The convex transform order is one way to make precise comparison between the skewness of probability distributions on the real line. We establish a simple and complete characterisation of when one Beta distribution is smaller than another according to the convex transform order. As an application, we derive monotonicity properties for the probability of Beta distributed random variables exceeding the mean or mode of their distribution. Moreover, we obtain a simple alternative proof of the mode-median-mean inequality for unimodal distributions that are skewed in a sense made precise by the convex transform order. This new proof also gives an analogous inequality for the anti-mode of distributions that have a unique anti-mode. Such inequalities for Beta distributions follow as special cases. Finally, some consequences for the values of distribution functions of Binomial distributions near to their means are mentioned.
The convex hull of several i.i.d. beta distributed random vectors in $\mathbb R^d$ is called the random beta polytope. Recently, the expected values of their intrinsic volumes, number of faces, normal and tangent angles and other quantities have been calculated, explicitly and asymptotically. In this paper, we aim to investigate the asymptotic behavior of the beta polytopes with extremal intrinsic volumes. We suggest a conjecture and solve it in dimension 2. To this end, we obtain some general limit relation for a wide class of $U$-$\max$ statistics whose kernels include the perimeter and the …
Random walk in a dynamic i.i.d. beta random environment, conditioned to escape at an atypical velocity, converges to a Doob transform of the original walk. The Doob-transformed environment is correlated in time, i.i.d. in space, and its marginal density function is a product of a beta density and a hypergeometric function. Under its averaged distribution the transformed walk obeys the wandering exponent 2/3 that agrees with Kardar-Parisi-Zhang universality. The harmonic function in the Doob transform comes from a Busemann-type limit and appears as an extremal in a variational problem for the q…
Let $X_1,\ldots,X_n$ be i.i.d.\ random points in the $d$-dimensional Euclidean space sampled according to one of the following probability densities: $$ f_{d,β} (x) = \text{const} \cdot (1-\|x\|^2)^β, \quad \|x\|\leq 1, \quad \text{(the beta case)} $$ and $$ \tilde f_{d,β} (x) = \text{const} \cdot (1+\|x\|^2)^{-β}, \quad x\in\mathbb{R}^d, \quad \text{(the beta' case).} $$ We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of $X_1,\ldots,X_n$. Asymptotic formulae where obtained previously by Affentranger [The convex hull of random points with …
The random beta polytope is defined as the convex hull of $n$ independent random points with the density proportional to $(1-\|x\|^2)^β$ on the $d$-dimensional unit ball, where $β>-1$ is a parameter. Similarly, the random beta' polytope is defined as the convex hull of $n$ independent random points with the density proportional to $(1+\|x\|^2)^{-β}$ on $\mathbb R^d$, where $β>\frac d2$. In a previous work [Angles of random simplices and face numbers of random polytopes, Adv. Math., 380 (2021), 107612], we established exact and explicit formulae for the expected $f$-vectors of these random poly…
We study random polytopes of the form $[X_1,\ldots,X_n]$ defined as convex hulls of independent and identically distributed random points $X_1,\ldots,X_n$ in $\mathbb{R}^d$ with one of the following densities: $$ f_{d,β} (x) = c_{d,β} (1-\|x\|^2)^β, \qquad \|x\| < 1, \quad \text{(beta distribution, $β>-1$)} $$ or $$ \tilde f_{d,β} (x) = \tilde{c}_{d,β} (1+\|x\|^2)^{-β}, \qquad x\in\mathbb{R}^d, \quad \text{(beta' distribution, $β>d/2$)}. $$ This setting also includes the uniform distribution on the unit sphere and the standard normal distribution as limiting cases. We derive exact and asymptot…
We develop a Mellin transform framework which allows us to simultaneously analyze the four known exactly solvable 1+1 dimensional lattice polymer models: the log-gamma, strict-weak, beta, and inverse-beta models. Using this framework we prove the conjectured fluctuation exponents of the free energy and the polymer path for the stationary point-to-point versions of these four models. The fluctuation exponent for the polymer path was previously unproved for the strict-weak, beta, and inverse-beta models.
