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Compact convex sets of the plane and probability theory

Jean-François Marckert; David Renault — 2014

ESAIM: Probability and Statistics

The Gauss−Minkowski correspondence in ℝ states the existence of a homeomorphism between the probability measures on [0,2] such that $\int_{0}^{2 π} e^{i x} d μ (x) = 0$ ∫ 0 2 π e ix d μ ( x ) = 0 and the compact convex sets (CCS) of the plane with perimeter 1. In this article, we bring out explicit formulas relating the border of a CCS to its probability measure. As a consequence, we show that some natural operations on CCS – for example, the Minkowski sum – have natural translations in terms of probability measure operations, and...

On the reduction of a random basis

Ali Akhavi; Jean-François Marckert; Alain Rouault — 2009

ESAIM: Probability and Statistics

For , let be independent random vectors in $ℝ^{n}$ with the same distribution invariant by rotation and without mass at the origin. Almost surely these vectors form a basis for the Euclidean lattice they generate. The topic of this paper is the property of reduction of this random basis in the sense of Lenstra-Lenstra-Lovász (LLL). If ${\hat{b}}_{1}^{(n)}, ..., {\hat{b}}_{p}^{(n)}$ is the basis obtained from by Gram-Schmidt orthogonalization, the quality of the reduction depends upon the sequence of ratios of squared lengths of consecutive vectors...

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Currently displaying 1 – 7 of 7

Compact convex sets of the plane and probability theory

On the reduction of a random basis

Parking functions, empirical processes, and the width of rooted labeled trees.

Uniform bounds for exponential moment of maximum of a Dyck path.

Some families of increasing planar maps.

Asymptotics of Bernoulli random walks, bridges, excursions and meanders with a given number of peaks.

Directed animals and gas models revisited.