Partial-Sum Processes

  • Aad W. van der Vaart
  • Jon A. Wellner
Part of the Springer Series in Statistics book series (SSS)


The name “Donsker class of functions” was chosen in honor of Donsker’s theorem on weak convergence of the empirical distribution function. A second famous theorem by Donsker concerns the partial-sum process \( {\mathbb{Z}_n}(s) = \frac{1}{{\sqrt n }}\sum\limits_{i = 1}^{\left[ {ns} \right]} {{Y_i}} = \frac{1}{{\sqrt n }}\sum\limits_{i = 1}^k {{Y_i}} ,\frac{k}{n} \leqslant s < \frac{{k + 1}}{n}, \) for i.i.d. random variables Y1, …, Y n with zero mean and variance 1. Donsker essentially proved that the sequence of processes {ℤn (t): 0 ≤ t ≤ 1} converges in distribution in the space [0,1] to a standard Brownian motion process [Donsker (1951)].


Lebesgue Measure Gaussian Process Empirical Process Empirical Distribution Function Dirac Measure 
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Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Aad W. van der Vaart
    • 1
  • Jon A. Wellner
    • 2
  1. 1.Department of Mathematics and Computer ScienceFree UniversityAmsterdamThe Netherlands
  2. 2.StatisticsUniversity of WashingtonSeattleUSA

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