Generalized zero-one laws for large-order statistics

Abstract

For a fixed integer $r\ge 1$, let $Z_{\mathrm{rn}}$ be the $r$th largest of $\{X_1,X_2,...,X_n\}$, where $X_1,X_2,...$ is a sequence of i.i.d. random variables with the common distribution fuction $F\left(x\right)$. We prove that $P\{Z_{\mathrm{rn}}\le u_n,$ i.o.}=$0$ or $1$ accordingly as the series ${\sum }_{n=1}^{\mathrm{\infty }}exp[-n\{1-F\left(u_n\right)\left\}\right]\left[n\right\{1-F\left(u_n\right)\}{]}^r/n<\mathrm{\infty }$ or $=\mathrm{\infty }$ for any real sequence $\left\{u_n\right\}$ such that ${lim}_{n\to \mathrm{\infty }}n\{1-F(u_n\left)\right\}=+\mathrm{\infty }$. This weakens the condition added on the sequence $\left[n\right\{1-F\left(u_n\right)\left\}\right]$ by Wang and Tomkins and generalizes the results of Klass to the case when $r\ge 1$.