Probability Theory and Related Fields

, Volume 149, Issue 1, pp 271–278

Concentration of measures via size-biased couplings


DOI: 10.1007/s00440-009-0253-3

Cite this article as:
Ghosh, S. & Goldstein, L. Probab. Theory Relat. Fields (2011) 149: 271. doi:10.1007/s00440-009-0253-3


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).


Large deviationsSize-biased couplingsStein’s method

Mathematics Subject Classification (2000)


Copyright information

© Springer-Verlag 2009

Authors and Affiliations

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