In this paper we prove that two heterogeneous parallel systems with independent exponentially distributed components are comparable via the star transform order while the comparison via the convex transform fails. The latter conclusion provides a partial answer to a problem that remained open for a decade.
In the literature of stochastic orders, one rarely finds results that can be considered as criteria for the non-comparability of random variables. In this paper, we provide results that enable researchers to use simple tools to conclude that two random variables are not comparable with respect to the convex transform order. The criteria are applied to prove the non-comparability of parallel systems with components that are either exponential, Weibull or Gamma distributed, providing a negative answer for a conjecture about comparability with respect to the convex transform order in a much broad…
Pick $d+1$ points uniformly at random on the unit sphere in $\mathbb R^d$. What is the expected value of the angle sum of the simplex spanned by these points? Choose $n$ points uniformly at random in the $d$-dimensional ball. What is the expected number of faces of their convex hull? We answer these and some related questions of stochastic geometry. To this end, we compute expected internal angles of random simplices whose vertices are independent random points sampled from one of the following $d$-dimensional distributions: (i) the beta distribution with the density proportional to $(1-\|x\|^…
We prove that, at every positive temperature, the infinite-volume free energy of the one dimensional log-gas, or beta-ensemble, has a unique minimiser, which is the Sine-beta process arising from random matrix theory. We rely on a quantitative displacement convexity argument at the level of point processes, and on the screening procedure introduced by Sandier-Serfaty.
The beta polytope $P_{n,d}^β$ is the convex hull of $n$ i.i.d. random points distributed in the unit ball of $\mathbb{R}^d$ according to a density proportional to $(1-\lVert{x}\rVert^2)^β$ if $β>-1$ (in particular, $β=0$ corresponds to the uniform distribution in the ball), or uniformly on the unit sphere if $β=-1$. We show that the expected normalized volumes of high-dimensional beta polytopes exhibit a phase transition and we describe its shape. We derive analogous results for the intrinsic volumes of beta polytopes and, when $β=0$, their number of vertices.
Asymptotic expansion of beta matrix models in the one-cut regime
我们证明了在β矩阵模型中存在一个1/N展开到所有的阶数，该模型具有一个封闭的、非临界的势，对应于一个具有连接支撑的平衡测度。因此，展开式的系数可以通过Chekhov和Eynard的“拓扑递归”来递归地获得。我们的方法依赖于结合的先验界上的扩散和Schwinger-Dyson方程的研究，由于使用经典的复分析技术。这些先验界可以根据Boutet de Monvel，Pastur和Shcherbina的方法导出，或者对于严格凸势，可以利用测度的集中性导出。这样做，我们扩展的战略Guionnet和Maurel-Segala，从埃尔米特模型（β = 2）和微扰电位，一般β模型。Johansson以前曾考虑过1/N中第一个修正的存在，最近Kriecherbauer和Shcherbina也考虑过。在这里，通过采取类似的假设，我们将结果推广到1/N中的所有阶。
We study the probability that one beta-distributed random variable exceeds the maximum of two others, allowing all three to have general parameters. This amounts to studying Euler transforms of products of two incomplete beta functions. We provide a closed form for the general problem in terms of Kampé de Fériet functions and a variety of simpler closed forms in special cases. The results are applied to derive the moments of the maximum of two independent beta-distributed random variables and to find inner products of incomplete beta functions. Restricted to positive integer parameters, our re…
We establish some identities in law for the convolution of a beta prime distribution with itself, involving the square root of beta distributions. The proof of these identities relies on transformations on generalized hypergeometric series obtained via Appell series of the first kind and Thomae's relationships for ${}_3F_2(1)$. Using a self-decomposability argument, the identities are applied to derive complete monotonicity properties for quotients of confluent hypergeometric functions having a doubling character. By means of probability, we also obtain a simple proof of Turán's inequality for…
The paper describes some probabilistic and combinatorial aspects of the nonlinear Fourier transform associated with the AKNS-ZS problems. In the first of the two main results, we show that a family of polytopes that appear in a power expansion of the nonlinear Fourier transforms is distributed according to the beta probability distribution. We establish this result by studying an Euler type discretization of the nonlinear Fourier transform. This approach provides our second main result, discovering a novel discrete probability distribution that approximates the beta distribution. The numbers o…
