On the minimum of several random variables

Gordon, Y.; Litvak, A.; Schütt, C.; Werner, E.

doi:10.1090/S0002-9939-06-08453-X

On the minimum of several random variables
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by Y. Gordon, A. E. Litvak, C. Schütt and E. Werner PDF

Proc. Amer. Math. Soc. 134 (2006), 3665-3675 Request permission

Abstract:

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 \[ c \max _{1 \leq j \leq k}\ \frac {k+1-j}{\sum _{i=j}^n 1/x_i } \leq \mathbb E k\mbox {-}\min _{1\leq i\leq n} |x_{i} \xi _{i}| \leq C \ln (k+1) \max _{1 \leq j \leq k}\ \frac {k+1-j}{\sum _{i=j}^n 1/x_i}, \] 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 ${k\mbox {-}\min }_{1\leq i\leq n} |x_{i} \xi _{i}|$ as well.

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