Let $X=(X_1,\ldots,X_n)$ be independent nonnegative random variables, not necessarily identically distributed. Let $D=(D_0,D_1,\ldots,D_n)\sim\operatorname{Dir}(1,\ldots,1)$ be independent of $X$, and define $K(x)=\mathbb{P}\{\sum_{i=1}^n x_iD_i\le1\}$. We prove that, for every $n\ge1$, whenever $\mathbb{E} X_i\le1$ for every $i$, $\mathbb{P}\{K(X)\le\alpha\}\le\alpha$ for all $0\le\alpha\le1$. Thus $K(X)$ is a finite-sample, distribution-free $p$-value for testing the null hypothesis $\mathbb{E}X_i \le 1$ for all $i$. This proves a conjecture of Gaffke (2005).
Let $(X_1,\ldots,X_n)$ be independent nonnegative random variables with $\mathbb{E} X_i\le1$, and write $S=\sum_iX_i$. For $\delta>0$, we prove that \[ \mathbb{P}\left(S<\mathbb{E} S+\delta\right)\ge b_{n,\delta}, \] where $b_{n,\delta}=\delta(n/(n+\delta))^n$ for $0<\delta<1$ and $b_{n,\delta}=(1-1/(n+\delta))^n$ for...
Weibo Fu, Yanjun Han, Guanyang Wang et al.· 7 citations· ⚡1
Let $X_1,\,X_2,\,\ldots,\,X_N$, $N\in\mathbb N$ be independent, discrete, integer-valued random variables. Assume that $X_j\geqslant m_j$ almost surely for each $j=1,\,2,\,\ldots,\,N$, where $m_1,\,m_2,\,\ldots,\,m_N\in\mathbb{Z}$ satisfy $m_1+\cdots+m_N<0$. Furthermore, suppose that the sequence $X_1,\,X_2,\,\ldots$ i...
Let $X=(X_1,\ldots,X_n)$ have independent coordinates with mean zero, variance one, and $\|X_i\|_{\psi_2}\le K$, and let $H_d=(\mathbb R^n)^{\otimes_2 d}$. Let $L>0$ and let $f:H_d\to\mathbb R$ be convex and $L$-Lipschitz. We prove that, for $0\le t\le c_KLn^{d/2}$, \[ \textsf{P}\left\{ \left\lvert f(X^{\otimes d})-\te...
Let $X_1,\ldots,X_N$ be independent random vectors in $\mathbb{R}^n$ with common isotropic log-concave distribution $\mu$ and set $P_{N,n}^{\mu}:=\operatorname{conv}\{\pm X_i:1\leqslant i\leqslant N\}$. Assume that $N/n=\gamma\geqslant \gamma_0$ where $\gamma_0>1$ is an absolute constant. We prove that with probability...
Let $X = [ \xi_1, \,\, \xi_2,...\,\, ,\xi_d]^\top$ be a zero-mean random vector of large dimension $d$ ($d \rightarrow \infty$) with (hidden) covariance matrix $M = (m_{ij})_{1 \leq i, j \leq d},$ where $m_{ij} = m_{ji} = \textbf{Cov}(\xi_i, \xi_j).$ Let $X_1, X_2, \dots, X_n$ be $n$ iid samples of $X$. Consider the sa...
Let $A=(\xi_{ij})$ be an $n\times n$ random matrix with independent, not necessarily identically distributed, real entries satisfying \[ \mathbb E\xi_{ij}=0,\qquad \mathbb E\xi_{ij}^{2}=1,\qquad \sup_{z\in\mathbb R}\mathbb P(|\xi_{ij}-z|0$ and $b\in(0,1)$. We prove that, for every $\delta\in(0,1)$, there are constants...
Manuel Fernández, Achintya Raya Polavarapu· 0 citations
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