nXi → 0 a.e. if the Xn are independent with zero expectation and either (i) Σn=0∞1/(n+1)2 Var(Xn)<∞ or (ii) Σn=0∞E(|Xn|1+δ)<∞ for some δ>0 or (iii) the Xn are identically distributed.
The Central Limit Theorem
Normally distributed r.v.s; Lindeberg's condition for the Central Limit Theorem; corollaries; estimating ∫α∞exp(-x2/2)dx.
Sequences of σ-algebras, and martingales adapted to them; up-crossings; Doob's Martingale Convergence Theorem; uniform integrability, ∥ ∥1-convergence and martingales as sequences of conditional expectations; reverse martingales; stopping times.
Martingale difference sequences
Martingale difference sequences; strong law of large numbers for m.d.ss.; Komlós' theorem.
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