arXiv subjects

Ze-Chun Hu

Publications and source records attributed to Ze-Chun Hu.

At least 19 recordsLinked to original sources

arXiv2026

The infimum values of three probability functions for the Laplace distribution and the student's $t$ distribution

Let $\{X_\alpha\}$ be a family of random variables satisfying some distribution with a parameter $\alpha$, $E(X_{\alpha})$ be the expectation, and $Var(X_{\alpha})$ be the variance. In this paper, we study the infimum values of three probability functions: $P(X_{\alpha}\leq y E(X_{\alpha}))$, $P\left(|X_{\alpha}-E(X_{\alpha})|\leq y\sqrt{Var(X_{\alpha})}\right)$ and $P\left(|X_{\alpha}-E(X_{\alpha})|\geq y\sqrt{Var(X_{\alpha})}\right), \forall y>0$, with respect to the parameter $\alpha$ for the Laplace distribution and the student's $t$ distribution. Our motivation comes from three former conjectures: Chv\'{a}tal's conjecture, Tomaszewski's conjecture and Hitczenko-Kwapie\'{n}'s conjecture.

math.PR↗

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↗

arXiv2025

A note on the binomial distribution motivated by Chv\'{a}tal's theorem and Tomasewski's theorem

Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$. Chv\'{a}tal's theorem says that for any fixed $n\geq 2$, as $m$ ranges over $\{0,1,\ldots,n\}$, the probability $q_m:=P(B(n,m/n)\leq m)$ is the smallest when $m$ is closest to $2n/3$. Let $\mathcal{R}$ be the family of random variables of the form $X=\sum^n_{k=1}a_k\varepsilon_k$, where $n\ge 1$, $a_k, k=1, \dots, n,$ are real numbers with $\sum^n_{k=1} a_k^2=1$, and $\varepsilon_k$, $k=1, 2, \dots$, are independent Rademacher random variables (i.e., $P(\varepsilon_k=1)=P(\varepsilon_k=-1)=1/2$). Tomaszewski's theorem says that $\inf_{X\in \mathcal{R}}P(|X|\leq 1)=1/2$. Motivated by Chv\'{a}tal's Theorem and Tomasewski's Theorem, in this note, we study the minimum value of the probability $f_n(k):=P(|B(n,k/n)-k|\leq \sqrt{{\rm Var} (B(n,k/n))})$ when $k$ ranges over $\{0,1,\ldots,n\}$ for any fixed $n\geq 1$, where ${\rm Var} (\cdot)$ denotes the variance, and prove that it is the smallest when $k=1$ and $n-1$.

math.PR↗

arXiv2024

On the anti-concentration functions of some familiar families of distributions

Let $\{X_α\}$ be a family of random variables following a certain type of distributions with finite expectation $\mathbf{E}[X_α]$ and finite variance ${\rm Var}(X_α)$, where $α$ is a parameter. Motivated by the recent paper of Hollom and Portier (arXiv: 2306.07811v1), we study the anti-concentration function $(0, \infty)\ni y\to \inf_α\mathbf{P}\left(|X_α-\mathbf{E}[X_α]|\geq y \sqrt{{\rm Var}(X_α)}\right)$ and find its explicit expression. We show that, for certain familiar families of distributions, including uniform distributions, exponential distributions, non-degenerate Gaussian distributions and student's $t$-distribution, the anti-concentration function is not identically zero, while for some other familiar families of distributions, including binomial, Poisson, negative binomial, hypergeometric, Gamma, Pareto, Weibull, log-normal and Beta distributions, the anti-concentration function is identically zero.

math.PR↗

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

A study on the negative binomial distribution 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 note we consider the infimum value of the probability $P(X\leq E[X])$, where $X$ is a negative binomial random variable. As a consequence, we give an affirmative answer to the conjecture posed in [Statistics and Probability Letters, 200 (2023) 109871].

math.PR↗

arXiv2023

A note on the Hitczenko-Kwapien conjecture about a Rademacher sequence

Let $(ε_i)$ be a Rademacher sequence, i.e., a sequence of independent and identically distributed random variables satisfying $P(ε_i=1)=P(ε_i=-1)=1/2$. Set $S_n=a_1ε_1+\cdots+a_nε_n$ for $a=(a_1,\dots,a_n)\in \mathbb{R}^n$. The Hitczenko-Kwapien conjecture says that $P\left(\left|S_n\right|\geq\|a\|\right)\geq {7}/{32}$ for all $a\in \mathbb{R}^n$ and $n\in \mathbb{N}$. Up to now, we know that it holds when $n\leq 7$. In this note, we show that it holds when $n=8$.

math.PR↗

arXiv2023

The asymptotic behavior of rarely visited edges of the simple random walk

In this paper, we study the asymptotic behavior of the number of rarely visited edges (i.e., edges that visited only once) of a simple symmetric random walk on $\mathbb{Z}$. Let $\alpha(n)$ be the number of rarely visited edges up to time $n$. First, we evaluate $\mathbb{E}(\alpha(n))$, show that $n\to \mathbb{E}(\alpha(n))$ is non-decreasing in $n$ and that $\lim\limits_{n\to+\infty}\mathbb{E}(\alpha(n))=2$. Then we study the asymptotic behavior of $\mathbb{P} (\alpha(n)>a(\log n)^2)$ for any $a>0$ and use it to show that there exists a constant $C\in(1/32,1/2]$ such that $\limsup\limits_{n\to+\infty}\frac{\alpha(n)}{(\log n)^2}=C$ almost surely.

