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Sample path large deviations for order statistics. (English) Zbl 1229.62058

Let \(\left\{X_{n}:n\geq1\right\}\) be a sequence of independent and identically distributed real-valued random variables with a common distribution function \(F\) that is assumed to be strictly increasing, but possibly having discontinuities. Let \(a=\inf\left\{ x:F(x)>0\right\} \in[ -\infty,\infty)\) and \(b=\inf\left\{ x:F(x)=1\right\} \in(-\infty,\infty].\) For each \(n\geq1\), let \(X_{1,n}\leq X_{2,n}\leq\dots\leq X_{n,n}\) denote the ascending order statistics of \(X_{1},\dots,X_{n}\) and define \(X_{0,n}=a\) and \(X_{n+1,n}=b\). Define the sample paths of the order statistics by \(X_{n}(t)=X_{\left[ (n+1)t\right] ,n}\) for all \(t\in[0,1]\), where \([x]\) is the greatest integer that is less than \(x\).
The authors prove that the sample paths of the order statistics satisfy the large deviation principle in the Skorokhod \(M_{1}\) topology. Sanov’s theorem is derived in the Skorokhod \(M_{1}^{\prime}\) topology as a corollary to this result. A number of illustrative examples are presented, including applications to the sample paths of trimmed means and Hill plots.

MSC:

62G30 Order statistics; empirical distribution functions
60F10 Large deviations
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