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### Poisson Approximation

Rudolf Grubel, A. D. Barbour, Lars Holst, Svante Janson
1993 Journal of the American Statistical Association
We give several examples for 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).  ...

### Poisson approximation

S. Y. Novak
2019 Probability Surveys
The accuracy of shifted 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 ).  ...

### Poisson approximation [article]

S.Y.Novak
2019 arXiv   pre-print
We overview results on the topic of 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  ...

### Poisson–Voronoi approximation

Matthias Heveling, Matthias Reitzner
2009 The Annals of Applied Probability
Precise estimates for the variance of both quantities are obtained which follow from a new jackknife inequality for the variance of functionals of a 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!  ...

S. Y. Novak
2021 Probability Surveys
An important feature of a 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).  ...

### Topics in poisson approximation [chapter]

A.D. Barbour
2001 Handbook of Statistics

### Conditional Poisson process approximation [article]

H. L. Gan
2015 arXiv   pre-print
The basic starting point for point processes is usually the 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.  ...

### Compound Poisson process approximation

A. D. Barbour, Marianne Månsson
2002 The Annals of Probability
Compound 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.  ...

### The Poisson Approximation to the Poisson Binomial Distribution

J. L. Hodges, Lucien Le Cam
1960 The Annals of Mathematical Statistics
THE 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.  ...

### Poisson and Compound Poisson Approximations in a Nonconventional Setup [article]

Yuri Kifer, Ariel Rapaport
2012 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 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  ...

### Exchangeable pairs and Poisson approximation

Sourav Chatterjee, Persi Diaconis, Elizabeth Meckes
2005 Probability Surveys
This is a survey paper on 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.  ...

### Poisson approximations for telecommunications networks

T. C. Brown, P. K. Pollett
1994 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 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.  ...

### Poisson approximations for telecommunications networks

T. C. Brown, P. K. Pollett
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 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.  ...

### A Compound Poisson Approximation Inequality

Erol A. Peköz
2006 Journal of Applied Probability
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 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.  ...

### POISSON APPROXIMATION FOR RANDOM SUMS OF POISSON RANDOM VARIABLES

K. Teerapabolarn
2014 International Journal of Pure and Applied Mathematics
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 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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