Suppose $X_1,X_2,...,X_m$ are iid uniform on ${0,1/n,2/n,...,1}.$ Let $S_l$ be the $l$ th highest among all $X_1,...,X_m$. I want to show that Pr($S_l > S_{l+1}$) goes to $0$ at the rate $1/m.$ Please advise. Also I would like to know how can I generalize the result for a more general non-uniform distributions.
Space between order statistics
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0If $l$ is fixed, then $l/m$ is going to zero as $m$ grows, and eventually the top $l+1$ variables will all be equal to $1$ with probability very nearly $1$ (with error decaying exponentially). So this won't be right. On the other hand, the expected value of Pr($S_l > S_{l+1}$) taken over all $l$ does go to zero as $1/m$. – 2012-07-20