arXiv subjects

Guolie Lan

Publications and source records attributed to Guolie Lan.

5 recordsLinked to original sources

arXiv2025

Results related to the Gaussian product inequality conjecture for mixed-sign exponents in arbitrary dimension

This paper studies Gaussian product inequalities (GPIs) for centered Gaussian random vectors when positive and negative exponents appear simultaneously. We prove a quantitative lower bound for arbitrary mixed-sign patterns, conditional on the validity of a lower-dimensional GPI, and an upper bound for products of negative powers with convex functions of the remaining Gaussian components. The latter yields an explicit upper bound for mixed-sign products whenever the total positive-side exponent is at least $1/2$.

math.PR↗

arXiv2020

The 4-D Gaussian Random Vector Maximum Conjecture and the 3-D Simplex Mean Width Conjecture

We prove the four-dimensional Gaussian random vector maximum conjecture. This conjecture asserts that among all centered Gaussian random vectors $X=(X_1,X_2,X_3,X_4)$ with $E[X_i^2]=1$, $1\le i\le 4$, the expectation $E[\max(X_1,X_2,X_3,X_4)]$ is maximal if and only if all off-diagonal elements of the covariance matrix equal $-\frac{1}{3}$. As a direct consequence, we resolve the three-dimensional simplex mean width conjecture. This latter conjecture is a long-standing open problem in convex geometry, which asserts that among all simplices inscribed into the three-dimensional unit Euclidean ball the regular simplex has the maximal mean width.

math.PR↗

arXiv2019

Some explorations on two conjectures about Rademacher sequences

In this paper, we explore two conjectures about Rademacher sequences. Let $(ε_i)$ be a Rademacher sequence, i.e., a sequence of independent $\{-1,1\}$-valued symmetric random variables. Set $S_n=a_1ε_1+\cdots+a_nε_n$ for $a=(a_1,\dots,a_n)\in \mathbb{R}^n$. The first conjecture says that $P\ (\ |S_n\ |\leq \|a\|\ )\geq\frac{1}{2}$ for all $a\in \mathbb{R}^n$ and $n\in \mathbb{N}$. The second conjecture says that $P\ (\ |S_n\ |\geq\|a\|\ )\geq \frac{7}{32}$ for all $a\in \mathbb{R}^n$ and $n\in \mathbb{N}$. Regarding the first conjecture, we present several new equivalent formulations. These include a topological view, a combinatorial version and a strengthened version of the conjecture. Regarding the second conjecture, we prove that it holds true when $n\leq 7$.

math.PR↗

arXiv2019

Products of Conditional Expectation Operators: Convergence and Divergence

In this paper, we investigate the convergence of products of conditional expectation operators. We show that if $(Ω,\cal{F},P)$ is a probability space that is not purely atomic, then divergent sequences of products of conditional expectation operators involving 3 or 4 sub-$σ$-fields of $\cal{F}$ can be constructed for a large class of random variables in $L^2(Ω,\cal{F},P)$. This settles in the negative a long-open conjecture. On the other hand, we show that if $(Ω,\cal{F},P)$ is a purely atomic probability space, then products of conditional expectation operators involving any finite set of sub-$σ$-fields of $\cal{F}$ must converge for all random variables in $L^1(Ω,\cal{F},P)$.

math.PR↗

arXiv2019

The Three-Dimensional Gaussian Product Inequality

We prove the 3-dimensional Gaussian product inequality, i.e., for any real-valued centered Gaussian random vector $(X,Y,Z)$ and $m\in \mathbb{N}$, it holds that ${\mathbf{E}}[X^{2m}Y^{2m}Z^{2m}]\geq{\mathbf{E}}[X^{2m}]{\mathbf{E}}[Y^{2m}]{\mathbf{E}}[Z^{2m}]$. Our proof is based on some improved inequalities on multi-term products involving 2-dimensional Gaussian random vectors. The improved inequalities are derived using the Gaussian hypergeometric functions and have independent interest. As by-products, several new combinatorial identities and inequalities are obtained.

math.PR↗