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Let Y be a nonnegative random variable with mean and nite positive variance 2 , and let Y s , dened on the same space as Y , have theY size biased distribution, that is, the distribution characterized were shown to hold for some explicit A and B in (14), and were applied to the lightbulb process of (34). Those concentration of measure results are applied here to a number of new examples where one can construct a bounded coupling of Y s to Y : 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 R d . Concentration of measure results are also provided for certain instances where the coupling of Y s to Y is not bounded, in particular, for the number of isolated vertices in the Erdos- R enyi random graph model, and the innitely divisible and compound Poisson distributions that satisfy a bounded moment generating function condition, the latter with applications to the generalized variance of a random matrix.

Applications of size biased couplings for concentration of measures