Index

absolutely continuous, 237

almost everywhere, 77

convergence, 78

almost surely, 110

arc sine law, 26

Azuma’s inequality, 232

Bayes’s formula, 28

Bernoulli measure, 33

Bernstein polynomial, 132

Berry–Esseen theorem, 142

biased random walk, 35

binomial, 11

coeﬃcient, 11

N choose m, 11

distribution

parameters (n, p), 34

Borel

measurable function, 56

measurable set, 56

measure, 57

Borel–Cantelli lemma, 32, 244

branching process, 178

extinction, 178

Brownian motion, 206

Feynman–Kac formula, 269

L´ evy’s characterization, 263

relative to {Ft : t ≥ 0}, 210

scaling property, 209

strong law, 213

time inversion invariance, 213

centered Gaussian family, 152

central limit theorem, 140

De Moivre’s, 18

Lindeberg’s, 137

Chapman–Kolmogorov equation, 194

Chebychev’s inequality, 44

complement, 4

concave function, 85

conditional expectation, 110

conditional probability, 27

contraction, 171

convergence

μ-almost everywhere, 78

in μ-measure, 79

convex

function, 85

set, 84

convolution, 121

countably additive, 57

covariance, 143

cumulant of a random variable, 51

De Morgan’s law, 6

decreasing events, 5

density of a distribution, 117

difference, 4

discrete arcsine measure, 33

disjoint sets, 4

distribution, 59

having density, 117

of a random variable, 32

of a stochastic process, 193

distribution function, 70

Doeblin’s

281