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Asymptotic Behavior of Partial Sums: A More Robust Approach Via Trimming and Self-Normalization

  • Marjorie G. Hahn
  • Jim Kuelbs
  • Daniel C. Weiner
Part of the Progress in Probability book series (PRPR, volume 23)

Abstract

If X1, X2, X3, ...,are independent, identically distributed (i.i.d.) random variables and \(S_n = \sum\nolimits_{i = 1}^n {X_i }\) otes the nth partial sum, then limit theorems such as the law of large numbers (LLN), the central limit theorem (CLT), and the law of the iterated logarithm (LIL) all involve a strong interplay between the maximal terms of the sample.{|X 1|,...,|X n|} and the asymptotic behavior of the partial sum S n . Indeed, what is shown in the proofs of the classical formulation of each of these results is that the maximal or extreme terms of the sample are negligible in a sense required for the corresponding theorem. Furthermore, since the assumptions sufficient to prove the classical version of each of these results are also necessary, we see that extensions of these limit theorems will likely require methods that nullify or at least limit the effect of the extreme terms.

Keywords

Asymptotic Normality Moment Equation Partial Attraction Maximal Term Extreme Term 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 1991

Authors and Affiliations

  • Marjorie G. Hahn
    • 1
  • Jim Kuelbs
    • 2
  • Daniel C. Weiner
    • 3
  1. 1.Depart. of MathematicsTufts UniversityMedfordUSA
  2. 2.Depart. of MathematicsUniversity of WisconsinMadisonUSA
  3. 3.Depart. of MathematicsBoston UniversityBostonUSA

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