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Qian-Qian Zhou

Publications and source records attributed to Qian-Qian Zhou.

8 recordsLinked to original sources
arXiv2025

A lemma on a finite union-closed family of finite sets and its applications

Suppose that $\mathscr{F}$ is a finite union-closed family of sets with $\cup_{A\in \mathscr{F}}A=\{1,2,\ldots,m\}$ and $m\geq 2$. Fix $i\in \{1,2,\ldots,m\}$ and denote $\mathscr{G}:=\{A\backslash \{i\}: A\in \mathscr{F}\}$. For $j\in \{1,2,\ldots,m\}\backslash\{i\}$, let $\mathscr{G}_j:=\{A\in\mathscr{G}: j\in A\}$ and $\mathscr{F}_j:=\{A\in\mathscr{F}: j\in A\}$. In this note, we will prove a lemma which says that if $\frac{|\mathscr{G}_j|}{|\mathscr{G}|}\geq c\,(c\in (0,1])$, then $\frac{|\mathscr{F}_j|}{|\mathscr{F}|}\geq \frac{1}{1+2(1-c)/c}$. Several applications of this lemma will be given.

math.CO↗
arXiv2024

The infimum values of the probability functions for some infinitely divisible distributions motivated by Chvátal's theorem

Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$. Chvátal's theorem says that for any fixed $n\geq 2$, as $m$ ranges over $\{0,\ldots,n\}$, the probability $q_m:=P(B(n,m/n)\leq m)$ is the smallest when $m$ is closest to $\frac{2n}{3}$. Motivated by this theorem, in this paper we consider the infimum value of the probability $P(X\leq κE[X])$, where $κ$ is a positive real number, and $X$ is a random variable whose distribution belongs to some infinitely divisible distributions including the inverse Gaussian, log-normal, Gumbel and logistic distributions.

math.PR↗
arXiv2023

Three-dimensional solitons in Rydberg-Dressed cold atomic gases with spin-orbit coupling

We present numerical results for three-dimensional (3D) solitons with symmetries of the semi-vortex (SV) and mixed-mode (MM) types, which can be created in spinor Bose-Einstein condensates of Rydberg atoms under the action of the spin-orbit coupling (SOC). By means of systematic numerical computations, we demonstrate that the interplay of SOC and long-range spherically symmetric Rydberg interactions stabilize the 3D solitons, improving their resistance to collapse. We find how the stability range depends on the strengths of the SOC and Rydberg interactions and the soft-core atomic radius.

cond-mat.quant-gas↗
arXiv2023

Some New Results on Gaussian Product Inequalities

The long-standing Gaussian product inequality (GPI) conjecture states that, for any centered $\mathbb{R}^n$-valued Gaussian random vector $(X_1, \dots, X_n)$ and any positive reals $α_1, \dots, α_n$, ${\bf E}[\prod_{j=1}^{n}|X_j|^{α_j}]\ge \prod_{j=1}^{n}{\bf E}[|X_j|^{α_j}]$. In this paper, we present some related inequalities for centered $\mathbb{R}^n$-valued Gaussian random vector $(X_1, \dots, X_n)$ when $\{α_1, \dots, α_n\}$ contains both positive and negative numbers.

math.PR↗
arXiv2023

A study on the Weibull and Pareto distributions motivated by Chvátal's theorem

Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$. Chvátal's theorem says that for any fixed $n\geq 2$, as $m$ ranges over $\{0,\ldots,n\}$, the probability $q_m:=P(B(n,m/n)\leq m)$ is the smallest when $m$ is closest to $\frac{2n}{3}$. Motivated by this theorem, we consider the minimum value problem on the probability that a random variable is at most its expectation, when its distribution is the Weibull distribution or the Pareto distribution in this note.

math.PR↗
arXiv2022

Quantitative Versions of the Two-dimensional Gaussian Product Inequalities

The Gaussian product inequality (GPI) conjecture is one of the most famous inequalities associated with Gaussian distributions and has attracted a lot of concerns. In this note, we investigate the quantitative versions of the two-dimensional Gaussian product inequalities. For any centered non-degenerate two-dimensional Gaussian random vector $(X_1, X_2)$ with variances $σ_1^2, σ_2^2$ and the correlation coefficient $ρ$, we prove that for any real numbers $α_1, α_2\in (-1,0)$ or $α_1, α_2\in (0,\infty)$, it holds that %there exist functions of $α_1, α_2$ and $ρ$ such that $${\bf E}[|X_1|^{α_1}|X_2|^{α_2}]-{\bf E}[|X_1|^{α_1}]{\bf E}[|X_2|^{α_2}]\ge f(σ_1,σ_2,α_1, α_2, ρ)\ge 0, $$ where the function $f(σ_1,σ_2,α_1, α_2, ρ)$ will be given explicitly by Gamma function and is positive when $ρ\neq 0$. When $-1<α_1<0$ and $α_2>0,$ Russell and Sun (arXiv: 2205.10231v1) proved the "opposite Gaussian product inequality", of which we will also give a quantitative version. These quantitative inequalities are derived by employing the hypergeometric functions and the generalized hypergeometric functions.

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↗