Electronic Communications in Probability

Applications of size biased couplings for concentration of measures

Subhankar Ghosh and Larry Goldstein

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Let $Y$ be a nonnegative random variable with mean $\mu$ and finite positive variance $\sigma^2$, and let $Y^s$, defined on the same space as $Y$, have the $Y$ size biased distribution, that is, the distribution characterized by $$ E[Yf(Y)]=\mu E f(Y^s) \quad \mbox{for all functions $f$ for which these expectations exist.} $$ Under a variety of conditions on the coupling of $Y$ and $Y^s$, including combinations of boundedness and monotonicity, concentration of measure inequalities such as $$ P\left(\frac{Y-\mu}{\sigma}\ge t\right)\le \exp\left(-\frac{t^2}{2(A+Bt)}\right) \quad \mbox{for all $t \ge 0$} $$ are shown to hold for some explicit $A$ and $B$ in \cite{cnm}. Such concentration of measure results are applied to a number of new examples: the number of relatively ordered subsequences of a random permutation, sliding window statistics including the number of $m$-runs in a sequence of coin tosses, the number of local maxima of a random function on a lattice, the number of urns containing exactly one ball in an urn allocation model, and the volume covered by the union of $n$ balls placed uniformly over a volume $n$ subset of $\mathbb{R}^d$.

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Electron. Commun. Probab., Volume 16 (2011), paper no. 7, 70-83.

Accepted: 23 January 2011
First available in Project Euclid: 7 June 2016

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Ghosh, Subhankar; Goldstein, Larry. Applications of size biased couplings for concentration of measures. Electron. Commun. Probab. 16 (2011), paper no. 7, 70--83. doi:10.1214/ECP.v16-1605. https://projecteuclid.org/euclid.ecp/1465261963

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