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Concentration of measures via size-biased couplings
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  • Published: 14 November 2009

Concentration of measures via size-biased couplings

  • Subhankar Ghosh1 &
  • Larry Goldstein1 

Probability Theory and Related Fields volume 149, pages 271–278 (2011)Cite this article

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  • 22 Citations

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Abstract

Let Y be a nonnegative random variable with mean μ and finite positive variance σ 2, and let Y s, defined on the same space as Y, have the Y size-biased distribution, characterized by

$$ E[Yf(Y)]=\mu E f(Y^s) \quad {\rm for\,all\,functions}\,f\,{\rm for\,which\,these\,expectations\,exist}. $$

Under a variety of conditions on Y and the coupling of Y and Y s, including combinations of boundedness and monotonicity, one sided concentration of measure inequalities such as

$$ P\left(\frac{Y-\mu}{\sigma} \ge t\right)\le {\rm exp}\left(-\frac{t^2}{2(A+Bt)} \right) \quad {\rm for\,all}\,t\, > 0 $$

hold for some explicit A and B. The theorem is applied to the number of bulbs switched on at the terminal time in the so called lightbulb process of Rao et al. (Sankhyā 69:137–161, 2007).

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References

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Authors and Affiliations

  1. Department of Mathematics KAP 108, University of Southern California, Los Angeles, CA, 90089-2532, USA

    Subhankar Ghosh & Larry Goldstein

Authors
  1. Subhankar Ghosh
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  2. Larry Goldstein
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Correspondence to Subhankar Ghosh.

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Cite this article

Ghosh, S., Goldstein, L. Concentration of measures via size-biased couplings. Probab. Theory Relat. Fields 149, 271–278 (2011). https://doi.org/10.1007/s00440-009-0253-3

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  • Received: 23 June 2009

  • Revised: 14 October 2009

  • Published: 14 November 2009

  • Issue Date: February 2011

  • DOI: https://doi.org/10.1007/s00440-009-0253-3

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Keywords

  • Large deviations
  • Size-biased couplings
  • Stein’s method

Mathematics Subject Classification (2000)

  • 60E15
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