Some random series of functions

Title page
Preface
1 A few tools from probability theory
1 Introduction
2 The basic notions
3 Distribution and similarity
4 Product probability space
5 The standard model; independence; Steinhaus and Rademacher sequences
6 Integration: the main tools
7 Symmetric random vectors
8 Random functions and analytic sets
2 Random series in a Banach space
1 Introduction
2 Summability methods
3 Sums of symmetric random vectors; two lemmas
4 Proof of theorem 1
5 Rademacher series Σ ±u_n
6 A principle of contraction
7 The strong integrability for Rademacher series
8 Exercises
3 Random series in a Hilbert space
1 Introduction
2 The Kolmogorov inequality
3 The Paley-Zygmund inequalities
4 Positive random series
5 Necessary and sufficient conditions for convergence and boundedness
6 Exercises
4 Random Taylor series
1 Introduction
2 Singular points
3 The symmetric case
4 The general case
5 Random Taylor series in two complex variables
6 Random Dirichlet series
7 Complements and exercises
5 Random Fourier series
1 Introduction
2 Auxiliary results on trigonometric series
3 Rademacher series: the case Σ x²_n = ∞
4 Rademacher series: the case Σ x²_n < ∞
5 The general Paley-Zygmund theorem
6 Auxiliary results on series of translates
7 Convergence and boundedness in C or L^∞
8 Convergence everywhere; the Billard theorem
9 An application: Fourier coefficients of continuous functions
10 Exercises
6 A bound for random trigonometric polynomials and applications
1 Introduction
2 Distribution of M = ||P||_∞
3 Applications; a theorem of Littlewood and Salem; Sidon and Helson sets
4 Another application: generalized almost periodic sequences
5 Polynomials with unimodular coefficients
6 Sums of sinuses
7 Exercises
7 Conditions on coefficients for regularity
1 Introduction
2 A sufficient condition for (1)∈C
3 Estimates for the modulus of continuity (subgaussian case)
4 A sufficient condition for (1)∈Λ_α
5 An application
6 Exercises
8 Conditions on coefficients for irregularity
1 Introduction
2 Unboundedness: the Paley-Zygmund approach
3 Unboundedness: a particular case
4 Unboundedness: the general case
5 Irregularity almost everywhere
6 Irregularity everywhere
7 Simultaneous inequalities
8 Irregularity everywhere (continued)
9 Divergence everywhere
10 Exercises
9 Random point-masses on the circle
1 Introduction
2 Two theorems on Fourier-Stieltjes series
3 Proof of theorem 2
4 An almost everywhere divergent Fourier series
5 Poisson transform of etc.
6 A theorem on conjugate harmonic functions
7 More about the case Σ m²_j = 1
8 Exercises
10 A few geometric notions
1 Introduction
2 Hausdorff measures and dimensions; Frostman's lemma
3 Energy and capacity; Frostman's theorem
4 ε-covering numbers
5 Helices
6 Quasi-helices; von Koch and Assouad curves
7 More on dimensions
8 Exercises
11 Random translates and covering
1 Introduction
2 Covering the circle: a sufficient condition
3 Covering the circle: a necessary condition
4 Covering the circle: the necessary and sufficient condition
5 Covering a subset of T^q by random sets: a necessary condition
6 Covering a subset of T^q: a sufficient condition; the case of convex g_n
7 The case of non-flattening convex g_n; covering a set of given Hausdorff dimension
8 The case of non-flattening convex g_n (continued); dimension of the non-covered set
9 Concluding remarks
10 Exercises
12 Gaussian variables and gaussian series
1 Introduction
2 Formulas on Fourier transforms
3 Gaussian random variables
4 Some more formulas
5 Around the Borel-Cantelli lemma
6 Transient and recurrent gaussian series
7 Gaussian series in a Banach space
8 Exercises
13 Gaussian Taylor series
1 Introduction
2 A review of previous results
3 The range of F(z)(|z| < 1)
4 The radial behavior: a recurrence condition
5 The radial behavior: transience conditions
6 Non-radial behavior: recurrence conditions
7 Transience on circular sets
8 Exercises
14 Gaussian Fourier series
1 Introduction
2 Review of known results
3 Capacities and Hausdorff dimension reviewed
4 Range of F
5 The zeros of F
6 A definition of δ^(q)(F)
7 The Malliavin theorem on spectral synthesis
8 Exercises
15 Boundedness and continuity for gaussian processes
1 Introduction
2 Slepian's lemma
3 Marcus and Shepp's theorem; the Pisier algebra
4 Dudley's theorem
5 Fernique's theorem
6 Non-gaussian Fourier series
7 Exercises
16 The brownian motion
1 Introduction
2 The Wiener process
3 The Fourier-Wiener series
4 More on local properties
5 Stopping times, polar sets and newtonian capacity
6 Self-crossing
17 Brownian images in harmonic analysis
1 Introduction
2 Brownian images
3 Brownian image of a measure; proof of theorem 1
4 Arithmetical properties of brownian images; proof of theorem 2
5 Image of a measure by a gaussian Fourier series
6 A construction of H. Cartan; proof of lemma 6
7 A generalization of theorems 1 and 2
8 Exercises
18 Fractional brownian images and level sets
1 Introduction
2 The gaussian processes (n,d,γ)
3 Fractional brownian image of a measure; new Salem sets
4 Fractional brownian images (continued); occupation density
5 Level sets
6 Uniqueness and continuity of δ(X-x)
7 Graphs
8 Exercises
Notes
Bibliography
Index