Jean-Pierre Kahane

Some random series of functions

1. A few tools from probability theory 2. Random series in a Banach space 3. Random series in a Hilbert space 4. Random Taylor series 5. Random Fourier series 6. A bound for random trigonometric polynomials 7. Conditions on coefficients for regularity 8. Conditions on coefficients for irregularity 9. Random point masses on the circle 10. A few geometric notions 11. Random translates and coverings 12. Gaussian variables and Gaussian series 13. Gaussian Taylor series 14. Gaussian Fourier series 15. Boundedness and continuity for Gaussian processes 16. The Brownian motion 17. Brownian images in harmonic analysis 18. Fractional Brownian images and level sets.

Une inegalite du type de Slepian et Gordon sur les processus gaussiens

A new proof and extension of the Slepian-Gordon inequality is given.

Produits de poids aléatoires indépendants et applications

Le recouvrement par des arcs disposes au hasard, les cascades self-similaires de Benoit Mandelbrot, et le chaos multiplicatif sont des cas particuliers d’une theorie generale des produits de poids aleatoires independants. Un tel produit definit un operateur Q qui transforme les mesures en mesures aleatoires. Les resultats les plus precis sont obtenus pour les recouvrements. La caracterisation, en termes de capacite nulle par rapport a un noyau convenable, des ensembles presque surement recouverts, est exposee sous une forme plus compacte et plus claire que precedemment. Elle s’explique d’ailleurs par un theoreme de Fitzsimmons, Fristedt et Shepp dont une demonstration rapide est donnee a la fin du cours.

Random Coverings and Multiplicative Processes

The problems and results on random coverings are described as they arose in the course of history. Then it is explained how the method of multiplicative processes applies in that connection and further examples and generalizations are given.

A Century of Interplay Between Taylor Series, Fourier Series and Brownian Motion

This article is an extended version of a lecture given in Oxford on 12 May 1995 at the invitation of the London Mathematical Society and the British Society for the History of Mathematics.Contents1. A few figures2. Taylor series before 1900. A strange statement of Borel3. Fourier series before 1900. A strange field4. Brownian motion around 1900. A rising subject5. Fourier and Taylor series after 1900. A revival6. Lacunarity and randomness7. The appearance of random series of functions8. The Wiener theory of Brownian motion9. The merging of Brownian motion and random Fourier series10. The non-differentiability and local behaviour of Brownian motion11. Three ways to figure out the Brownian motion12. The plane Brownian motion13. Applications of Brownian motion to Taylor series and analytic functions14. Applications of Brownian motion to Fourier series and harmonic analysis

Moyennes uniformes et moyennes suivant une marche aléatoire

SummaryLet ϕ be a bounded function on ℤ such that

From Riesz Products to Random Sets

\frac{{\text{1}}}{n}\sum\limits_{j = 1}^n {\varphi {\text{(}}m - j{\text{)}}}

Une Inégalité de Heisenberg Inverse

converges towards l as n goes to infinity, uniformly with respect to m. Let {Xn} be a random walk on ℤ, not concentrated on a proper subgroup of ℤ Then, with probability 1,