On the minimum of several random variables

For a given sequence of real numbers $a_{1}, \dots, a_{n}$, we denote the $k$th smallest one by ${k\mbox{-}\min} _{1\leq i\leq n}a_{i}$. Let $\mathcal{A}$ be a class of random variables satisfying certain distribution conditions (the class contains $N(0, 1)$ Gaussian random variables). We show that there exist two absolute positive constants $c$ and $C$ such that for every sequence of real numbers $0< x_{1}\leq \ldots \leq x_{n}$ and every $k\leq n$, one has

$c \max_{1 \leq j \leq k} \frac {k+1-j}{\sum_{i=j}^n 1/x_i } \leq \mathbb{E} \, \, k$-$\min_{1\leq i\leq n} \vert x_{i} \xi_{i}\vert \leq C\, \ln(k+1)\, \max_{1 \leq j \leq k} \frac{k+1-j}{\sum_{i=j}^n 1/x_i},$

where $\xi_1, \dots, \xi_n$ are independent random variables from the class $\mathcal{A}$. Moreover, if $k=1$, then the left-hand side estimate does not require independence of the $\xi_i$'s. We provide similar estimates for the moments of ${k\mbox{-}\min}_{1\leq i\leq n} \vert x_{i} \xi_{i}\vert$ as well.