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Poisson Approximation

1993
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Journal of the American Statistical Association
*

We give several examples for

doi:10.2307/2290818
fatcat:pb7cdmra7fcnzftqribkh2zxce
*Poisson**approximation*of quantities of interest in the analysis of algorithms: the distribution of node depth in a binary search tree, the distribution of the number of losers ... A simple and well-known upper bound for the total variation distance between the distribution of a sum of independent Bernoulli variables and the*Poisson*distribution with the same mean turns out to be ... normal density function may be a hint to*Poisson**approximation*(as we pointed out in Section 2, normal*approximation*can be a corollary to*Poisson**approximation*). ...##
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Poisson approximation

2019
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Probability Surveys
*

The accuracy of shifted

doi:10.1214/18-ps318
fatcat:3de4c4lcubbwhkvtmtnahwtepi
*Poisson**approximation*can be sharper than that of pure*Poisson**approximation*. ... Check if the correct rate of convergence in Compound*Poisson**approximation*The topic of compound*Poisson*(CP)*approximation*is vast. ... in [41] ), the total variation distance between { n i=1 1{Y n,i < t}, t ∈ [0; T ]} and the*approximating**Poisson*process coincides with d T V (B(n, p); Π(np)), where p = P(Y n,1 ≤ T ). ...##
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Poisson approximation
[article]

2019
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arXiv
*
pre-print

We overview results on the topic of

arXiv:1901.01847v2
fatcat:udjmeya4gzhtdi6r2lolcxwste
*Poisson**approximation*that are missed in existing surveys. ... The topic of*Poisson**approximation*to the distribution of a sum of integer-valued random variables is presented as well. ...*Poisson**approximation*...##
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Poisson–Voronoi approximation

2009
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The Annals of Applied Probability
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Precise estimates for the variance of both quantities are obtained which follow from a new jackknife inequality for the variance of functionals of a

doi:10.1214/08-aap561
fatcat:sqexle5nn5abzn26ntcztzan2i
*Poisson*point process. ... Let X be a*Poisson*point process and K⊂R^d a measurable set. Construct the Voronoi cells of all points x∈ X with respect to X, and denote by v_X(K) the union of all Voronoi cells with nucleus in K. ...*POISSON*-VORONOI*APPROXIMATION*9 For X i (RB d ) ≥ 2 we use (A2) with D = X i , the estimate ∞ n=2 µ n n! ...##
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Poisson approximation. Addendum

2021
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Probability Surveys
*

An important feature of a

doi:10.1214/21-ps2
fatcat:vqj2w6slrrbdvnefompbmlvkhe
*Poisson*limit theorem in [4] is the absence of the traditional assumption (A.2). ... The purpose of this addendum is to explain why assumption (A.2) is not required, and compare the assumptions of the*Poisson*limit theorem in [4] with traditional ones. ... We denote by π λ a*Poisson*Π(λ) random variable. A*Poisson*limit theorem states that as n →∞, S n ⇒ π λ (∃λ>0). ...##
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Topics in poisson approximation
[chapter]

2001
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Handbook of Statistics
*

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Conditional Poisson process approximation
[article]

2015
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arXiv
*
pre-print

The basic starting point for point processes is usually the

arXiv:1511.03251v1
fatcat:vnlxjunt6jccxg36cnous5gq64
*Poisson*process. Over the years, Stein's method has been developed with a great deal of success for*Poisson*point process*approximation*. ... In this paper, we develop Stein's method for conditional*Poisson*point process*approximation*, and closely examine what sort of difficulties that this conditioning entails. ... We seek to*approximate*a conditional Bernoulli process Ξ (1) with a conditional*Poisson*process Pn (1) (λ), conditioning upon at least 1 atom. ...##
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Compound Poisson process approximation

2002
*
The Annals of Probability
*

Compound

doi:10.1214/aop/1029867135
fatcat:igzs6e3wrbdezgu7qjv3poyfxm
*Poisson*processes are often useful as*approximate*models, when describing the occurrence of rare events. In this paper, we develop a method for showing how close such*approximations*are. ... Our approach is to use Stein's method directly, rather than by way of declumping and a marked*Poisson*process; this has conceptual advantages, but entails technical difficulties. ... However, a compound*Poisson*process*approximation*contains much more information. ...##
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The Poisson Approximation to the Poisson Binomial Distribution

1960
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The Annals of Mathematical Statistics
*

