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Ze-Chun Hu

Publications and source records attributed to Ze-Chun Hu.

At least 37 records · Page 2Linked to original sources
arXiv2020

Unlinking Theorem for Symmetric Quasi-convex Polynomials

Let $μ_n$ be the standard Gaussian measure on $\mathbb{R}^n$ and $X$ be a random vector on $\mathbb{R}^n$ with the law $μ_n$. U-conjecture states that if $f$ and $g$ are two polynomials on $\mathbb{R}^n$ such that $f(X)$ and $g(X)$ are independent, then there exist an orthogonal transformation $L$ on $\mathbb{R}^n$ and an integer $k$ such that $f\circ L$ is a function of $(x_1,\cdots,x_k)$ and $g\circ L$ is a function of $(x_{k+1},\cdots,x_n)$. In this case, $f$ and $g$ are said to be unlinked. In this note, we prove that two symmetric, quasi-convex polynomials $f$ and $g$ are unlinked if $f(X)$ and $g(X)$ are independent.

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↗
arXiv2020

A Note on the Gaussian Minimum Conjecture

Let $n\geq 2$ and $(X_i,1\leq i\leq n)$ be a centered Gaussian random vector. The Gaussian minimum conjecture says that $E\left(\min_{1\leq i\leq n}|X_i|\right)\geq E\left(\min_{1\leq i\leq n}|Y_i|\right)$, where $Y_1,\ldots,Y_n$ are independent centered Gaussian random variables with $E(X_i^2)=E(Y_i^2)$ for any $i=1,\ldots,n$. In this note, we will show that this conjecture holds if and only if $n=2$.

math.PR↗
arXiv2020

A Note on Convergence of Random Variables

In this note, convergence of random variables will be revisited. We will give the answers to 5 questions among the 6 open questions introduced in (Convergence rates in the law of large numbers and new kinds of convergence of random variables, {\it Communication in Statistics - Theory and Methods}, DOI: 10.1080/03610926.2020.1716248), and make some related discussions.

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

Hunt's Hypothesis (H) for Markov Processes: Survey and Beyond

The goal of this paper is threefold. First, we survey the existing results on Hunt's hypothesis (H) for Markov processes and Getoor's conjecture for Lévy processes. Second, we investigate (H) for multidimensional Lévy processes from the viewpoints of projections and energy, respectively. Third, we present a few open questions for further study.

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

Two Stronger Versions of the Union-closed Sets Conjecture

The union-closed sets conjecture (Frankl's conjecture) says that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in the family. In this paper, we introduce two stronger versions of Frankl's conjecture and give a partial proof. Three related questions are introduced.

math.CO↗
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↗
arXiv2019

Two Theorems on Hunt's Hypothesis (H) for Markov Processes

Hunt's hypothesis (H) and the related Getoor's conjecture is one of the most important problems in the basic theory of Markov processes. In this paper, we investigate the invariance of Hunt's hypothesis (H) for Markov processes under two classes of transformations, which are change of measure and subordination. Our first theorem shows that for two standard processes $(X_t)$ and $(Y_t)$, if $(X_t)$ satisfies (H) and $(Y_t)$ is locally absolutely continuous with respect to $(X_t)$, then $(Y_t)$ satisfies (H). Our second theorem shows that a standard process $(X_t)$ satisfies (H) if and only if $(X_{τ_t})$ satisfies (H) for some (and hence any) subordinator $(τ_t)$ which is independent of $(X_t)$ and has a positive drift coefficient. Applications of the two theorems are given.

math.PR↗
arXiv2018

On the length of the longest consecutive switches

An unbiased coin is tossed $n$ times independently and sequentially. In this paper, we will study the length of the longest consecutive switches, and prove that the limit behaviors are similar to that of the length of the longest head-run.

math.PR↗
arXiv2018

The 6-element case of S-Frankl conjecture (I)

