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21
Logconcavity and the maximum entropy property of the Poisson distribution
 Stochastic Processes and their Applications
"... We prove that the Poisson distribution maximises entropy in the class of ultralogconcave distributions, extending a result of Harremoës. The proof uses ideas concerning logconcavity, and a semigroup action involving adding Poisson variables and thinning. We go on to show that the entropy is a conc ..."
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Cited by 36 (15 self)
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We prove that the Poisson distribution maximises entropy in the class of ultralogconcave distributions, extending a result of Harremoës. The proof uses ideas concerning logconcavity, and a semigroup action involving adding Poisson variables and thinning. We go on to show that the entropy is a concave function along this semigroup. 1 Maximum entropy distributions It is wellknown that the distributions which maximise entropy under certain very natural conditions take a simple form. For example, among random variables with fixed mean and variance the entropy is maximised by the normal distribution. Similarly, for random variables with positive support and fixed mean, the entropy is maximised by the exponential distribution. The standard technique for proving such results uses the Gibbs inequality, and exploits the fact that, given a function f(x), and fixing Λ(p) = ∫ p(x)f(x)dx, the maximum entropy density is of the form α exp(−βf(x)). Example 1.1 For a density p with mean µ and variance σ2, write φµ,σ2 for the density of a N(µ, σ2) random variable, and define the function Λ(p) = − ∫ p(x) log φ µ,σ2(x)dx.
Poissontype deviation inequalities for curved continuous time Markov chains
 Bernoulli 13 (2007), n
"... In this paper, we present new Poissontype deviation inequalities for continuous time Markov chains whose Wasserstein curvature or Γcurvature is bounded below. Although these two curvatures are equivalent for Brownian motion on Riemannian manifolds, they are not comparable in discrete settings and ..."
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Cited by 20 (3 self)
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In this paper, we present new Poissontype deviation inequalities for continuous time Markov chains whose Wasserstein curvature or Γcurvature is bounded below. Although these two curvatures are equivalent for Brownian motion on Riemannian manifolds, they are not comparable in discrete settings and yield different deviation bounds. In the case of birthdeath processes, we provide some conditions on the transition rates of the associated generator for such curvatures to be bounded below, and we extend the deviation inequalities established by Ané and Ledoux (2000) for continuous time random walks, seen as models in null curvature. Some applications of these tail estimates are given to Brownian driven OrnsteinUhlenbeck processes and M/M/1 queues.
A new Poissontype deviation inequality for Markov jump processes with positive Wasserstein curvature
, 2008
"... The purpose of this paper is to extend the investigation of Poissontype deviation inequalities started by Joulin (2007) to the empirical mean of positively curved Markov jump processes. In particular, our main result generalizes the tail estimates given by Lezaud (1998, 2001). An application to bir ..."
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Cited by 19 (4 self)
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The purpose of this paper is to extend the investigation of Poissontype deviation inequalities started by Joulin (2007) to the empirical mean of positively curved Markov jump processes. In particular, our main result generalizes the tail estimates given by Lezaud (1998, 2001). An application to birthdeath processes completes this work.
Convex entropy decay via the BochnerBakryEmery approach
 ANN. INST. HENRI POINCARÉ PROBAB. STAT
, 2007
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Monotonic Convergence in an InformationTheoretic Law Of Small Numbers
, 2009
"... An “entropy increasing to the maximum” result analogous to the entropic central limit theorem (Barron 1986; Artstein et al. 2004) is obtained in the discrete setting. This involves the thinning operation and a Poisson limit. Monotonic convergence in relative entropy is established for general discr ..."
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Cited by 11 (5 self)
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An “entropy increasing to the maximum” result analogous to the entropic central limit theorem (Barron 1986; Artstein et al. 2004) is obtained in the discrete setting. This involves the thinning operation and a Poisson limit. Monotonic convergence in relative entropy is established for general discrete distributions, while monotonic increase of Shannon entropy is proved for the special class of ultralogconcave distributions. Overall we extend the parallel between the informationtheoretic central limit theorem and law of small numbers explored by Kontoyiannis et al. (2005) and Harremoës et al. (2007, 2008). Ingredients in the proofs include convexity, majorization, and stochastic orders.
Phientropy inequalities for diffusion semigroups. Prépublication, 2009. [BGL01] [CE02
 J. Math. Pures Appl
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The LévyFokkerPlanck equation: Φentropies and convergence to equilibrium
, 2007
"... Abstract. In this paper, we study a FokkerPlanck equation of the form ut = I[u] + div(xu) where the operator I, which is usually the Laplacian, is replaced here with a general Lévy operator. We prove by the entropy production method the exponential decay in time of the solution to the only steady s ..."
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Cited by 8 (1 self)
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Abstract. In this paper, we study a FokkerPlanck equation of the form ut = I[u] + div(xu) where the operator I, which is usually the Laplacian, is replaced here with a general Lévy operator. We prove by the entropy production method the exponential decay in time of the solution to the only steady state of the associated stationnary equation.
Logconcavity, ultralogconcavity and a maximum entropy property of discrete compound Poisson measures
, 2009
"... Abstract Sufficient conditions are developed, under which the compound Poisson distribution has maximal entropy within a natural class of probability measures on the nonnegative integers. Recently, one of the authors [O. Johnson, Stoch. Proc. Appl., 2007] used a semigroup approach to show that the ..."
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Cited by 4 (1 self)
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Abstract Sufficient conditions are developed, under which the compound Poisson distribution has maximal entropy within a natural class of probability measures on the nonnegative integers. Recently, one of the authors [O. Johnson, Stoch. Proc. Appl., 2007] used a semigroup approach to show that the Poisson has maximal entropy among all ultralogconcave distributions with fixed mean. We show via a nontrivial extension of this semigroup approach that the natural analog of the Poisson maximum entropy property remains valid if the compound Poisson distributions under consideration are logconcave, but that it fails in general. A parallel maximum entropy result is established for the family of compound binomial measures. Sufficient conditions for compound distributions to be logconcave are discussed and applications to combinatorics are examined; new bounds are derived on the entropy of the cardinality of a random independent set in a clawfree graph, and a connection is drawn to Mason's conjecture for matroids. The present results are primarily motivated by the desire to provide an informationtheoretic foundation for compound Poisson approximation and associated limit theorems, analogous to the corresponding developments for the central limit theorem and for Poisson approximation. Our results also demonstrate new links between some probabilistic methods and the combinatorial notions of logconcavity and ultralogconcavity, and they add to the growing body of work exploring the applications of maximum entropy characterizations to problems in discrete mathematics.
INTERTWINING RELATIONS FOR ONEDIMENSIONAL DIFFUSIONS AND APPLICATION TO FUNCTIONAL INEQUALITIES
, 2013
"... Abstract. Following the recent work [13] fulfilled in the discrete case, we provide in this paper new intertwining relations for semigroups of onedimensional diffusions. Various applications of these results are investigated, among them the famous variational formula of the spectral gap derived by ..."
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Cited by 3 (2 self)
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Abstract. Following the recent work [13] fulfilled in the discrete case, we provide in this paper new intertwining relations for semigroups of onedimensional 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. 1.