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Order of magnitude bounds for expectations of Δ2-functions of generalized random bilinear forms
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  • Published: December 1998

Order of magnitude bounds for expectations of Δ2-functions of generalized random bilinear forms

  • Michael J. Klass1 &
  • Krzysztof Nowicki2 

Probability Theory and Related Fields volume 112, pages 457–492 (1998)Cite this article

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  • 2 Citations

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Abstract.

Let Φ be a symmetric function, nondecreasing on [0,∞) and satisfying a Δ2 growth condition, (X 1,Y 1), (X 2,Y 2),…,(X n ,Y n ) be arbitrary independent random vectors such that for any given i either Y i =X i or Y i is independent of all the other variates. The purpose of this paper is to develop an approximation of

valid for any constants {a ij }1≤ i,j≤n , {b i } i =1 n, {c j } j =1 n and d. Our approach relies primarily on a chain of successive extensions of Khintchin's inequality for decoupled random variables and the result of Klass and Nowicki (1997) for non-negative bilinear forms of non-negative random variables. The decoupling is achieved by a slight modification of a theorem of de la Peña and Montgomery–Smith (1995).

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Authors and Affiliations

  1.  Departments of Statistics and Mathematics, 367 Evans Hall # 3860, University of California, Berkeley, CA 94720–3860, USA, , , , , , US

    Michael J. Klass

  2. Department of Statistics, Lund University, Box 743, S-22007 Lund, Sweden e-mail: Krzysztof.Nowicki@stat.lu.se, , , , , , SE

    Krzysztof Nowicki

Authors
  1. Michael J. Klass
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  2. Krzysztof Nowicki
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Received: 25 March 1997 /  Revised version: 5 December 1997

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Klass, M., Nowicki, K. Order of magnitude bounds for expectations of Δ2-functions of generalized random bilinear forms. Probab Theory Relat Fields 112, 457–492 (1998). https://doi.org/10.1007/s004400050197

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  • Issue Date: December 1998

  • DOI: https://doi.org/10.1007/s004400050197

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  • Mathematics Subject Classification (1991): Primary 60E15
  • 60F25
  • 60G50
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