3,764 research outputs found
Detecting independence of random vectors: generalized distance covariance and Gaussian covariance
Distance covariance is a quantity to measure the dependence of two random
vectors. We show that the original concept introduced and developed by
Sz\'{e}kely, Rizzo and Bakirov can be embedded into a more general framework
based on symmetric L\'{e}vy measures and the corresponding real-valued
continuous negative definite functions. The L\'{e}vy measures replace the
weight functions used in the original definition of distance covariance. All
essential properties of distance covariance are preserved in this new
framework. From a practical point of view this allows less restrictive moment
conditions on the underlying random variables and one can use other distance
functions than Euclidean distance, e.g. Minkowski distance. Most importantly,
it serves as the basic building block for distance multivariance, a quantity to
measure and estimate dependence of multiple random vectors, which is introduced
in a follow-up paper [Distance Multivariance: New dependence measures for
random vectors (submitted). Revised version of arXiv: 1711.07775v1] to the
present article.Comment: Published at https://doi.org/10.15559/18-VMSTA116 in the Modern
Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA)
by VTeX (http://www.vtex.lt/
Extreme value laws for fractal intensity functions in dynamical systems: Minkowski analysis
Typically, in the dynamical theory of extremal events, the function that
gauges the intensity of a phenomenon is assumed to be convex and maximal, or
singular, at a single, or at most a finite collection of points in
phase--space. In this paper we generalize this situation to fractal landscapes,
i.e. intensity functions characterized by an uncountable set of singularities,
located on a Cantor set. This reveals the dynamical r\^ole of classical
quantities like the Minkowski dimension and content, whose definition we extend
to account for singular continuous invariant measures. We also introduce the
concept of extremely rare event, quantified by non--standard Minkowski
constants and we study its consequences to extreme value statistics. Limit laws
are derived from formal calculations and are verified by numerical experiments.Comment: 20 pages, 13 figure
Spacelike distance from discrete causal order
Any discrete approach to quantum gravity must provide some prescription as to
how to deduce continuum properties from the discrete substructure. In the
causal set approach it is straightforward to deduce timelike distances, but
surprisingly difficult to extract spacelike distances, because of the unique
combination of discreteness with local Lorentz invariance in that approach. We
propose a number of methods to overcome this difficulty, one of which
reproduces the spatial distance between two points in a finite region of
Minkowski space. We provide numerical evidence that this definition can be used
to define a `spatial nearest neighbor' relation on a causal set, and conjecture
that this can be exploited to define the length of `continuous curves' in
causal sets which are approximated by curved spacetime. This provides evidence
in support of the ``Hauptvermutung'' of causal sets.Comment: 32 pages, 16 figures, revtex4; journal versio
An effective theory of initial conditions in inflation
We examine the renormalization of an effective theory description of a
general initial state set in an isotropically expanding space-time, which is
done to understand how to include the effects of new physics in the calculation
of the cosmic microwave background power spectrum. The divergences that arise
in a perturbative treatment of the theory are of two forms: those associated
with the properties of a field propagating through the bulk of space-time,
which are unaffected by the choice of the initial state, and those that result
from summing over the short-distance structure of the initial state. We show
that the former have the same renormalization and produce the same subsequent
scale dependence as for the standard vacuum state, while the latter correspond
to divergences that are localized at precisely the initial time hypersurface on
which the state is defined. This class of divergences is therefore renormalized
by adding initial-boundary counterterms, which render all of the perturbative
corrections small and finite. Initial states that approach the standard vacuum
at short distances require, at worst, relevant or marginal boundary
counterterms. States that differ from the vacuum at distances below that at
which any new, potentially trans-Planckian, physics becomes important are
renormalized with irrelevant boundary counterterms.Comment: 21 pages, 3 eps figures, uses RevTe
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