Chapter 27: Probability theory
271
Distributions
Terminology; distributions as Radon measures; distribution
functions; densities; transformations of random variables;
*distribution functions and convergence in measure.
272
Independence
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.
273
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.
274
The Central Limit Theorem
Normally distributed r.v.s; Lindeberg's condition for the Central
Limit Theorem; corollaries; estimating
∫α∞exp(-x2/2)dx.
275
Martingales
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.
276
Martingale difference sequences
Martingale difference sequences; strong law of large numbers for
m.d.ss.; Komlós' theorem.
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