217 research outputs found
Strong approximation results for the empirical process of stationary sequences
We prove a strong approximation result for the empirical process associated
to a stationary sequence of real-valued random variables, under dependence
conditions involving only indicators of half lines. This strong approximation
result also holds for the empirical process associated to iterates of expanding
maps with a neutral fixed point at zero, as soon as the correlations decrease
more rapidly than for some positive . This shows that
our conditions are in some sense optimal.Comment: Published in at http://dx.doi.org/10.1214/12-AOP798 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Concentration inequalities for mean field particle models
This article is concerned with the fluctuations and the concentration
properties of a general class of discrete generation and mean field particle
interpretations of nonlinear measure valued processes. We combine an original
stochastic perturbation analysis with a concentration analysis for triangular
arrays of conditionally independent random sequences, which may be of
independent interest. Under some additional stability properties of the
limiting measure valued processes, uniform concentration properties, with
respect to the time parameter, are also derived. The concentration inequalities
presented here generalize the classical Hoeffding, Bernstein and Bennett
inequalities for independent random sequences to interacting particle systems,
yielding very new results for this class of models. We illustrate these results
in the context of McKean-Vlasov-type diffusion models, McKean collision-type
models of gases and of a class of Feynman-Kac distribution flows arising in
stochastic engineering sciences and in molecular chemistry.Comment: Published in at http://dx.doi.org/10.1214/10-AAP716 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A Bernstein type inequality and moderate deviations for weakly dependent sequences
In this paper we present a tail inequality for the maximum of partial sums of
a weakly dependent sequence of random variables that are not necessarily
bounded. The class considered includes geometrically and subgeometrically
strongly mixing sequences. The result is then used to derive asymptotic
moderate deviations results. Applications include classes of Markov chains,
functions of linear processes with absolutely regular innovations and ARCH
model
Upper bounds for superquantiles of martingales
Let be a discrete martingale in for in or . In this note, we give upper bounds on the superquantiles of and the quantiles and superquantiles of
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