Christian Houdré

Bio: Christian Houdré is an academic researcher from Georgia Institute of Technology. The author has contributed to research in topics: Longest increasing subsequence & Longest common subsequence problem. The author has an hindex of 23, co-authored 110 publications receiving 1523 citations. Previous affiliations of Christian Houdré include University of Paris.

Isoperimetric constants for product probability measures

Abstract: A dimension free lower bound is found for isoperimetric constants of product probability measures. From this, some analytic inequalities are derived.

Covariance Identities and Inequalities for Functionals on Wiener and Poisson Spaces

Abstract: We present covariance identities and inequalities for functionals of the Wiener and the Poisson processes. Using Malliavin calculus techniques, an expansion with a remainder term is obtained for the covariance of such functionals. Our results extend known identities and inequalities for functions of multivariate random vectors.

Remarks on deviation inequalities for functions of infinitely divisible random vectors

Abstract: We obtain deviation inequalities for some classes of functions of infinitely divisible random vectors having finite exponential moments.

Small-time expansions for the transition distributions of L\'evy processes

Abstract: Let $X$ be a Levy process with absolutely continuous Levy measure $ u$. Small time polynomial expansions of order $n$ in $t$ are obtained for the tails $P(X_{t}\geq{}y)$ of the process, assuming smoothness conditions on the Levy density away from the origin. By imposing additional regularity conditions on the transition density $p_{t}$ of $X_{t}$, an explicit expression for the remainder of the approximation is also given. As a byproduct, polynomial expansions of order $n$ in $t$ are derived for the transition densities of the process. The conditions imposed on $p_{t}$ require that its derivatives remain uniformly bounded away from the origin, as $t\to{}0$; such conditions are shown to be satisfied for symmetric stable Levy processes as well as for other related Levy processes of relevance in mathematical finance. The expansions seem to correct asymptotics previously reported in the literature.

On Gaussian and Bernoulli covariance representations

Abstract: We discuss several applications, to large deviations for smooth functions of Gaussian random vectors, of a covariance representation in Gauss space. The existence of this type of representation characterizes Gaussian measures. New representations for Bernoulli measures are also derived, recovering some known inequalities.

Convergence of Probability Measures

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

The concentration of measure phenomenon

Abstract: Concentration functions and inequalities Isoperimetric and functional examples Concentration and geometry Concentration in product spaces Entropy and concentration Transportation cost inequalities Sharp bounds of Gaussian and empirical processes Selected applications References Index

Lévy processes and infinitely divisible distributions

Abstract: Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

An Introduction to Random Matrices

Abstract: The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.