Comparison and anti-concentration bounds for maxima of Gaussian random vectors

Victor Chernozhukov, Denis Chetverikov () and Kengo Kato
Additional contact information
Denis Chetverikov: Institute for Fiscal Studies and UCLA
Kengo Kato: Institute for Fiscal Studies

No CWP40/16, CeMMAP working papers from Centre for Microdata Methods and Practice, Institute for Fiscal Studies

Abstract: Slepian and Sudakov-Fernique type inequalities, which compare expectations of maxima of Gaussian random vectors under certain restrictions on the covariance matrices, play an important role in probability theory, especially in empirical process and extreme value theories. Here we give explicit comparisons of expectations of smooth functions and distribution functions of maxima of Gaussian random vectors without any restriction on the covariance matrices. We also establish an anti-concentration inequality for the maximum of a Gaussian random vector, which derives a useful upper bound on the Lévy concentration function for the Gaussian maximum. The bound is dimension-free and applies to vectors with arbitrary covariance matrices. This anti-concentration inequality plays a crucial role in establishing bounds on the Kolmogorov distance between maxima of Gaussian random vectors. These results have immediate applications in mathematical statistics. As an example of application, we establish a conditional multiplier central limit theorem for maxima of sums of independent random vectors where the dimension of the vectors is possibly much larger than the sample size.

Keywords: Slepian inequality; anti-concentration; Levy concentration function; maximum of Gaussian random vector; conditional multiplier central limit theorem. 1 (search for similar items in EconPapers)
Date: 2016-08-26
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.ifs.org.uk/uploads/cemmap/wps/cwp401616.pdf (application/pdf)
Our link check indicates that this URL is bad, the error code is: 404 Not Found (https://www.ifs.org.uk/uploads/cemmap/wps/cwp401616.pdf [302 Found]--> https://ifs.org.uk/uploads/cemmap/wps/cwp401616.pdf)

Related works:
Working Paper: Comparison and anti-concentration bounds for maxima of Gaussian random vectors (2016)
Working Paper: Comparison and anti-concentration bounds for maxima of Gaussian random vectors (2013)
Working Paper: Comparison and anti-concentration bounds for maxima of Gaussian random vectors (2013)
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:ifs:cemmap:40/16

Ordering information: This working paper can be ordered from
The Institute for Fiscal Studies 7 Ridgmount Street LONDON WC1E 7AE

Access Statistics for this paper

More papers in CeMMAP working papers from Centre for Microdata Methods and Practice, Institute for Fiscal Studies The Institute for Fiscal Studies 7 Ridgmount Street LONDON WC1E 7AE. Contact information at EDIRC.
Bibliographic data for series maintained by Emma Hyman ().