12,318 research outputs found
Sample path properties of the stochastic flows
We consider a stochastic flow driven by a finite-dimensional Brownian motion. We show that almost every realization of such a flow exhibits strong statistical properties such as the exponential convergence of an initial measure to the equilibrium state and the central limit theorem. The proof uses new estimates of the mixing rates of the multi-point motion
Concentration of Measure Inequalities in Information Theory, Communications and Coding (Second Edition)
During the last two decades, concentration inequalities have been the subject
of exciting developments in various areas, including convex geometry,
functional analysis, statistical physics, high-dimensional statistics, pure and
applied probability theory, information theory, theoretical computer science,
and learning theory. This monograph focuses on some of the key modern
mathematical tools that are used for the derivation of concentration
inequalities, on their links to information theory, and on their various
applications to communications and coding. In addition to being a survey, this
monograph also includes various new recent results derived by the authors. The
first part of the monograph introduces classical concentration inequalities for
martingales, as well as some recent refinements and extensions. The power and
versatility of the martingale approach is exemplified in the context of codes
defined on graphs and iterative decoding algorithms, as well as codes for
wireless communication. The second part of the monograph introduces the entropy
method, an information-theoretic technique for deriving concentration
inequalities. The basic ingredients of the entropy method are discussed first
in the context of logarithmic Sobolev inequalities, which underlie the
so-called functional approach to concentration of measure, and then from a
complementary information-theoretic viewpoint based on transportation-cost
inequalities and probability in metric spaces. Some representative results on
concentration for dependent random variables are briefly summarized, with
emphasis on their connections to the entropy method. Finally, we discuss
several applications of the entropy method to problems in communications and
coding, including strong converses, empirical distributions of good channel
codes, and an information-theoretic converse for concentration of measure.Comment: Foundations and Trends in Communications and Information Theory, vol.
10, no 1-2, pp. 1-248, 2013. Second edition was published in October 2014.
ISBN to printed book: 978-1-60198-906-
Local limit theorem in deterministic systems
We show that for every ergodic and aperiodic probability preserving system,
there exists a valued, square integrable function such that
the partial sums process of the time series satisfies the lattice local limit theorem.Comment: 17 page
Gaussian limit for determinantal random point fields
We prove that under fairly general conditions properly rescaled determinantal
random point field converges to a generalized Gaussian random process.Comment: This is the revised version accepted for publication in the Annals of
Probability. The results of Theorems 1 and 2 are extended, minor misprints
are correcte
Towards zero variance estimators for rare event probabilities
Improving Importance Sampling estimators for rare event probabilities
requires sharp approximations of conditional densities. This is achieved for
events E_{n}:=(f(X_{1})+...+f(X_{n}))\inA_{n} where the summands are i.i.d. and
E_{n} is a large or moderate deviation event. The approximation of the
conditional density of the real r.v's X_{i} 's, for 1\leqi\leqk_{n} with repect
to E_{n} on long runs, when k_{n}/n\to1, is handled. The maximal value of k
compatible with a given accuracy is discussed; algorithms and simulated results
are presented
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