59 research outputs found
Moment inequalities for U-statistics
We present moment inequalities for completely degenerate Banach space valued
(generalized) U-statistics of arbitrary order. The estimates involve suprema of
empirical processes which, in the real-valued case, can be replaced by simpler
norms of the kernel matrix (i.e., norms of some multilinear operators
associated with the kernel matrix). As a corollary, we derive tail inequalities
for U-statistics with bounded kernels and for some multiple stochastic
integrals.Comment: Published at http://dx.doi.org/10.1214/009117906000000476 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A note on the Hanson-Wright inequality for random vectors with dependencies
We prove that quadratic forms in isotropic random vectors in
, possessing the convex concentration property with constant ,
satisfy the Hanson-Wright inequality with constant , where is an
absolute constant, thus eliminating the logarithmic (in the dimension) factors
in a recent estimate by Vu and Wang. We also show that the concentration
inequality for all Lipschitz functions implies a uniform version of the
Hanson-Wright inequality for suprema of quadratic forms (in the spirit of the
inequalities by Borell, Arcones-Gin\'e and Ledoux-Talagrand). Previous results
of this type relied on stronger isoperimetric properties of and in some
cases provided an upper bound on the deviations rather than a concentration
inequality.
In the last part of the paper we show that the uniform version of the
Hanson-Wright inequality for Gaussian vectors can be used to recover a recent
concentration inequality for empirical estimators of the covariance operator of
-valued Gaussian variables due to Koltchinskii and Lounici
Modified log-Sobolev inequalities for convex functions on the real line. Sufficient conditions
We provide a mild sufficient condition for a probability measure on the real
line to satisfy a modified log-Sobolev inequality for convex functions,
interpolating between the classical log-Sobolev inequality and a Bobkov-Ledoux
type inequality. As a consequence we obtain dimension-free two-level
concentration results for convex function of independent random variables with
sufficiently regular tail decay. We also provide a link between modified
log-Sobolev inequalities for convex functions and weak transport-entropy
inequalities, complementing recent work by Gozlan, Roberto, Samson, and Tetali.Comment: 25 pages; changes: references and comments about recent results by
other Authors added, hypercontractive estimates in Section 3 added, a few
typos corrected; accepted for publication in Studia Mathematic
U-statistics of Ornstein-Uhlenbeck branching particle system
We consider a branching particle system consisting of particles moving
according to the Ornstein-Uhlenbeck process in \Rd and undergoing a binary,
supercritical branching with a constant rate . This system is known
to fulfil a law of large numbers (under exponential scaling). Recently the
question of the corresponding central limit theorem has been addressed. It
turns out that the normalization and form of the limit in the CLT fall into
three qualitatively different regimes, depending on the relation between the
branching intensity and the parameters of the Orstein-Uhlenbeck process. In the
present paper we extend those results to -statistics of the system proving a
law of large numbers and a central limit theorem.Comment: References update. arXiv admin note: substantial text overlap with
arXiv:1007.171
Orlicz integrability of additive functionals of Harris ergodic Markov chains
For a Harris ergodic Markov chain , on a general state space,
started from the so called small measure or from the stationary distribution we
provide optimal estimates for Orlicz norms of sums ,
where is the first regeneration time of the chain. The estimates are
expressed in terms of other Orlicz norms of the function (wrt the
stationary distribution) and the regeneration time (wrt the small
measure). We provide applications to tail estimates for additive functionals of
the chain generated by unbounded functions as well as to classical
limit theorems (CLT, LIL, Berry-Esseen)
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