Abstract
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 ℓ ∞ [0,1] to a standard Brownian motion process [Donsker (1951)].
Keywords
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer Science+Business Media New York
About this chapter
Cite this chapter
van der Vaart, A.W., Wellner, J.A. (1996). Partial-Sum Processes. In: Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2545-2_24
Download citation
DOI: https://doi.org/10.1007/978-1-4757-2545-2_24
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4757-2547-6
Online ISBN: 978-1-4757-2545-2
eBook Packages: Springer Book Archive