2,299 research outputs found

### Stein meets Malliavin in normal approximation

Stein's method is a method of probability approximation which hinges on the
solution of a functional equation. For normal approximation the functional
equation is a first order differential equation. Malliavin calculus is an
infinite-dimensional differential calculus whose operators act on functionals
of general Gaussian processes. Nourdin and Peccati (2009) established a
fundamental connection between Stein's method for normal approximation and
Malliavin calculus through integration by parts. This connection is exploited
to obtain error bounds in total variation in central limit theorems for
functionals of general Gaussian processes. Of particular interest is the fourth
moment theorem which provides error bounds of the order
$\sqrt{\mathbb{E}(F_n^4)-3}$ in the central limit theorem for elements
$\{F_n\}_{n\ge 1}$ of Wiener chaos of any fixed order such that
$\mathbb{E}(F_n^2) = 1$. This paper is an exposition of the work of Nourdin and
Peccati with a brief introduction to Stein's method and Malliavin calculus. It
is based on a lecture delivered at the Annual Meeting of the Vietnam Institute
for Advanced Study in Mathematics in July 2014.Comment: arXiv admin note: text overlap with arXiv:1404.478

### Poisson process approximation: From Palm theory to Stein's method

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 immigration-death process and Palm theory. The latter approach also
enables us to define local dependence of point processes [Chen and Xia (2004)]
and use it to study Poisson process approximation for locally dependent point
processes and for dependent superposition of point processes.Comment: Published at http://dx.doi.org/10.1214/074921706000001076 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Stein's method, Malliavin calculus, Dirichlet forms and the fourth moment theorem

The fourth moment theorem provides error bounds of the order $\sqrt{{\mathbb
E}(F^4) - 3}$ in the central limit theorem for elements $F$ of Wiener chaos of
any order such that ${\mathbb E}(F^2) = 1$. It was proved by Nourdin and
Peccati (2009) using Stein's method and the Malliavin calculus. It was also
proved by Azmoodeh, Campese and Poly (2014) using Stein's method and Dirichlet
forms. This paper is an exposition on the connections between Stein's method
and the Malliavin calculus and between Stein's method and Dirichlet forms, and
on how these connections are exploited in proving the fourth moment theorem

### Stein's method, Palm theory and Poisson process approximation

The framework of Stein's method for Poisson process approximation is
presented from the point of view of Palm theory, which is used to construct
Stein identities and define local dependence. A general result (Theorem
\refimportantproposition) in Poisson process approximation is proved by taking
the local approach.
It is obtained without reference to any particular metric, thereby allowing
wider applicability. A Wasserstein pseudometric is introduced for measuring the
accuracy of point process approximation. The pseudometric provides a
generalization of many metrics used so far, including the total variation
distance for random variables and the Wasserstein metric for processes as in
Barbour and Brown [Stochastic Process. Appl. 43 (1992) 9-31]. Also, through the
pseudometric, approximation for certain point processes on a given carrier
space is carried out by lifting it to one on a larger space, extending an idea
of Arratia, Goldstein and Gordon [Statist. Sci. 5 (1990)
403-434]. The error bound in the general result is similar in form to that
for Poisson approximation. As it yields the Stein factor 1/\lambda as in
Poisson approximation, it provides good approximation, particularly in cases
where \lambda is large. The general result is applied to a number of problems
including Poisson process modeling of rare words in a DNA sequence.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Probability
(http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000002

### Normal approximation for nonlinear statistics using a concentration inequality approach

Let $T$ be a general sampling statistic that can be written as a linear
statistic plus an error term. Uniform and non-uniform Berry--Esseen type bounds
for $T$ are obtained. The bounds are the best possible for many known
statistics. Applications to U-statistics, multisample U-statistics,
L-statistics, random sums and functions of nonlinear statistics are discussed.Comment: Published at http://dx.doi.org/10.3150/07-BEJ5164 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

### Stein's method via induction

Applying an inductive technique for Stein and zero bias couplings yields
Berry-Esseen theorems for normal approximation for two new examples. The
conditions of the main results do not require that the couplings be bounded.
Our two applications, one to the Erd\H{o}s-R\'enyi, random graph with a fixed
number of edges, and one to Jack measure on tableaux, demonstrate that the
method can handle non-bounded variables with non-trivial global dependence, and
can produce bounds in the Kolmogorov metric with the optimal rate.Comment: 59 page

### 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

### Normal approximation under local dependence

We establish both uniform and nonuniform error bounds of the Berry-Esseen
type in normal approximation under local dependence. These results are of an
order close to the best possible if not best possible. They are more general or
sharper than many existing ones in the literature. The proofs couple Stein's
method with the concentration inequality approach.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Probability
(http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000045

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