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

Author： Hashorva Enkelejd Macci Claudio Pacchiarotti Barbara

ISSN： 1387-5841

Source： Methodology And Computing In Applied Probability, Vol.15, Iss.4, 2013-12, pp. : 875-896

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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 $lim_{ntoinfty}frac{h_n}{n}=lambda$ for some λ ∈ [0,1], then ${K_{h_n:n}(D)/n:ngeq 1}$ satisfies the large deviation principle. As a by product we derive the large deviation principle for order statistics ${X_{h_n:n}:ngeq 1}$ . 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 ${K_{h_n:n}(D)/n:ngeq 1}$ and some related quantities.