高斯不等式

伯杰龙猜想与两个二项式系数的故事

Bergeron's conjecture & a tale of two binomial coefficients

研究伯杰龙猜想，即整数\(1\leq a<b<c<d\)且\(ad=bc\)时高斯多项式系数关系\({\binom{b+c}b}_q \ge {\binom{a+d}a}_q\)。给出两个证明并推广，还给出特殊情况的组合证明。

A simplified second-order Gaussian Poincaré inequality in discrete setting with applications

In this paper, a simplified second-order Gaussian Poincaré inequality for normal approximation of functionals over infinitely many Rademacher random variables is derived. It is based on a new bound for the Kolmogorov distance between a general Rademacher functional and a Gaussian random variable, which is established by means of the discrete Malliavin-Stein method and is of independent interest. As an application, the number of vertices with prescribed degree and the subgraph counting statistic in the Erdös-Rényi random graph are discussed. The number of vertices of fixed degree is also studie…

A combinatorial proof of the Gaussian product inequality beyond the MTP${}_2$ case

A combinatorial proof of the Gaussian product inequality (GPI) is given under the assumption that each component of a centered Gaussian random vector $\boldsymbol{X} = (X_1, \ldots, X_d)$ of arbitrary length can be written as a linear combination, with coefficients of identical sign, of the components of a standard Gaussian random vector. This condition on $\boldsymbol{X}$ is shown to be strictly weaker than the assumption that the density of the random vector $(|X_1|, \ldots, |X_d|)$ is multivariate totally positive of order $2$, abbreviated MTP${}_2$, for which the GPI is already known to ho…