Abstract
Rényi's thinning operation on a discrete random variable is a natural discrete analog of the scaling operation for continuous random variables. The properties of thinning are investigated in an information-theoretic context, especially in connection with information-theoretic inequalities related to Poisson approximation results. The classical Binomial-to-Poisson convergence (sometimes referred to as the “law of small numbers”) is seen to be a special case of a thinning limit theorem for convolutions of discrete distributions. A rate of convergence is provided for this limit, and nonasymptotic bounds are also established. This development parallels, in part, the development of Gaussian inequalities leading to the information-theoretic version of the central limit theorem. In particular, a “thinning Markov chain” is introduced, and it is shown to play a role analogous to that of the Ornstein-Uhlenbeck process in connection to the entropy power inequality.
Translated title of the contribution | Thinning, Entropy and the Law of Thin Numbers |
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Original language | English |
Pages (from-to) | 4228 - 4244 |
Number of pages | 17 |
Journal | IEEE Transactions on Information Theory |
Volume | 56 |
Issue number | 9 |
DOIs | |
Publication status | Published - Sept 2010 |