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1
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Sur le caractere gaussien de la convergence presque partout

100%
Weber M.
Fundamenta Mathematicae
|
2001
|
tom 167
|
nr 1
23-54
EN
We establish functional type inequalities linking the regularity properties of sequences of operators S = (Sₙ) acting on L²-spaces with those of the canonical Gaussian process on the associated subsets of L² defined by (Sₙ(f)), f ∈ L². These inequalities allow us to easily deduce as corollaries Bourgain's famous entropy criteria in the theory of almost everywhere convergence. They also provide a better understanding of the role of the Gaussian processes in the study of almost everywhere convergence. A partial converse path to Bourgain's entropy criteria is also proposed.
2
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Moment convergence and the law of iterated logarithm for additive functions

64%
Berkes I., Weber M.
Acta Arithmetica
|
2006
|
tom 123
|
nr 1
43-55
3
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On the supremum of random Dirichlet polynomials

64%
Lifshits M., Weber M.
Studia Mathematica
|
2007
|
tom 182
|
nr 1
41-65
EN
We study the supremum of some random Dirichlet polynomials $D_{N}(t) = ∑_{n=2}^{N} εₙdₙn^{-σ-it}$, where (εₙ) is a sequence of independent Rademacher random variables, the weights (dₙ) are multiplicative and 0 ≤ σ < 1/2. Particular attention is given to the polynomials $∑_{n∈ 𝓔_{τ}} εₙn^{-σ-it}$, $𝓔_{τ} = {2 ≤ n ≤ N : P⁺(n) ≤ p_{τ}}$, P⁺(n) being the largest prime divisor of n. We obtain sharp upper and lower bounds for the supremum expectation that extend the optimal estimate of Halász-Queffélec, $𝔼 sup_{t∈ ℝ} |∑_{n=2}^{N} εₙn^{-σ-it}| ≈ (N^{1-σ})/(log N)$. The proofs are entirely based on methods of stochastic processes, in particular the metric entropy method.
4
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On the convergence of moments in the CLT for triangular arrays with an application to random polynomials

64%
Cuny C., Weber M.
Colloquium Mathematicum
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2006
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tom 106
|
nr 1
147-160
EN
We give a proof of convergence of moments in the Central Limit Theorem (under the Lyapunov-Lindeberg condition) for triangular arrays, yielding a new estimate of the speed of convergence expressed in terms of νth moments. We also give an application to the convergence in the mean of the pth moments of certain random trigonometric polynomials built from triangular arrays of independent random variables, thereby extending some recent work of Borwein and Lockhart.
5
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Counting occurrences in almost sure limit theorems

64%
Giuliano-Antonini R., Weber M.
Colloquium Mathematicum
|
2005
|
tom 102
|
nr 2
271-290
EN
Let X,X₁,X₂,... be a sequence of i.i.d. random variables with $X ∈ L^{p}$, 0 < p ≤ 2. For n ≥ 1, let Sₙ = X₁ + ⋯ + Xₙ. Developing a preceding work concerning the L²-case only, we compare, under strictly weaker conditions than those of the central limit theorem, the deviation of the series $∑_{n} wₙ 1_{Sₙ
6
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Convergence of series of dilated functions and spectral norms of GCD matrices

51%
Aistleitner C., Berkes I., Seip K., Weber M.
Acta Arithmetica
|
2015
|
tom 168
|
nr 3
221-246
EN
We establish a connection between the L² norm of sums of dilated functions whose jth Fourier coefficients are $𝓞(j^{-α})$ for some α ∈ (1/2,1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L² and for the almost everywhere convergence of series of dilated functions.
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