Slepian’s inequality for Gaussian processes with respect to weak majorization
© Fang; licensee Springer 2013
Received: 12 July 2012
Accepted: 15 December 2012
Published: 4 January 2013
In this paper, we obtain some sufficient conditions for Slepian’s inequality for Gaussian processes with respect to weak majorization. For our results, we also provide an application example.
1 Introduction and main results
Gaussian processes are natural extensions of multivariate Gaussian random variables to infinite (countably or continuous) index sets. For Gaussian processes, strong and weak stationarity are the same concept. Gaussian processes are by far the most accessible and well-understood processes (on uncountable index sets), which are important in statistical modeling because of properties inherited from the normal one, and many deep theoretical analyses of various properties are available.
Slepian’s inequality and its modifications are an essential ingredient in the proofs of many results being concerned with sample path properties of Gaussian processes. See, e.g., Adler and Taylor  and Maurer . Some sufficient conditions for Slepian’s inequality with respect to majorization for two Gaussian random vectors have been given in Fang and Zhang .
Majorization is a pre-ordering on vectors by sorting all components in nonincreasing order, which is a very interesting topic in various fields of mathematics and statistics. The history of investigating majorization should date back to Schur  and Hardy et al. . The reader can find that majorization has been connected with combinatorics, analytic inequalities, numerical analysis, matrix theory, probability and statistics in Marshall and Olkin . Recent research on majorization with respect to matrix inequalities and norm inequalities has been carried out by Ando .
In this paper, we establish four Slepian’s inequalities for Gaussian processes with respect to weak majorization, with their proofs and an application given in Section 2. Firstly, we recall the definitions of majorization and weak majorization.
Definition 1.1 (Marshall and Olkin )
- (1)is said to be majorized by λ, in symbols , if
- (2)is said to be weak lower majorized by λ, in symbols , if
- (3)is said to be weak upper majorized by λ, in symbols , if
for , and .
The main results of the paper are stated as follows.
- (1)If , for all , and , then
- (2)If , for all , and , then
- (3)If , for all , and , then
- (4)If , for all , and , then
for all .
In Theorem 1.2, after setting , we can easily get the following result.
- (1)If , then
- (2)If , then
for all .
2 Proof and application
for all .
It is easy to check that and satisfy the conditions in Theorem 1.2.
This research is supported by the National Statistical Science Research Project of China (No. 2012LY158), Natural Science Foundation of Anhui-Province (No. 1208085MA11).
- Slepian D: The one-sided barrier problem for Gaussian processes. Bell Syst. Tech. J. 1962, 41: 463–501.MathSciNetView ArticleGoogle Scholar
- Adler RJ, Taylor JE: Random Fields and Geometry. Springer, New York; 2007.MATHGoogle Scholar
- Maurer A: Transfer bounds for linear feature learning. Mach. Learn. 2009, 75: 327–350. 10.1007/s10994-009-5109-7View ArticleGoogle Scholar
- Fang L, Zhang X: Slepian’s inequality with respect to majorization. Linear Algebra Appl. 2010, 434: 1107–1118.MathSciNetView ArticleGoogle Scholar
- Schur I: Über eine klasse von mittelbildungen mit anwendungen auf die determinantentheorie. 22. Sitzungsberichte der Berliner Mathematischen Gesellschaft 1923, 9–20.Google Scholar
- Hardy GH, Littlewood JE, Pólya G: Some simple inequalities satisfied by convex functions. Messenger Math. 1929, 58: 145–152.Google Scholar
- Marshall AW, Olkin I: Inequalities: Theory of Majorization and Its Applications. Academic Press, New York; 1979.MATHGoogle Scholar
- Ando T: Majorization and inequalities in matrix theory. Linear Algebra Appl. 1989, 199: 17–67.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.