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

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.