Two remarks on generalized entropy power inequalities

This note contributes to the understanding of generalized entropy power inequalities. Our main goal is to construct a counter-example regarding monotonicity and entropy comparison of weighted sums of independent identically distributed log-concave random variables. We also present a complex analogue of a recent dependent entropy power inequality of Hao and Jog, and give a very simple proof.Comment: arXiv:1811.00345 is split into 2 papers, with this being on

Intertwining relations for one-dimensional diffusions and application to functional inequalities

International audienceFollowing the recent work [13] fulfilled in the discrete case, we pro- vide in this paper new intertwining relations for semigroups of one-dimensional diffusions. Various applications of these results are investigated, among them the famous variational formula of the spectral gap derived by Chen and Wang [15] together with a new criterion ensuring that the logarithmic Sobolev inequality holds. We complete this work by revisiting some classical examples, for which new estimates on the optimal constants are derived

Dimension-independent Harnack inequalities for subordinated semigroups

Dimension-independent Harnack inequalities are derived for a class of subordinate semigroups. In particular, for a diffusion satisfying the Bakry-Emery curvature condition, the subordinate semigroup with power $\alpha$ satisfies a dimension-free Harnack inequality provided $\alpha \in(1/2, 1)$, and it satisfies the log-Harnack inequality for all $\alpha \in (0,1).$ Some infinite-dimensional examples are also presented

On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures

This note, mostly expository, is devoted to Poincar{\'e} and log-Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interaction. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from convexity. We prove and characterize optimality in the case of quadratic confinement via a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gaussian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact solvability by reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting the Hermite-Lassalle orthogonal polynomials as a complete set of eigenfunctions. We also discuss the consequence of the log-Sobolev inequality in terms of concentration of measure for Lipschitz functions such as maxima and linear statistics.Comment: Minor improvements. To appear in Geometric Aspects of Functional Analysis -- Israel Seminar (GAFA) 2017-2019", Lecture Notes in Mathematics 225

Derivative based global sensitivity measures

International audienceThe method of derivative based global sensitivity measures (DGSM) has recently become popular among practitioners. It has a strong link with the Morris screening method and Sobol' sensitivity indices and has several advantages over them. DGSM are very easy to implement and evaluate numerically. The computational time required for numerical evaluation of DGSM is generally much lower than that for estimation of Sobol' sensitivity indices. This paper presents a survey of recent advances in DGSM concerning lower and upper bounds on the values of Sobol' total sensitivity indices $S_{i}^{tot}$. Using these bounds it is possible in most cases to get a good practical estimation of the values of $S_{i}^{tot}$. Several examples are used to illustrate an application of DGSM

Evolutionary Modeling of Rate Shifts Reveals Specificity Determinants in HIV-1 Subtypes

A hallmark of the human immunodeficiency virus 1 (HIV-1) is its rapid rate of evolution within and among its various subtypes. Two complementary hypotheses are suggested to explain the sequence variability among HIV-1 subtypes. The first suggests that the functional constraints at each site remain the same across all subtypes, and the differences among subtypes are a direct reflection of random substitutions, which have occurred during the time elapsed since their divergence. The alternative hypothesis suggests that the functional constraints themselves have evolved, and thus sequence differences among subtypes in some sites reflect shifts in function. To determine the contribution of each of these two alternatives to HIV-1 subtype evolution, we have developed a novel Bayesian method for testing and detecting site-specific rate shifts. The RAte Shift EstimatoR (RASER) method determines whether or not site-specific functional shifts characterize the evolution of a protein and, if so, points to the specific sites and lineages in which these shifts have most likely occurred. Applying RASER to a dataset composed of large samples of HIV-1 sequences from different group M subtypes, we reveal rampant evolutionary shifts throughout the HIV-1 proteome. Most of these rate shifts have occurred during the divergence of the major subtypes, establishing that subtype divergence occurred together with functional diversification. We report further evidence for the emergence of a new sub-subtype, characterized by abundant rate-shifting sites. When focusing on the rate-shifting sites detected, we find that many are associated with known function relating to viral life cycle and drug resistance. Finally, we discuss mechanisms of covariation of rate-shifting sites

On the central limit theorem along subsequences of noncorrelated observations

Bobkov SG, Götze F. On the central limit theorem along subsequences of noncorrelated observations. THEORY OF PROBABILITY AND ITS APPLICATIONS. 2003;48(4):604-621.We study the asymptotic behavior of distributions of normalized and self-normalized sums along suitable subsequences of noncorrelated random variables

On moments of polynomials

Bobkov SG, Götze F. On moments of polynomials. THEORY OF PROBABILITY AND ITS APPLICATIONS. 1998;42(3):518-A520.The equivalence of L-p-norms of polynomials of random variables is investigated