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Estimating Averages of Order Statistics of Bivariate Functions

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Abstract

We prove uniform estimates for the expected value of averages of order statistics of bivariate functions in terms of their largest values by a direct analysis. As an application, uniform estimates for the expected value of averages of order statistics of sequences of independent random variables in terms of Orlicz norms are obtained. In the case where the bivariate functions are matrices, we provide a “minimal” probability space which allows us to C-embed certain Orlicz spaces \(\ell _M^n\) into \(\ell _1^{cn^3}\), with \(c,C>0\) being absolute constants.

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Acknowledgments

R. Lechner is supported by the Austrian Science Fund, FWF P23987 and FWF P22549. M. Passenbrunner is supported by the Austrian Science Fund, FWF P27723. J. Prochno is supported by the Austrian Science Fund, FWFM 1628000.

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Authors and Affiliations

  1. Institute of Analysis, Johannes Kepler University Linz, Altenbergerstrasse 69, 4040, Linz, Austria

    Richard Lechner & Markus Passenbrunner

  2. Department of Mathematics, University of Hull, Cottingham Road, Hull, HU6 7RX, UK

    Joscha Prochno

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  1. Richard Lechner
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  2. Markus Passenbrunner
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  3. Joscha Prochno
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Correspondence to Joscha Prochno.

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Lechner, R., Passenbrunner, M. & Prochno, J. Estimating Averages of Order Statistics of Bivariate Functions. J Theor Probab 30, 1445–1470 (2017). https://doi.org/10.1007/s10959-016-0702-8

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  • DOI: https://doi.org/10.1007/s10959-016-0702-8

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