, Volume 149, Issue 1-2, pp 271-278

Concentration of measures via size-biased couplings

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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).