Abstract
The theme of Chap. 5 is to provide upper bounds in the L ∞, or Kolmogorov distance, that can be applied when certain bounded couplings can be constructed between an auxiliary random variable \(\widetilde{W}\) and the variable W. Important cases considered include when the variable \(\widetilde{W}\) has the same distribution as W, or has the zero bias or size bias distribution of W. The bounds shown in this chapter are often interpretable, sometimes directly, as a distance between W and \(\widetilde{W}\), a small bound being a reflection of a small distance. The chapter concludes with the use of smoothing inequalities to obtain distances between W and the normal over general function classes, one special case being the derivation of Kolmogorov distance bounds when bounded size bias couplings exist.
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Chen, L.H.Y., Goldstein, L., Shao, QM. (2011). L ∞ by Bounded Couplings. In: Normal Approximation by Stein’s Method. Probability and Its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15007-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-15007-4_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15006-7
Online ISBN: 978-3-642-15007-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)