The convergence rate for the normal approximation of extreme sums

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Abstract

This paper derives the normal convergence rate for sums of extreme values. The exact rates are obtained under some regularity and smoothing conditions on the tail of the distribution of the sample.

Section snippets

Introduction and main results

Let {X,X1,X2,} be a sequence of independent and identically distributed random variables with distribution F, and for each n1 let Xn,1Xn,n denote the order statistics of X1,,Xn. Define Sn,k=j=1kXn,nj+1 as an extreme sum, where k=kn satisfies limnkn=andlimnknn=0. 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 j=1nXj 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 γR , thenlimtV(tx)V(t)tV(t)=xγ1γ,x>0with convention xγ1γ|γ=0=logx for x>0 . Furthermore, for any δ>0 , there exists a tδ>0 such that(1δ)xγδ1γδV(tx)V(t)tV(t)(1+δ)xγ+δ1γ+δholds for all x1 and ttδ .

Proof

Note that V(tx)V(t)tV(t)=1xV(tu)V(t)du,x>0. By the properties of regularly varying functions, the convergence in (3) is locally uniform. Therefore, we have limtV(tx)V(t)tV(t)=1xuγ1du=xγ1γ, which is

Proof of Theorem 2

For n=1,2, and t,x,uR, let ρ(γ)=2(1+γ)12γ13γ;G(x)=Φ(x)ρ(γ)(x21)ϕ(x)3kn;Fn(x,u)=P(Sˆn,kn(u)x);fn(t,u)=Eexp{itZˆn,1(u)}(the characteristic function of Zˆn,1(u)) ;ψn(t,u)=logfn(tkn,u)+t22kn.

Lemma 7

Suppose(1), (3)hold. Thenlimnsup|u|kn1/6|EZˆn,13(u)ρ(γ)|=0,and for any positive p with γp<13γlim supnsup|u|kn1/6E|Zˆn,1(u)|3+p<.

Proof

Write E[V(ln(u)Z)EV(ln(u)Z)]3=Tn,3(u)3Tn,2(u)[EV(ln(u)Z)V(ln)]+2[EV(ln(u)Z)V(ln)]3+E[V(lnZ)V(ln)]33[EV(ln(u)Z)V(ln)]E[V(lnZ)V(ln)]2. 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.

References (11)

  • G.S. Lo

    A note on the asymptotic normality of sums of extreme values

    J. Statist. Plann. Inference

    (1989)
  • S. Csörgö et al.

    What portion of the sample makes a partial sum asymptotically stable or normal?

    Probab. Theory Related Fields

    (1986)
  • S. Csörgö et al.

    The asymptotic distribution of sums of extreme values from a regularly varying distribution

    Ann. Probab.

    (1986)
  • E. Haeusler et al.

    A law of the iterated logaritheorem for sums of extreme values from a distribution with a regularly varying upper tail

    Ann. Probab.

    (1987)
  • S. Csörgö et al.

    The asymptotic distribution of extreme sums

    Ann. Probab.

    (1991)
There are more references available in the full text version of this article.

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