We prove a sharp inequality conjectured by Bobkov on the measure of dilations of Borel sets in the Euclidean space by a s-concave probability measure. Our result gives a common generalization of an inequality of Nazarov, Sodin and Volberg and a concentration inequality of Guédon. Applying our inequality to the level sets of functions satisfying a Remez type inequality, we deduce, as it is class...

Several maximal inequalities of Kahane-Khintchine’s type in certain Orlicz spaces are proved. The method relies upon Lévy’s inequality and the technique established in [14] which is obtained by Haagerup-Young-Stechkin’s best possible constants in the classical Khintchine inequalities. Moreover by using Donsker’s invariance principle it is shown that the numerical constant in the inequality dedu...

Over the last thirty years, the use of child safety seats in motor vehicles has increased dramatically, fueled by well publicized information campaigns and legal mandates. In spite of this movement, there is relatively little empirical evidence regarding the efficacy of child safety seats relative to the much cheaper alternative of traditional seat belts. Using data from the Fatality Analysis R...

The Kahane–Salem–Zygmund inequality for multilinear forms in ?? spaces claims that, all positive integers m,n1,...,nm, there exists an m-linear form A:??n1×?×??nm?K (K=R or C) of the typeA(z(1),...,z(m))=?j1=1n1??jm=1nm±zj1(1)?zjm(m), satisfying?A??Cmmax?{n11/2,…,nm1/2}?j=1mnj1/2, forCm??mlog?mm! and a certain ?>0. Our main result shows that given any ?>0 integer m, N such thatCm<1+?, when we c...

S. Adams, W. Ambrose, A. Andretta, H. Becker, R. Camerlo, C. Champetier, J.P.R. Christensen, D.E. Cohen, A. Connes. C. Dellacherie, R. Dougherty, R.H. Farrell, F. Feldman, A. Furman, D. Gaboriau, S. Gao, V. Ya. Golodets, P. Hahn, P. de la Harpe, G. Hjorth, S. Jackson, S. Kahane, A.S. Kechris, A. Louveau,, R. Lyons, P.-A. Meyer, C.C. Moore, M.G. Nadkarni, C. Nebbia, A.L.T. Patterson, U. Krengel,...

It would be quite impossible to give a survey of these subjects in a short article or lecture, and I will only succeed by making some arbitrary restrictions on the topics with which I will deal . First of all, I will restrict myself to problems and results on which I worked, and secondly, I will not discuss subjects which have been discussed in recently appeared review articles [1] . Probabilis...

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with lognormal weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation; these measures fall under the scope of the Gaussian multiplicative chaos theory developed by J....

The distribution νλ of the random series ∑ ±λ is the infinite convolution product of 1 2 (δ−λn + δλn). These measures have been studied since the 1930’s, revealing connections with harmonic analysis, the theory of algebraic numbers, dynamical systems, and Hausdorff dimension estimation. In this survey we describe some of these connections, and the progress that has been made so far on the funda...