Nonlinear Analysis: Theory, Methods & Applications
The convergence rate for the normal approximation of extreme sums
Section snippets
Introduction and main results
Let be a sequence of independent and identically distributed random variables with distribution , and for each let denote the order statistics of . Define as an extreme sum, where satisfies The extreme sums have been widely studied in the literature. On one hand, the study is motivated by the interest of how the asymptotic properties of partial sums are influenced by the extreme sums. On the other
Proof of Theorem 1
The following two lemmas are related to condition (3).
Lemma 1 If(3)holds for , thenwith convention for . Furthermore, for any , there exists a such thatholds for all and .
Proof Note that By the properties of regularly varying functions, the convergence in (3) is locally uniform. Therefore, we have which is
Proof of Theorem 2
For and , let
Lemma 7 Suppose(1), (3)hold. Thenand for any positive p with
Proof Write It follows from (11), (12)
Acknowledgements
The authors would like to thank the two referees for their careful reading of the manuscript and for their constructive suggestions that have improved the presentation of the paper. Qi’s research was supported by NSF grant DMS 0604176.
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