451 research outputs found
A Renewal version of the Sanov theorem
Large deviations for the local time of a process are investigated,
where for and are i.i.d.\ random
variables on a Polish space, is the -th arrival time of a renewal
process depending on . No moment conditions are assumed on the arrival
times of the renewal process.Comment: 13 page
Large deviations for intersection local times in critical dimension
Let be a continuous time simple random walk on
(), and let be the time spent by on the site
up to time . We prove a large deviations principle for the -fold
self-intersection local time in the
critical case . When is integer, we obtain similar results
for the intersection local times of independent simple random walks.Comment: Published in at http://dx.doi.org/10.1214/09-AOP499 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Testing the order of a model
This paper deals with order identification for nested models in the i.i.d.
framework. We study the asymptotic efficiency of two generalized likelihood
ratio tests of the order. They are based on two estimators which are proved to
be strongly consistent. A version of Stein's lemma yields an optimal
underestimation error exponent. The lemma also implies that the overestimation
error exponent is necessarily trivial. Our tests admit nontrivial
underestimation error exponents. The optimal underestimation error exponent is
achieved in some situations. The overestimation error can decay exponentially
with respect to a positive power of the number of observations. These results
are proved under mild assumptions by relating the underestimation (resp.
overestimation) error to large (resp. moderate) deviations of the
log-likelihood process. In particular, it is not necessary that the classical
Cram\'{e}r condition be satisfied; namely, the -densities are not
required to admit every exponential moment. Three benchmark examples with
specific difficulties (location mixture of normal distributions, abrupt changes
and various regressions) are detailed so as to illustrate the generality of our
results.Comment: Published at http://dx.doi.org/10.1214/009053606000000344 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Nonconventional Large Deviations Theorems
We obtain large deviations theorems for nonconventional sums with underlying
process being a Markov process satisfying the Doeblin condition or a dynamical
system such as subshift of finite type or hyperbolic or expanding
transformation
A Gibbsian model for message routeing in highly dense multihop networks
We investigate a probabilistic model for routeing of messages in
relay-augmented multihop ad-hoc networks, where each transmitter sends one
message to the origin. Given the (random) transmitter locations, we weight the
family of random, uniformly distributed message trajectories by an exponential
probability weight, favouring trajectories with low interference (measured in
terms of signal-to-interference ratio) and trajectory families with little
congestion (measured in terms of the number of pairs of hops using the same
relay). Under the resulting Gibbs measure, the system targets the best
compromise between entropy, interference and congestion for a common welfare,
instead of an optimization of the individual trajectories.
In the limit of high spatial density of users, we describe the totality of
all the message trajectories in terms of empirical measures. Employing large
deviations arguments, we derive a characteristic variational formula for the
limiting free energy and analyse the minimizer(s) of the formula, which
describe the most likely shapes of the trajectory flow. The empirical measures
of the message trajectories well describe the interference, but not the
congestion; the latter requires introducing an additional empirical measure.
Our results remain valid under replacing the two penalization terms by more
general functionals of these two empirical measures.Comment: 40 page
Full metastable asymptotic of the Fisher information
We establish an expansion by Gamma-convergence of the Fisher information
relative to the reference measure exp(-beta V), where V is a generic multiwell
potential and beta goes to infinity. The expansion reveals a hierarchy of
multiple scales reflecting the metastable behavior of the underlying overdamped
Langevin dynamics: distinct scales emerge and become relevant depending on
whether one considers probability measures concentrated on local minima of V,
probability measures concentrated on critical points of V, or generic
probability measures on R^d. We thus fully describe the asymptotic behavior of
minima of the Fisher information over regular sets of probabilities. The
analysis mostly relies on spectral properties of diffusion operators and the
related semiclassical Witten Laplacian and covers also the case of a compact
smooth manifold as underlying space.Comment: 24 pages. Typos correcte
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