Large Deviations for Proportions of Observations Which Fall in Random Sets Determined by Order Statistics

Enkelejd Hashorva (), Claudio Macci () and Barbara Pacchiarotti ()
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Enkelejd Hashorva: Université de Lausanne
Claudio Macci: Università di Roma Tor Vergata
Barbara Pacchiarotti: Università di Roma Tor Vergata

Methodology and Computing in Applied Probability, 2013, vol. 15, issue 4, 875-896

Abstract: Abstract Let {X n :n ≥ 1} be independent random variables with common distribution function F and consider $K_{h:n}(D)=\sum_{j=1}^n1_{\{X_j-X_{h:n}\in D\}}$ , where h ∈ {1,...,n}, X 1:k ≤ ⋯ ≤ X k:k are the order statistics of the sample X 1,...,X k and D is some suitable Borel set of the real line. In this paper we prove that, if F is continuous and strictly increasing in the essential support of the distribution and if $\lim_{n\to\infty}\frac{h_n}{n}=\lambda$ for some λ ∈ [0,1], then $\{K_{h_n:n}(D)/n:n\geq 1\}$ satisfies the large deviation principle. As a by product we derive the large deviation principle for order statistics $\{X_{h_n:n}:n\geq 1\}$ . We also present results for the special case of Bernoulli distributed random variables with mean p ∈ (0,1), and we see that the large deviation principle holds only for p ≥ 1/2. We discuss further almost sure convergence of $\{K_{h_n:n}(D)/n:n\geq 1\}$ and some related quantities.

Keywords: Almost sure convergence; Bernoulli law; Near maximum; Point process; Relative entropy; 60F10; 60G70; 62G30 (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s11009-012-9290-y

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