Philip Lamkin, Tomasz Tkocz
We show that the sequence of moments of order less than 1 of averages of i.i.d. positive random variables is log-concave. For moments of order at least 1, we conjecture that the sequence is log-convex and show that this holds eventually for integer moments (after neglecting the first $p^2$ terms of the sequence).
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We define, in the frame of an abstract Wiener space, the notions of convexity and of concavity for the equivalence classes of random variables. As application we show that some important inequalities of the finite dimensional case have their natural counterparts in this setting.
Log-convexity and log-concavity of noncentral gamma sums and differences
研究非中心伽马随机变量和与差的密度函数的对数凸性与对数凹性，给出相关完整分类及准则，还推导出特殊情况下正态、方差-伽马及麦凯I型分布密度的对数凸凹性分类。
In this paper, we prove that for the doubly symmetric binary distribution, the lower increasing envelope and the upper envelope of the minimum-relative-entropy region are respectively convex and concave. We also prove that another function induced the minimum-relative-entropy region is concave. These two envelopes and this function were previously used to characterize the optimal exponents in strong small-set expansion problems and strong Brascamp--Lieb inequalities. The results in this paper, combined with the strong small-set expansion theorem derived by Yu, Anantharam, and Chen (2021), and …
We extend the exponential formula by Bender and Canfield (1996), which relates log-concavity and the cycle index polynomials. The extension clarifies the log-convexity relation. The proof is by noticing the property of a compound Poisson distribution together with its moment generating function. We also give a combinatorial proof of extended "log-convex part" referring Bender and Canfield's approach, where the formula by Bruijn and Erdös (1953) is additionally exploited. The combinatorial approach yields richer structural results more than log-convexity. Furthermore, we consider normal and bin…
We introduce the antipodal pairs property for probability measures on finite Boolean algebras and prove that conditional versions imply strong forms of log-concavity. We give several applications of this fact, including improvements of some results of Wagner; a new proof of a theorem of Liggett stating that ultra-log-concavity of sequences is preserved by convolutions; and some progress on a well-known log-concavity conjecture of J. Mason.
In this paper we will develop a systematic method to answer the questions $(Q1)(Q2)(Q3)(Q4)$ (stated in Section 1) with complete generality. As a result, we can solve the difficulties $(D1)(D2)$ (discussed in Section 1) without uncertainty. For these purposes we will introduce certain classes of growth functions $u$ and apply the Legendre transform to obtain a sequence which leads to the weight sequence $\{\a(n)\}$ first studied by Cochran et al. \cite{cks}. The notion of (nearly) equivalent functions, (nearly) equivalent sequences and dual Legendre functions will be defined in a very natural …
Log-concavity in planar random walks
该研究证明了在特定平面区域内，格点随机游走的退出概率满足对数凹性，为相关概率性质提供了新结果。
We give counterexamples and a few positive results related to several conjectures of R. Pemantle and D. Wagner concerning negative correlation and log-concavity properties for probability measures and relations between them. Most of the negative results have also been obtained, independently but somewhat earlier, by Borcea et al. We also give short proofs of a pair of results due to Pemantle and Borcea et al.; prove that "almost exchangeable" measures satisfy the "Feder-Mihail" property, thus providing a "non-obvious" example of a class of measures for which this important property can be show…
In this paper we develop tools for studying limit theorems by means of convexity. We establish bounds for the discrepancy in total variation between probability measures $μ$ and $ν$ such that $ν$ is log-concave with respect to $μ$. We discuss a variety of applications, which include geometric and binomial approximations to sums of random variables, and discrepancy between Gamma distributions. As special cases we obtain a law of rare events for intrinsic volumes, quantitative bounds on proximity to geometric for infinitely divisible distributions, as well as binomial and Poisson approximation f…
We disprove a recent conjecture regarding discrete distributions and their generating polynomials stating that strong log-concavity implies log-submodularity.
We give a self-contained proof of the strongest version of Mason's conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave. To do this, we introduce a class of polynomials, called completely log-concave polynomials, whose bivariate restrictions have ultra log-concave coefficients. At the heart of our proof we show that for any matroid, the homogenization of the generating polynomial of its independent sets is completely log-concave.
