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Bounds for Stochastic Processes on Product Index Spaces

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Book cover High Dimensional Probability VII

Part of the book series: Progress in Probability ((PRPR,volume 71))

Abstract

In this paper we discuss the question of how to bound the supremum of a stochastic process with an index set of a product type. It is tempting to approach the question by analyzing the process on each of the marginal index sets separately. However it turns out that it is necessary to also study suitable partitions of the entire index set. We show what can be done in this direction and how to use the method to reprove some known results. In particular we point out that all known applications of the Bernoulli Theorem can be obtained in this way. Moreover we use the shattering dimension to slightly extend the application to VC classes. We also show some application to the regularity of paths of processes which take values in vector spaces. Finally we give a short proof of the Mendelson–Paouris result on sums of squares for empirical processes.

Mathematics Subject Classification (2010). Primary 60G15; Secondary 60G17

Research partially supported by MNiSW Grant N N201 608740 and MNiSW program Mobility Plus.

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Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Warsaw, Banacha 2, 02-097, Warsaw, Poland

    Witold Bednorz

Authors
  1. Witold Bednorz
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Correspondence to Witold Bednorz .

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Editors and Affiliations

  1. Georgia Institute of Technology, Atlanta, Georgia, USA

    Christian Houdré

  2. University of Delaware, Department of Applied Economics and Stat University of Delaware, Newark, Delaware, USA

    David M. Mason

  3. Centre national de la recherche scientifique, Laboratoire J.A. Dieudonné, Université Côte d’Azur, Nice, France

    Patricia Reynaud-Bouret

  4. Department of Mathematics, University of Tennessee Department of Mathematics, Knoxville, Tennessee, USA

    Jan Rosiński

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Bednorz, W. (2016). Bounds for Stochastic Processes on Product Index Spaces. In: Houdré, C., Mason, D., Reynaud-Bouret, P., Rosiński, J. (eds) High Dimensional Probability VII. Progress in Probability, vol 71. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-40519-3_14

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