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))Σ_{i=0}^{n}*X*_{i} → 0 a.e. if the *X*_{n} are
independent with zero expectation and either (i)
Σ_{n=0}^{∞}1/(*n*+1)^{2}
Var(*X*_{n})<∞ or (ii)
Σ_{n=0}^{∞}**E**(|*X*_{n}|^{1+δ})<∞ for some
δ>0 or (iii) the *X*_{n} 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(-*x*^{2}/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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Revised 30.8.13