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

Kontoyiannis, I.; Madiman, M.

Electron. Commun. Probab. 11, 45-57 (2006).

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)].

MSC:

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

Citations:

Zbl 1017.60018; Zbl 0970.60093