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Thinning, Entropy, and the Law of Thin Numbers
"... 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 informationtheoretic context, especially in connection with informationtheoretic inequalities r ..."
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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 informationtheoretic context, especially in connection with informationtheoretic inequalities related to Poisson approximation results. The classical BinomialtoPoisson 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 informationtheoretic 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 OrnsteinUhlenbeck process in connection to the entropy power inequality. Index Terms—Binomial distribution, compound Poisson distribution, entropy, information divergence, law of small numbers, law of thin numbers, Poisson distribution, PoissonCharlier polynomials, thinning. I.
Poisson process approximation for dependent superposition of point processes
, 2006
"... Although the study of weak convergence of superpositions of point processes to the Poisson process dates back to the work of Grigelionis in 1963, it was only recently that Schuhmacher [Stochastic Process. Appl. 115 (2005) 18191837] obtained error bounds for the weak convergence. Schuhmacher cons ..."
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Although the study of weak convergence of superpositions of point processes to the Poisson process dates back to the work of Grigelionis in 1963, it was only recently that Schuhmacher [Stochastic Process. Appl. 115 (2005) 18191837] obtained error bounds for the weak convergence. Schuhmacher considered dependent superposition, truncated the individual point processes to 01 point processes and then applied Stein's method to the latter. In this paper, we adopt a different approach to the problem by using Palm theory and Stein's method, thereby expressing the error bounds in terms of the mean measures of the individual point processes, which is not possible with Schuhmacher's approach. We consider locally dependent superposition as a generalization of the locally dependent point process introduced in Chen and Xia [Ann. Probab. 32 (2004) and apply the main theorem to the superposition of thinned point processes and of renewal processes.
Poisson Process Approximation: From Palm Theory to Stein's Method
"... Abstract: This exposition explains the basic ideas of Stein's method for Poisson random variable approximation and Poisson process approximation from the point of view of the immigrationdeath process and Palm theory. The latter approach also enables us to define local dependence of point proc ..."
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Abstract: This exposition explains the basic ideas of Stein's method for Poisson random variable approximation and Poisson process approximation from the point of view of the immigrationdeath process and Palm theory. The latter approach also enables us to define local dependence of point processes
Notes on the Poisson point process
, 2015
"... This work is licensed under a “CC BYSA 3.0 ” license. The Poisson process is a type of random object known as a point process that has been the focus of much study and application. This survey aims to give an accessible but detailed account of the Poisson point process by covering its history, mat ..."
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This work is licensed under a “CC BYSA 3.0 ” license. The Poisson process is a type of random object known as a point process that has been the focus of much study and application. This survey aims to give an accessible but detailed account of the Poisson point process by covering its history, mathematical definitions in a number of settings, and key properties as well detailing various terminology and applications of the process, with recommendations for further reading. 1