The union-closed sets conjecture (Frankl's conjecture) says that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in the family. In [3], a stronger version of Frankl's conjecture (S-Frankl conjecture for short) was introduced and a partial proof was given. In particular, it was proved in \cite{CH17} that S-Frankl conjecture holds when $n\leq 5$, where $n$ is the number of all the elements in the family of sets. Now, we want to prove that it holds when $n=6$. Since the paper is very long, we split it into two parts. This is the first part.

math.CO↗
arXiv2018

Convergence rates in the law of large numbers and new kinds of convergence of random variables

In this paper, we first study convergence rates in the law of large numbers for independent and identically distributed random variables. We obtain a strong $L^p$-convergence version and a strongly almost sure convergence version of the law of large numbers. Second, we investigate several new kinds of convergence of random variables and discuss their relations and properties.

math.PR↗
arXiv2017

A system of nonlinear equations with application to large deviations for Markov chains with finite lifetime

In this paper, we first show the existence of solutions to the following system of nonlinear equations \begin{eqnarray*}\left\{\begin{array}{l} a_{11}x_1+a_{12}x_2+a_{13}x_3+\cdots+a_{1n}x_{n} = b_{11}\frac{1}{x_1}+b_{12}\frac{1}{x_2}+b_{13}\frac{1}{x_3}+\cdots+b_{1n}\frac{1}{x_{n}},\\ a_{21}\frac{1}{x_1}+a_{22}\frac{x_2}{x_1}+a_{23}\frac{x_3}{x_1}+\cdots+a_{2n}\frac{x_{n}}{x _1}=b_{21}x_1+b_{22}\frac{x_1}{x_2}+b_{23}\frac{x_1}{x_3}+\cdots+b_{2n}\frac{x_1}{x_{n}},\\ a_{31}\frac{x_1}{x_2}+a_{32}\frac{1}{x_2}+a_{33}\frac{x_3}{x_2}+\cdots+a_{3n}\frac{x_{n}}{x _2}=b_{31}\frac{x_2}{x_1}+b_{32}x_2+b_{33}\frac{x_2}{x_3}+\cdots+b_{3n}\frac{x_2}{x_{n}},\\ \cdots\cdots\\ a_{n1}\frac{x_1}{x_{n-1}}+a_{n2}\frac{x_2}{x_{n-1}}+a_{n3}\frac{x_3}{x_{n-1}}+ \cdots+a_{n,n-1}\frac{1}{x_{n-1}}+a_{nn}\frac{x_{n}}{x_{n-1}}\\ =b_{n1}\frac{x_{n-1}}{x_1}+b_{n2}\frac{x_{n-1}}{x_2}+b_{n3}\frac{x_{n-1}}{x_3}+\cdots+b_{n, n-1}x_{n-1} +b_{nn}\frac{x_{n-1}}{x_{n}}, \end{array} \right. \end{eqnarray*} where $n\ge 3$ and $a_{ij},b_{ij},1\le i,j\le n$, are positive constants. Then, we make use of this result to obtain the large deviation principle for the occupation time distributions of continuous-time finite state Markov chains with finite lifetime.

math.PR↗
arXiv2017

Convergences of Random Variables under Sublinear Expectations

In this note, we will survey the existing convergence results for random variables under sublinear expectations, and prove some new results. Concretely, under the assumption that the sublinear expectation has the monotone continuity property, we will prove that $L^p$ convergence is stronger than convergence in capacity, convergence in capacity is stronger than convergence in distribution, and give some equivalent characterizations of convergence in distribution. In addition, we give a dominated convergence theorem under sublinear expectations, which may have its own interest.

math.PR↗
arXiv2017

Hunt's Hypothesis (H) for the Sum of Two Independent Levy Processes

Which Levy processes satisfy Hunt's hypothesis (H) is a long-standing open problem in probabilistic potential theory. The study of this problem for one-dimensional Levy processes suggests us to consider (H) from the point of view of the sum of Levy processes. In this paper, we present theorems and examples on the validity of (H) for the sum of two independent Levy processes. We also give a novel condition on the Levy measure which implies (H) for a large class of one-dimensional Levy processes.

math.PR↗