NDLI: Concentration and relative entropy for compound Poisson distributions

Content Provider	IEEE Xplore Digital Library
Author	Madiman, M. Kontoyiannis, I.
Copyright Year	2005
Description	Author affiliation: Div. of Appl. Math., Brown Univ., Providence, RI (Madiman, M.; Kontoyiannis, I.)
Abstract	Using a simple inequality about the relative entropy, its so-called "tensorization property," we give a simple proof of a functional inequality which is satisfied by any compound Poisson distribution. This functional inequality belongs to the class of modified logarithmic Sobolev inequalities. We use it to obtain measure concentration bounds for compound Poisson distributions under a variety of assumptions on their tail behavior. In particular, we show how the celebrated "Herbst argument" can be modified to yield sub-exponential concentration bounds. For example, suppose Z is a compound Poisson random variable with values on the nonnegative integers, and let f be a function such that \|f(k+1) - f(k)\| les 1 for all k. Then, if the base distribution of Z does not have a finite moment-generating function but has finite moments up to some order L > 1, we show that the probability that f(Z) exceeds its mean by a positive amount t or more decays approximately like $(const)middott^{-L},$ where the constant is explicitly identified. This appears to be one of the very first examples of concentration bounds with power-law decay
Starting Page	1833
Ending Page	1837
File Size	161760
Page Count	5
File Format	PDF
ISBN	0780391519
DOI	10.1109/ISIT.2005.1523662
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2005-09-04
Publisher Place	Australia
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Entropy Random variables Mathematics Computer science Probability distribution Information geometry Functional analysis Algorithm design and analysis Physics Statistical analysis
Content Type	Text
Resource Type	Article

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