THE

doi:10.1214/aoms/1177705799
fatcat:dbh6jogydrgzpp3a6kuubbfiwq
*POISSON**APPROXIMATION*TO THE*POISSON*BINOMIAL DISTRIBUTION' By J. L. Hopegs, Jr. anp Lucien Le Cam University of California, Berkeley 1. Introduction. ... It has been observed empirically that in many situations the number S of events of a specified kind has*approximately*a*Poisson*dis- tribution. ...##
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Poisson and Compound Poisson Approximations in a Nonconventional Setup
[article]

2012
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arXiv
*
pre-print

We show that for all ψ-mixing shifts distributions of the numbers of multiple recurrencies to shrinking cylindrical neighborhoods of all points are close either to

arXiv:1211.5238v2
fatcat:kdxmlv4klzedho66tet3w7ofja
*Poisson*or to compound*Poisson*distributions ... In Sections 4 and 5 our*Poisson*and compound*Poisson**approximations*results will be proved. ... In the "conventional" case ℓ = 1*Poisson**approximation*estimates were obtained in [4] and [5] while compound*Poisson**approximation*estimates for periodic sequences were derived in [15] but even in ...##
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Exchangeable pairs and Poisson approximation

2005
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Probability Surveys
*

This is a survey paper on

doi:10.1214/154957805100000096
fatcat:yimcnjr5d5ax7pqzukkryhrtce
*Poisson**approximation*using Stein's method of exchangeable pairs. ... We illustrate using*Poisson*-binomial trials and many variations on three classical problems of combinatorial probability: the matching problem, the coupon collector's problem, and the birthday problem. ... Sections 7 and 8 give multivariate and*Poisson*process*approximations*. The*Poisson*heuristic and the Chen-Stein approach to*Poisson**approximation*have been actively developed since 1970. ...##
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Poisson approximations for telecommunications networks

1994
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The Journal of the Australian Mathematical Society Series B Applied Mathematics
*

We explain how this can be used to delimit circumstances under which traffic is

doi:10.1017/s0334270000010298
fatcat:njt2nhivqbdl5gpc5ohlsynn3e
*approximately*Poissonian. ... The first of these allows one to identify*Poisson*traffic via the notion of "deterministic past-conditional arrival rate". ...*Poisson**approximations*in teletraffic networks Apart from those which emanate from outside, traffic processes in telecommunications networks are rarely*Poisson*. ...##
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Poisson approximations for telecommunications networks

1991
*
The Journal of the Australian Mathematical Society Series B Applied Mathematics
*

We explain how this can be used to delimit circumstances under which traffic is

doi:10.1017/s0334270000006913
fatcat:mswa2uub2relznn75gq63azj6i
*approximately*Poissonian. ... The first of these allows one to identify*Poisson*traffic via the notion of "deterministic past-conditional arrival rate". ...*Poisson**approximations*in teletraffic networks Apart from those which emanate from outside, traffic processes in telecommunications networks are rarely*Poisson*. ...##
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A Compound Poisson Approximation Inequality

2006
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Journal of Applied Probability
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We give conditions under which the number of events which occur in a sequence of m-dependent events is stochastically smaller than a suitably defined compound

doi:10.1239/jap/1143936260
fatcat:oclw4jfmcfhnrnheu2nrrbzukm
*Poisson*random variable. ... The study of*approximations*for this classical problem goes back at least to von Mises (1921) , and is often used to test the effectiveness of*Poisson*or compound*Poisson**approximations*; see, for example ... To get the stochastic inequality we must have P(S > 0) ≤ P(C > 0), and so computing the*Poisson*parameter for the*approximating*compound*Poisson*distribution may be complicated in practice. ...##
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POISSON APPROXIMATION FOR RANDOM SUMS OF POISSON RANDOM VARIABLES

2014
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International Journal of Pure and Applied Mathematics
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In this paper, we use the Stein-Chen method to determine a bound for the total variation distance between the distribution of random sums of independent

doi:10.12732/ijpam.v95i4.6
fatcat:j7hvzndhwfbgvaj2ul6girjrnq
*Poisson*random variables and an appropriate*Poisson*... This method was developed and applied in the setting of*Poisson**approximation*by Chen [2] . ... In this paper, we are interested to*approximate*the distribution of S N by a*Poisson*distribution with mean λ = E(µ N ), in terms of the total variation distance between the distributions of S N and P ...
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