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