We extend Hoggar's theorem that the sum of two independent discrete-valued log-concave random variables is itself log-concave. We introduce conditions under which the result still holds for dependent variables. We argue that these conditions are natural by giving some applications. Firstly, we use our main theorem to give simple proofs of the log-concavity of the Stirling numbers of the second kind and of the Eulerian numbers. Secondly, we prove results concerning the log-concavity of the sum of independent (not necessarily log-concave) random variables.
We prove that the Poisson distribution maximises entropy in the class of ultra-log-concave distributions, extending a result of Harremoës. The proof uses ideas concerning log-concavity, 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.
Log-concavity in one-dimensional Coulomb gases and related ensembles
本文证明了年轻图顶层行长、beta集合最大元素以及随机空气算子最小特征值的对数凹性，确认了特雷西-威多姆分布的对数凹性，并揭示了相关分布的正相关性。
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, {\em Stoch. Proc. Appl.}, 2007] used a semigroup approach to show that the Poisson has maximal entropy among all ultra-log-concave distributions with fixed mean. We show via a non-trivial 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 log-concave, b…
We study the Monge and Kantorovich transportation problems on $\mathbb{R}^{\infty}$ within the class of exchangeable measures. With the help of the de Finetti decomposition theorem the problem is reduced to an unconstrained optimal transportation problem on the Hilbert space. We find sufficient conditions for convergence of finite-dimensional approximations to the Monge solution. The result holds, in particular, under certain analytical assumptions involving log-concavity of the target measure. As a by-product we obtain the following result: any uniformly log-concave exchangeable sequence of r…
MRL order, log-concavity and an application to peacocks
本文提出实值过程MRL序等价于对数凹性条件，结合二阶全正性构造新的MRL过程族，并指出常数均值MRL过程为孔雀过程，可由Azéma-Yor算法构造显式鞅。
We show that a mixture of Beta distributions has log-concave density whenever the mixing weights are themselves log-concave. Some economic and statistical applications are provided in the last section.
We prove that the (B) conjecture and the Gardner-Zvavitch conjecture are true for all log-concave measures that are rotationally invariant, extending previous results known for Gaussian measures. Actually, our result apply beyond the case of log-concave measures, for instance to Cauchy measures as well. For the proof, new sharp weighted Poincaré inequalities are obtained for even probability measures that are log-concave with respect to a rotationally invariant measure.
Bi-log-concavity of probability measures is a univariate extension of the notion of log-concavity that has been recently proposed in a statistical literature. Among other things, it has the nice property from a modelisation perspective to admit some multimodal distributions, while preserving some nice features of log-concave measures. We compute the isoperimetric constant for a bi-log-concave measure, extending a property available for log-concave measures. This implies that bi-log-concave measures have exponentially decreasing tails. Then we show that the convolution of a bi-log-concave measu…
On the log-concavity of the Wright function
该文研究概率框架下Wright函数在半直线上的对数凹性，给出其在构造广义熵中的应用并提出相关猜想，解决了两类参数情形下的问题且得到临界参数，还证明了该类函数的单峰性与乘法强单峰性的充要条件。
We establish a discrete analog of the Rényi entropy comparison due to Bobkov and Madiman. For log-concave variables on the integers, the min entropy is within log e of the usual Shannon entropy. Additionally we investigate the entropic Rogers-Shephard inequality studied by Madiman and Kontoyannis, and establish a sharp Rényi version for certain parameters in both the continuous and discrete cases
Log-concavity and discrete degrees of freedom
该研究提出离散对数凹序列的自由度概念，证明固定方差下几何分布最小化离散对数凹随机变量的无穷阶Rényi熵，且固定整数均值下Poisson分布最大化超对数凹随机变量取均值的概率。
In this paper, we present a comprehensive study of the monotonicity and log-concavity of the generalized Marcum and Nuttall Q-functions. More precisely, a simple probabilistic method is firstly given to prove the monotonicity of these two functions. Then, the log-concavity of the generalized Marcum Q-function and its deformations is established with respect to each of the three parameters. Since the Nuttall Q-function has similar probabilistic interpretations as the generalized Marcum Q-function, we deduce the log-concavity of the Nuttall Q-function. By exploiting the log-concavity of these tw…
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