math.PR↗

arXiv2023

On the measure concentration of infinitely divisible distributions

Let ${\cal I}$ be the set of all infinitely divisible random variables\ with finite second moments, ${\cal I}_0=\{X\in{\cal I}:{\rm Var}(X)>0\}$, $P_{\cal I}=\inf_{X\in{\cal I}}P\{|X-E[X]|\le \sqrt{{\rm Var}(X)}\}$ and $P_{{\cal I}_0}=\inf_{X\in{\cal I}_0} P\{|X-E[X]|< \sqrt{{\rm Var}(X)}\}$. Firstly, we prove that $P_{\cal I}\ge P_{{\cal I}_0}>0$. Secondly, we find the exact values of $\inf_{X\in{\cal J}}P\{|X-E[X]|\le \sqrt{{\rm Var}(X)}\}$ and $\inf_{X\in\cal J} P\{|X-E[X]|< \sqrt{{\rm Var}(X)}\}$ for the cases that $\cal J$ is the set of all geometric random variables, symmetric geometric random variables, Poisson random variables and symmetric Poisson random variables, respectively. As a consequence, we obtain that $P_{\cal I}\le e^{-1}\sum_{k=0}^{\infty}\frac{1}{2^{2k}(k!)^2}\approx 0.46576$ and $P_{{\cal I}_0}\le e^{-1}\approx 0.36788$.

math.PR↗

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

Variation comparison between the $F$-distribution and the normal distribution

Let $X_{d_1,d_2}$ be an $F$-random variable with numerator and denominator degrees of freedom $d_1$ and $d_2$, respectively. We investigate the inequality: $P\{|X_{d_1,d_2}-E[X_{d_1,d_2}]|\le \sqrt{{\rm Var}(X_{d_1,d_2})}\}\ge P\{|W-E[W]|\le \sqrt{{\rm Var}(W)}\}$, where $W$ is a standard normal random variable or a $χ^2(d_1)$ random variable. We prove that this inequality holds for $d_1\in\{1,2,3,4\}$ and $5\le d_2\in\mathbb{N}$.

math.PR↗

arXiv2023

Variation comparison between infinitely divisible distributions and the normal distribution

Let $X$ be a random variable with finite second moment. We investigate the inequality: $P\{|X-E[X]|\le \sqrt{{\rm Var}(X)}\}\ge P\{|Z|\le 1\}$, where $Z$ is a standard normal random variable. We prove that this inequality holds for many familiar infinitely divisible continuous distributions including the Laplace, Gumbel, Logistic, Pareto, infinitely divisible Weibull, log-normal, student's $t$ and inverse Gaussian distributions. Numerical results are given to show that the inequality with continuity correction also holds for some infinitely divisible discrete distributions.

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↗

arXiv2023

A study on the Poisson, geometric and Pascal distributions motivated by Chvátal's conjecture

Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$. Vasěk Chvátal conjectured 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 $\frac{2n}{3}$. This conjecture has been solved recently. Motivated by this conjecture, in this paper, we consider the corresponding minimum value problem on the probability that a random variable is not more than its expectation, when its distribution is the Poisson distribution, the geometric distribution or the Pascal distribution.

math.PR↗

arXiv2023

The extreme values of two probability functions for the Gamma distribution

Motivated by Chvátal's conjecture and Tomaszewaki's conjecture, we investigate the extreme value problem of two probability functions for the Gamma distribution. Let $α,β$ be arbitrary positive real numbers and $X_{α,β}$ be a Gamma random variable with shape parameter $α$ and scale parameter $β$. We study the extreme values of functions $P\{X_{α,β}\le E[X_{α,β}]\}$ and $P\{|X_{α,β}-E[X_{α,β}]|\le \sqrt{{\rm Var}(X_{α,β})}\}$. Among other things, we show that $ \inf_{α,β}P\{X_{α,β}\le E[X_{α,β}]\}=\frac{1}{2}$ and $\inf_{α,β}P\{|X_{α,β}-E[X_{α,β}]|\le \sqrt{{\rm Var}(X_{α,β})}\}=P\{|Z|\le 1\}\approx 0.6826$, where $Z$ is a standard normal random variable.

math.PR↗

arXiv2022

Favorite Downcrossing Sites of One-Dimensional Simple Random Walk

Random walk is a very important Markov process and has important applications in many fields.For a one-dimensional simple symmetric random walk $(S_n)$, a site $x$ is called a favorite downcrossing site at time $n$ if its downcrossing local time at time $n$ achieves the maximum among all sites. In this paper, we study the cardinality of the favorite downcrossing site set, and will show that with probability 1 there are only finitely many times at which there are at least four favorite downcrossing sites and three favorite downcrossing sites occurs infinitely often. Some related open questions will be introduced.

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↗

arXiv2022

Three Favorite Edges Occurs Infinitely Often for One-Dimensional Simple Random Walk

For a one-dimensional simple symmetric random walk $(S_n)$, an edge $x$ (between points $x-1$ and $x$) is called a favorite edge at time $n$ if its local time at $n$ achieves the maximum among all edges. In this paper, we show that with probability 1 three favorite edges occurs infinitely often. Our work is inspired by Tóth and Werner [Combin. Probab. Comput. {\bf 6} (1997) 359-369], and Ding and Shen [Ann. Probab. {\bf 46} (2018) 2545-2561], disproves a conjecture mentioned in Remark 1 on page 368 of Tóth and Werner [Combin. Probab. Comput. {\bf 6} (1997) 359-369].

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