Concentration of Measure.

Concentration of measure for infinitely divisible distributions

In joint work with Ioannis Kontoyiannis [KM06], we gave a simple development of the concentration properties of infinitely divisible random vectors. A new modification of the Herbst argument is applied to a modified logarithmic Sobolev inequality of Wu [Wu00] to derive new concentration bounds. The point here is that the usual application of the Herbst argument leads to exponential concentration, but this may be impossible here because infinitely divisible distributions can have heavier tails. When the measure of interest does not have finite exponential moments but finite mean, our bounds exhibit polynomial decay of optimal order. A simple new proof is also given for earlier results of Houdré [Hou02], in the case where the measure does have finite exponential moments. All the results hold for infinitely divisible random vectors, but are stated in [KM06] only for compound Poisson distributions on the non-negative integers because in this special case, we can also give a simple proof of Wu's modified logarithmic Sobolev inequality. Concentration results of a similar flavour for infinitely divisible distributions (and more general settings) are also given by Breton, Houdré and Privault [BHP07], but the assumptions, techniques and form of the results are different, and it is not clear how to compare their bounds with ours.

Concentration inequalities for log-concave distributions

Description forthcoming...

References

[BHP07]    J.-C. Breton, C. Houdré, and N. Privault. Dimension free and infinite variance tail estimates on Poisson space. Acta Appl. Math., 95: 151–203, 2007.
[KM06]    I. Kontoyiannis and M. Madiman. Measure concentration for compound Poisson distributions. Elect. Comm. in Probab., 11:45–57, 2006.
[Hou02]    C. Houdré. Remarks on deviation inequalities for functions of infinitely divisible random vectors. Ann. Probab., 30(3):1223–1237, 2002.
[Wu00]    L. Wu. A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields, 118(3):427–438, 2000.

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