75,788 research outputs found
Strictly positive definite kernels on a product of circles
We supply a Fourier characterization for the real, continuous, isotropic and
strictly positive definite kernels on a product of circles
Modeling Temporally Evolving and Spatially Globally Dependent Data
The last decades have seen an unprecedented increase in the availability of
data sets that are inherently global and temporally evolving, from remotely
sensed networks to climate model ensembles. This paper provides a view of
statistical modeling techniques for space-time processes, where space is the
sphere representing our planet. In particular, we make a distintion between (a)
second order-based, and (b) practical approaches to model temporally evolving
global processes. The former are based on the specification of a class of
space-time covariance functions, with space being the two-dimensional sphere.
The latter are based on explicit description of the dynamics of the space-time
process, i.e., by specifying its evolution as a function of its past history
with added spatially dependent noise.
We especially focus on approach (a), where the literature has been sparse. We
provide new models of space-time covariance functions for random fields defined
on spheres cross time. Practical approaches, (b), are also discussed, with
special emphasis on models built directly on the sphere, without projecting the
spherical coordinate on the plane.
We present a case study focused on the analysis of air pollution from the
2015 wildfires in Equatorial Asia, an event which was classified as the year's
worst environmental disaster. The paper finishes with a list of the main
theoretical and applied research problems in the area, where we expect the
statistical community to engage over the next decade
Complex Analysis of Real Functions III: Extended Fourier Theory
In the context of the complex-analytic structure within the unit disk
centered at the origin of the complex plane, that was presented in a previous
paper, we show that the complete Fourier theory of integrable real functions is
contained within that structure, that is, within the structure of the space of
inner analytic functions on the open unit disk. We then extend the Fourier
theory beyond the realm of integrable real functions, to include for example
singular Schwartz distributions, and possibly other objects.Comment: 23 pgs. Small formatting corrections and bibliography updat
On bifurcation of eigenvalues along convex symplectic paths
We consider a continuously differentiable curve in the
space of real symplectic matrices, which is the solution of the
following ODE:
,
where and is a continuous in the space of real matrices which are
symmetric. Under certain convexity assumption (which includes the particular
case that is strictly positive definite for all ), we
investigate the dynamics of the eigenvalues of when varies,
which are closely related to the stability of such Hamiltonian dynamical
systems. We rigorously prove the qualitative behavior of the branching of
eigenvalues and explicitly give the first order asymptotics of the eigenvalues.
This generalizes classical Krein-Lyubarskii theorem on the analytic bifurcation
of the Floquet multipliers under a linear perturbation of the Hamiltonian. As a
corollary, we give a rigorous proof of the following statement of Ekeland:
is a discrete set.Comment: 8 figure
Fourier Theory on the Complex Plane IV: Representability of Real Functions by their Fourier Coefficients
The results presented in this paper are refinements of some results presented
in a previous paper. Three such refined results are presented. The first one
relaxes one of the basic hypotheses assumed in the previous paper, and thus
extends the results obtained there to a wider class of real functions. The
other two relate to a closer examination of the issue of the representability
of real functions by their Fourier coefficients. As was shown in the previous
paper, in many cases one can recover the real function from its Fourier
coefficients even if the corresponding Fourier series diverges almost
everywhere. In such cases we say that the real function is still representable
by its Fourier coefficients. Here we establish a very weak condition on the
Fourier coefficients that ensures the representability of the function by those
coefficients. In addition to this, we show that any real function that is
absolutely integrable can be recovered almost everywhere from, and hence is
representable by, its Fourier coefficients, regardless of whether or not its
Fourier series converges. Interestingly, this also provides proof for a
conjecture proposed in the previous paper.Comment: 13 pages, including 3 pages of appendices; there was some expansion
of the content in this version; a few improvements in the text and on some
equations were also made; improved the treatment of the concept of
integration in the tex
The Sine operator
We show that Sine, the bulk limit of the Gaussian -ensembles
is the spectrum of a self-adjoint random differential operator where is the
positive definite matrix representation of hyperbolic Brownian motion with
variance in logarithmic time. The result connects the
Montgomery-Dyson conjecture about the Sine process and the non-trivial
zeros of the Riemann zeta function, the Hilbert-P\'olya conjecture and de
Brange's attempt to prove the Riemann hypothesis. We identify the Brownian
carousel as the Sturm-Liouville phase function of this operator.
We provide similar operator representations for several other finite
dimensional random ensembles and their limits: finite unitary or orthogonal
ensembles, Hua-Pickrell ensembles and their limits, hard-edge
-ensembles, as well as the Schr\"odinger point process. In this more
general setting, hyperbolic Brownian motion is replaced by a random walk or
Brownian motion on the affine group.
Our approach provides a unified framework to study -ensembles that has
so far been missing in the literature. In particular, we connect It\^o's
classification of affine Brownian motions with the classification of limits of
random matrix ensembles.Comment: 51 pages, 2 figure
On monotonicity of zeros of paraorthogonal polynomials on the unit circle
The purpose of this note is to establish, in terms of the primary
coefficients in the framework of the tridiagonal theory developed by Delsarte
and Genin in the environment of nonnegative definite Toeplitz matrices,
necessary and sufficient conditions for the monotonicity with respect to a real
parameter of zeros of paraorthogonal polynomials on the unit circle. It is also
provided tractable sufficient conditions and an application example. These
polynomials can be regarded as the characteristic polynomials of any matrix
similar to an unitary upper Hessenberg matrix with positive subdiagonal
elements
Strictly positive definite kernels on a product of spheres
For the real, continuous, isotropic and positive definite kernels on a
product of spheres, one may consider not only its usual strict positive
definiteness but also strict positive definiteness restrict to the points of
the product that have distinct components. In this paper, we provide a
characterization for strict positive definiteness in these two cases, settling
all the cases but those in which one of the spheres is a circle.Comment: 17 page
Fourier Theory on the Complex Plane II: Weak Convergence, Classification and Factorization of Singularities
The convergence of DP Fourier series which are neither strongly convergent
nor strongly divergent is discussed in terms of the Taylor series of the
corresponding inner analytic functions. These are the cases in which the
maximum disk of convergence of the Taylor series of the inner analytic function
is the open unit disk. An essentially complete classification, in terms of the
singularity structure of the corresponding inner analytic functions, of the
modes of convergence of a large class of DP Fourier series, is established.
Given a weakly convergent Fourier series of a DP real function, it is shown how
to generate from it other expressions involving trigonometric series, that
converge to that same function, but with much better convergence
characteristics. This is done by a procedure of factoring out the singularities
of the corresponding inner analytic function, and works even for divergent
Fourier series. This can be interpreted as a resummation technique, which is
firmly anchored by the underlying analytic structure.Comment: 43 pages, including 20 pages of appendices with explicit calculations
and examples; updated cross-references; made a few improvements in the text
and in one equation; updated the reference
Strictly positive definite kernels on two-point compact homogeneous spaces
We present a necessary and sufficient condition for the strict positive
definiteness of a real, continuous, isotropic and positive definite kernel on a
two-point compact homogeneous space. The characterization adds to others
previously obtained by D. Chen at all (2003) in the case in which the space is
a sphere of dimension at least 2 and Menegatto at all (2006) in the case in
which the space is the unit circle. As an application, we use the
characterization to improve upon a recent result on the differentiability of
positive definite kernels on the spaces
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