426,961 research outputs found
Moderate deviations via cumulants
The purpose of the present paper is to establish moderate deviation
principles for a rather general class of random variables fulfilling certain
bounds of the cumulants. We apply a celebrated lemma of the theory of large
deviations probabilities due to Rudzkis, Saulis and Statulevicius. The examples
of random objects we treat include dependency graphs, subgraph-counting
statistics in Erd\H{o}s-R\'enyi random graphs and -statistics. Moreover, we
prove moderate deviation principles for certain statistics appearing in random
matrix theory, namely characteristic polynomials of random unitary matrices as
well as the number of particles in a growing box of random determinantal point
processes like the number of eigenvalues in the GUE or the number of points in
Airy, Bessel, and random point fields.Comment: 24 page
Moderate deviations for particle filtering
Consider the state space model (X_t,Y_t), where (X_t) is a Markov chain, and
(Y_t) are the observations. In order to solve the so-called filtering problem,
one has to compute L(X_t|Y_1,...,Y_t), the law of X_t given the observations
(Y_1,...,Y_t). The particle filtering method gives an approximation of the law
L(X_t|Y_1,...,Y_t) by an empirical measure \frac{1}{n}\sum_1^n\delta_{x_{i,t}}.
In this paper we establish the moderate deviation principle for the empirical
mean \frac{1}{n}\sum_1^n\psi(x_{i,t}) (centered and properly rescaled) when the
number of particles grows to infinity, enhancing the central limit theorem.
Several extensions and examples are also studied.Comment: Published at http://dx.doi.org/10.1214/105051604000000657 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Delta method in large deviations and moderate deviations for estimators
The delta method is a popular and elementary tool for deriving limiting
distributions of transformed statistics, while applications of asymptotic
distributions do not allow one to obtain desirable accuracy of approximation
for tail probabilities. The large and moderate deviation theory can achieve
this goal. Motivated by the delta method in weak convergence, a general delta
method in large deviations is proposed. The new method can be widely applied to
driving the moderate deviations of estimators and is illustrated by examples
including the Wilcoxon statistic, the Kaplan--Meier estimator, the empirical
quantile processes and the empirical copula function. We also improve the
existing moderate deviations results for -estimators and -statistics by
the new method. Some applications of moderate deviations to statistical
hypothesis testing are provided.Comment: Published in at http://dx.doi.org/10.1214/10-AOS865 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Moderate deviations for recursive stochastic algorithms
We prove a moderate deviation principle for the continuous time interpolation
of discrete time recursive stochastic processes. The methods of proof are
somewhat different from the corresponding large deviation result, and in
particular the proof of the upper bound is more complicated. The results can be
applied to the design of accelerated Monte Carlo algorithms for certain
problems, where schemes based on moderate deviations are easier to construct
and in certain situations provide performance comparable to those based on
large deviations.Comment: Submitted to Stochastic System
Option Pricing in the Moderate Deviations Regime
We consider call option prices in diffusion models close to expiry, in an
asymptotic regime ("moderately out of the money") that interpolates between the
well-studied cases of at-the-money options and out-of-the-money fixed-strike
options. First and higher order small-time moderate deviation estimates of call
prices and implied volatility are obtained. The expansions involve only simple
expressions of the model parameters, and we show in detail how to calculate
them for generic local and stochastic volatility models. Some numerical
examples for the Heston model illustrate the accuracy of our results
Moderate deviations and extinction of an epidemic
Consider an epidemic model with a constant flux of susceptibles, in a
situation where the corresponding deterministic epidemic model has a unique
stable endemic equilibrium. For the associated stochastic model, whose law of
large numbers limit is the deterministic model, the disease free equilibrium is
an absorbing state, which is reached soon or later by the process. However, for
a large population size, i.e. when the stochastic model is close to its
deterministic limit, the time needed for the stochastic perturbations to stop
the epidemic may be enormous. In this paper, we discuss how the Central Limit
Theorem, Moderate and Large Deviations allow us to give estimates of the
extinction time of the epidemic, depending upon the size of the population
Moderate deviations for diffusions with Brownian potentials
We present precise moderate deviation probabilities, in both quenched and
annealed settings, for a recurrent diffusion process with a Brownian potential.
Our method relies on fine tools in stochastic calculus, including Kotani's
lemma and Lamperti's representation for exponential functionals. In particular,
our result for quenched moderate deviations is in agreement with a recent
theorem of Comets and Popov [Probab. Theory Related Fields 126 (2003) 571-609]
who studied the corresponding problem for Sinai's random walk in random
environment.Comment: Published at http://dx.doi.org/10.1214/009117904000000829 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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