Measure concentration for compound Poisson distributions. (English) Zbl 1112.60008

Summary: We give a simple development of the concentration properties of compound Poisson measures on the nonnegative integers. A new modification of the Herbst argument is applied to an appropriate modified logarithmic-Sobolev inequality to derive new concentration bounds. When the measure of interest does not have finite exponential moments, these bounds exhibit optimal polynomial decay. Simple new proofs are also given for earlier results of C. Houdré [Ann. Probab. 30, No. 3, 1223–1237 (2002; Zbl 1017.60018)] and L. Wu [Probab. Theory Relat. Fields 118, No. 3, 427–438 (2000; Zbl 0970.60093)].


60E07 Infinitely divisible distributions; stable distributions
26D15 Inequalities for sums, series and integrals
39B62 Functional inequalities, including subadditivity, convexity, etc.
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
60E15 Inequalities; stochastic orderings
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