Moment inequality of the minimum for nonnegative negatively orthant dependent random variables

Abstract

Let $\{x_n,n\geq1\}$ be a sequence of positive numbers and
$\{\xi_n,n\geq1\}$ be a sequence of nonnegative negatively orthant
dependent random variables satisfying certain distribution
conditions. An exponential inequality for the minimum $\min_{1\leq
i\leq n} x_i\xi_i$ is given. In addition, the moment inequality of
the minimum for nonnegative negatively orthant dependent random
variables is established as follows: $$ \left(\mathbb{E}
~k-\min_{1\leq i\leq n}|x_i\xi_i|^p\right)^{1/p}\leq \beta^{-1}
C(p,k) \max_{1\leq j\leq k}\frac{k+1-j}{\sum_{i=j}^n1/x_i},~2\leq
k\leq n,$$ where $C(p,k)=4\sqrt{2}\max\{p, \ln (1+k)\}$. If $k=1$,
the following inequality is presented: $$\mathbb{E}\left(\min_{1\leq
i\leq n}|x_i\xi_i|^p\right)\leq \beta^{-p} \Gamma
(1+p)\left(\sum_{i=1}^n\frac{1}{|x_i|}\right)^{-p}.$$ Our results
generalize the corresponding ones for independent random variables
to the case of negatively orthant dependent random variables.