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Almost sure convergence of sample range

Authors:

Tan Zhongquan Peng Zuoxiang Saralees Nadarajah

Institution:

(1) Department of Mathematics, Zunyi Normal College, Zunyi, 563002, China;(2) School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China;(3) School of Mathematics, University of Manchester, Manchester, M60 1QD, UK

Abstract:

Let X ₁, X ₂, ... be i.i.d. random variables. The sample range is R _n = max {X _i, 1 ≤ i ≤ n} − min {X _i, 1 ≤ i ≤ n}. If $\alpha _{k} {\left( {R_{k} - \beta _{k} } \right)}\xrightarrow{w}G$ for a non-degenerate distribution G and some sequences (α _k), (β _k) then we have

${\mathop {\lim }\limits_{n \to \infty } }\frac{1}{{\log n}}{\sum\limits_{k = 1}^n {\frac{1}{k}I{\left( {\alpha _{k} {\left( {R_{k} - \beta _{k} } \right)} \leqslant x} \right)} = G{\left( x \right)}} }$

and

${\mathop {\lim }\limits_{n \to \infty } }\frac{1}{{\log n}}{\sum\limits_{k = 1}^n {\frac{1}{k}f{\left( {\alpha _{k} {\left( {R_{k} - \beta _{k} } \right)}} \right)} = {\int_{ - \infty }^\infty {f{\left( x \right)}dG{\left( x \right)}} }} }$

almost surely for any continuity point x of G and for any bounded Lipschitz function f: R → R.

Keywords:

Almost sure central limit theorem Domains of attractions I i d sequences Sample range

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