3,425 research outputs found

### Exchangeable pairs and Poisson approximation

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.
While many details are new, the results are closely related to a body of work
developed by Andrew Barbour, Louis Chen, Richard Arratia, Lou Gordon, Larry
Goldstein, and their collaborators. Some comparison with these other approaches
is offered.Comment: Published at http://dx.doi.org/10.1214/154957805100000096 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Two sufficient conditions for Poisson approximations in the ferromagnetic Ising model

A $d$-dimensional ferromagnetic Ising model on a lattice torus is considered.
As the size of the lattice tends to infinity, two conditions ensuring a Poisson
approximation for the distribution of the number of occurrences in the lattice
of any given local configuration are suggested. The proof builds on the
Stein--Chen method. The rate of the Poisson approximation and the speed of
convergence to it are defined and make sense for the model. Thus, the two
sufficient conditions are traduced in terms of the magnetic field and the pair
potential. In particular, the Poisson approximation holds even if both
potentials diverge.Comment: Published in at http://dx.doi.org/10.1214/1214/07-AAP487 the Annals
of Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### The lower tail: Poisson approximation revisited

The well-known "Janson's inequality" gives Poisson-like upper bounds for the
lower tail probability \Pr(X \le (1-\eps)\E X) when X is the sum of dependent
indicator random variables of a special form. We show that, for large
deviations, this inequality is optimal whenever X is approximately Poisson,
i.e., when the dependencies are weak. We also present correlation-based
approaches that, in certain symmetric applications, yield related conclusions
when X is no longer close to Poisson. As an illustration we, e.g., consider
subgraph counts in random graphs, and obtain new lower tail estimates,
extending earlier work (for the special case \eps=1) of Janson, Luczak and
Rucinski.Comment: 21 page

### Relaxation of monotone coupling conditions: Poisson approximation and beyond

It is well-known that assumptions of monotonicity in size-bias couplings may
be used to prove simple, yet powerful, Poisson approximation results. Here we
show how these assumptions may be relaxed, establishing explicit Poisson
approximation bounds (depending on the first two moments only) for random
variables which satisfy an approximate version of these monotonicity
conditions. These are shown to be effective for models where an underlying
random variable of interest is contaminated with noise. We also give explicit
Poisson approximation bounds for sums of associated or negatively associated
random variables. Applications are given to epidemic models, extremes, and
random sampling. Finally, we also show how similar techniques may be used to
relax the assumptions needed in a Poincar\'e inequality and in a normal
approximation result.Comment: 19 page

### Poisson approximation

This is a survey article on the topic of Poisson approximation

### Stochastic Combinatorial Optimization via Poisson Approximation

We study several stochastic combinatorial problems, including the expected
utility maximization problem, the stochastic knapsack problem and the
stochastic bin packing problem. A common technical challenge in these problems
is to optimize some function of the sum of a set of random variables. The
difficulty is mainly due to the fact that the probability distribution of the
sum is the convolution of a set of distributions, which is not an easy
objective function to work with. To tackle this difficulty, we introduce the
Poisson approximation technique. The technique is based on the Poisson
approximation theorem discovered by Le Cam, which enables us to approximate the
distribution of the sum of a set of random variables using a compound Poisson
distribution.
We first study the expected utility maximization problem introduced recently
[Li and Despande, FOCS11]. For monotone and Lipschitz utility functions, we
obtain an additive PTAS if there is a multidimensional PTAS for the
multi-objective version of the problem, strictly generalizing the previous
result.
For the stochastic bin packing problem (introduced in [Kleinberg, Rabani and
Tardos, STOC97]), we show there is a polynomial time algorithm which uses at
most the optimal number of bins, if we relax the size of each bin and the
overflow probability by eps.
For stochastic knapsack, we show a 1+eps-approximation using eps extra
capacity, even when the size and reward of each item may be correlated and
cancelations of items are allowed. This generalizes the previous work [Balghat,
Goel and Khanna, SODA11] for the case without correlation and cancelation. Our
algorithm is also simpler. We also present a factor 2+eps approximation
algorithm for stochastic knapsack with cancelations. the current known
approximation factor of 8 [Gupta, Krishnaswamy, Molinaro and Ravi, FOCS11].Comment: 42 pages, 1 figure, Preliminary version appears in the Proceeding of
the 45th ACM Symposium on the Theory of Computing (STOC13

### Poisson approximation of the length spectrum of random surfaces

Multivariate Poisson approximation of the length spectrum of random surfaces
is studied by means of the Chen-Stein method. This approach delivers simple and
explicit error bounds in Poisson limit theorems. They are used to prove that
Poisson approximation applies to curves of length up to order $o(\log\log g)$
with $g$ being the genus of the surface.Comment: 22 pages, 2 figures. To appear in Indiana Univ. Math.

### Moderate deviations in Poisson approximation: a first attempt

Poisson approximation using Stein's method has been extensively studied in
the literature. The main focus has been on bounding the total variation
distance. This paper is a first attempt on moderate deviations in Poisson
approximation for right-tail probabilities of sums of dependent indicators. We
obtain results under certain general conditions for local dependence as well as
for size-bias coupling. These results are then applied to independent
indicators, 2-runs, and the matching problem.Comment: 21 page

### Translated Poisson approximation using exchangeable pair couplings

It is shown that the method of exchangeable pairs introduced by Stein
[Approximate Computation of Expectations (1986) IMS, Hayward, CA] for normal
approximation can effectively be used for translated Poisson approximation.
Introducing an additional smoothness condition, one can obtain approximation
results in total variation and also in a local limit metric. The result is
applied, in particular, to the anti-voter model on finite graphs as analyzed by
Rinott and Rotar [Ann. Appl. Probab. 7 (1997) 1080--1105], obtaining the same
rate of convergence, but now for a stronger metric.Comment: Published in at http://dx.doi.org/10.1214/105051607000000258 the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org

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