Chapter
8 Laplace’s law of succession
9 Back to the future, the Copernican principle
10 Ambiguous communication
III A First Look at Independence
3 An application in genetics, Hardy’s law
5 Independent families, Dynkin’s π–λ theorem
1 Inclusion and exclusion
2 The sieve of Eratosthenes
3 On trees and a formula of Cayley
THEOREM 1 Every finite tree contains at least two leaves.
THEOREM 2 Every tree on n vertices contains n − 1 edges.
4 Boole’s inequality, the Borel–Cantelli lemmas
5 Applications in Ramsey theory
6 Bonferroni’s inequalities, Poisson approximation
7 Applications in random graphs, isolation
8 Connectivity, from feudal states to empire
9 Sieves, the Lovász local lemma
10 Return to Ramsey theory
11 Latin transversals and a conjecture of Euler
V Numbers Play a Game of Chance
2 Binary digits, Rademacher functions
3 The independence of the binary digits
4 The link to coin tossing
5 The binomial makes an appearance
6 An inequality of Chebyshev
7 Borel discovers numbers are normal
1 One curve to rule them all
2 A little Fourier theory I
3 A little Fourier theory II
5 Lévy suggests a thin sandwich, de Moivre redux
8 The limits of wireless cohabitation
VII Probabilities on the Real Line
1 Arithmetic distributions
4 Densities in one dimension
5 Densities in two and more dimensions
6 Randomisation, regression
7 How well can we estimate?
8 Galton on the heredity of height
9 Rotation, shear, and polar transformations
VIII The Bernoulli Schema
2 The binomial distribution
3 On the efficacy of polls
5 The arc sine laws, will a random walk return?
6 Law of small numbers, the Poisson distribution
7 Waiting time distributions
8 Run lengths, quality of dyadic approximation
9 The curious case of the tennis rankings
10 Population size, the hypergeometric distribution
IX The Essence of Randomness
1 The uniform density, a convolution formula
2 Spacings, a covering problem
3 Lord Rayleigh’s random flights
4 M. Poincaré joue à la roulette
5 Memoryless variables, the exponential density
7 Waiting times, the Poisson process
8 Densities arising in queuing theory
9 Densities arising in fluctuation theory
10 Heavy-tailed densities, self-similarity
2 Squared normals, the chi-squared density
3 A little linear algebra
4 The multivariate normal
5 An application in statistical estimation
7 The strange case of independence via mixing
8 A continuous, nowhere differentiable function
9 Brownian motion, from phenomena to models
10 The Haar system, a curious identity
11 A bare hands construction
12 The paths of Brownian motion are very kinky
XI Distribution Functions and Measure
2 Measure and its completion
3 Lebesgue measure, countable sets
5 From measure to outer measure, and back
4 Continuous distributions
6 Baire functions, coordinate transformations
7 Two and more dimensions
8 Independence, product measures
9 Do independent variables exist?
10 Remote events are either certain or impossible
1 Measures of central tendency
4 Approximation, monotone convergence
5 Arabesques of additivity
6 Applications of additivity
7 The expected complexity of Quicksort
8 Expectation in the limit, dominated convergence
XIV Variations on a Theme of Integration
1 UTILE ERIT SCRIBIT ∫PRO OMNIA
INTEGRALS WITH RESPECT TO GENERAL MEASURES
2 Change of variable, moments, correlation
3 Inequalities via convexity
5 Iterated integrals, a cautionary example
6 The volume of an n-dimensional ball
7 The asymptotics of the gamma function
8 A question from antiquity
9 How fast can we communicate?
10 Convolution, symmetrisation
11 Labeyrie ponders the diameter of stars
1 The transform of a distribution
3 The renewal equation and process
4 Gaps in the Poisson process
5 Collective risk and the probability of ruin
7 Ladder indices and a combinatorial digression
8 The amazing properties of fluctuations
XVI The Law of Large Numbers
1 Chebyshev’s inequality, reprise
2 Khinchin’s law of large numbers
3 A physicist draws inspiration from Monte Carlo
4 Triangles and cliques in random graphs
6 Some number-theoretic sums
7 The dance of the primes
8 Fair games, the St. Petersburg paradox
9 Kolmogorov’s law of large numbers
10 Convergence of series with random signs
11 Uniform convergence per Glivenko and Cantelli
12 What can be learnt per Vapnik and Chervonenkis
VAPNIK–CHERVONENKIS CLASSES
THE GEOMETRY OF THE SITUATION
A QUESTION OF IDENTIFICATION
XVII From Inequalities to Concentration
1 Exponential inequalities
2 Unreliable transcription, reliable replication
3 Concentration, the Gromov–Milman formulation
4 Talagrand views a distance
6 Sharpening, or the importance of convexity
7 The bin-packing problem
8 The longest increasing subsequence
9 Hilbert fills space with a curve
10 The problem of the travelling salesman
XVIII Poisson Approximation
1 A characterisation of the Poisson
3 Bounds from Stein’s equation
5 The local method, dependency graphs
6 Triangles and cliques in random graphs, reprise
7 Pervasive dependence, the method of coupling
8 Matchings, ménages, permutations
XIX Convergence in Law, Selection Theorems
3 Convolutional operators
4 An inversion theorem for characteristic functions
5 Vector spaces, semigroups
8 Equidistributed numbers, from Kronecker to Weyl
9 Walking around the circle
1 Identical distributions, the basic limit theorem
2 The value of a third moment
5 Varying distributions, triangular arrays
7 On the number of cycles
9 Random walks, random flights
10 A test statistic for aberrant counts
12 The strange case of Sir Cyril Burt, psychologist
XXI Sequences, Functions, Spaces
1 Sequences of real numbers
3 Some L2 function theory
The Theory of Probability
：Explorations and Applications
Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.