Philippe Marchal

On the Concentration of Measure Phenomenon for Stable and Related Random Vectors

where m(f(X)) is a median of f(X) and where Φ is the (one-dimensional) standard normal distribution function. The inequality (1) has seen many extensions and to date, most of the conditions under which these developments hold require the existence of finite exponential moments for the underlying vector X . It is thus natural to explore the robustness of this “concentration phenomenon” and to study the corresponding results for stable vectors. It is the purpose of these notes to initiate this study and to present a few concentration results for stable and related vectors, freeing us from the exponential moment requirement. Our main result will imply that if X is an α-stable random vector in Rd, then for all x> 0,

Concentration for Norms of Infinitely Divisible Vectors With Independent Components

We obtain dimension-free concentration inequalities for L p norms, p 2, of infinitely divisible random vectors with independent coordinates. The methods and results extend to some other classes of Lipschitz functions.

A class of special subordinators with nested ranges

We construct, on a single probability space, a class of regenerative sets R, indexed by all measurable functions α : [0, 1] → [0, 1]. For each function α, R, has the law of the range of a special subordinator. Constant functions correspond to stable subordinators. If α ≤ β, then R ⊂ R. Other examples of special subordinators are given in the lattice case.

On a new Wiener-Hopf factorization by Alili and Doney

1 Introduction 2 Proof 2.1 Two classes of paths 2.2 The path transform References

Nested Regenerative Sets and Their Associated Fragmentation Process

We give a fractal construction of nested,stable regenerative sets and study the associatedinhomogeneous fragmentation process.

A Combinatorial Approach to the Two-Sided Exit Problem for Left-Continuous Random Walks

We present a combinatorial lemma that provides a new approach to the two-sided exit problem and related questions for left-continuous random walks (i.e., random walks on the integers whose negative steps have size − 1). Some applications to random walks on the circle are also derived.

Periodic Pólya Urns and an Application to Young Tableaux.

P{o}lya urns are urns where at each unit of time a ball is drawn and is replaced with some other balls according to its colour. We introduce a more general model: The replacement rule depends on the colour of the drawn ball and the value of the time (mod p). We discuss some intriguing properties of the differential operators associated to the generating functions encoding the evolution of these urns. The initial partial differential equation indeed leads to ordinary linear differential equations and we prove that the moment generating functions are D-finite. For a subclass, we exhibit a closed form for the corresponding generating functions (giving the exact state of the urns at time n). When the time goes to infinity, we show that these periodic P{o}lya urns follow a rich variety of behaviours: their asymptotic fluctuations are described by a family of distributions, the generalized Gamma distributions, which can also be seen as powers of Gamma distributions. En passant, we establish some enumerative links with other combinatorial objects, and we give an application for a new result on the asymptotics of Young tableaux: This approach allows us to prove that the law of the lower right corner in a triangular Young tableau follows asymptotically a product of generalized Gamma distributions.