arXiv subjects

Ze-Yu Tao

Publications and source records attributed to Ze-Yu Tao.

2 recordsLinked to original sources

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↗