Probability Theory and Related Fields

, Volume 149, Issue 1–2, pp 271–278

Concentration of measures via size-biased couplings

Article

Abstract

Let Y be a nonnegative random variable with mean μ and finite positive variance σ2, and let Ys, 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 Ys, 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).

Keywords

Large deviations Size-biased couplings Stein’s method 

Mathematics Subject Classification (2000)

60E15 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Mathematics KAP 108University of Southern CaliforniaLos AngelesUSA

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