107 research outputs found
Persistence probabilities in centered, stationary, Gaussian processes in discrete time
Lower bounds for persistence probabilities of stationary Gaussian processes
in discrete time are obtained under various conditions on the spectral measure
of the process. Examples are given to show that the persistence probability can
decay faster than exponentially. It is shown that if the spectral measure is
not singular, then the exponent in the persistence probability cannot grow
faster than quadratically. An example that appears (from numerical evidence) to
achieve this lower bound is presented.Comment: 9 pages; To appear in a special volume of the Indian Journal of Pure
and Applied Mathematic
Towards a Mathematical Theory of Super-Resolution
This paper develops a mathematical theory of super-resolution. Broadly
speaking, super-resolution is the problem of recovering the fine details of an
object---the high end of its spectrum---from coarse scale information
only---from samples at the low end of the spectrum. Suppose we have many point
sources at unknown locations in and with unknown complex-valued
amplitudes. We only observe Fourier samples of this object up until a frequency
cut-off . We show that one can super-resolve these point sources with
infinite precision---i.e. recover the exact locations and amplitudes---by
solving a simple convex optimization problem, which can essentially be
reformulated as a semidefinite program. This holds provided that the distance
between sources is at least . This result extends to higher dimensions
and other models. In one dimension for instance, it is possible to recover a
piecewise smooth function by resolving the discontinuity points with infinite
precision as well. We also show that the theory and methods are robust to
noise. In particular, in the discrete setting we develop some theoretical
results explaining how the accuracy of the super-resolved signal is expected to
degrade when both the noise level and the {\em super-resolution factor} vary.Comment: 48 pages, 12 figure
On the Supremum of Random Dirichlet Polynomials
We study the supremum of some random Dirichlet polynomials and obtain sharp
upper and lower bounds for supremum expectation that extend the optimal
estimate of Hal\'asz-Queff\'elec and enable to cunstruct random polynomials
with unusually small maxima.
Our approach in proving these results is entirely based on methods of
stochastic processes, in particular the metric entropy method
Dirichlet polynomials: some old and recent results, and their interplay in number theory
In the first part of this expository paper, we present and discuss the
interplay of Dirichlet polynomials in some classical problems of number theory,
notably the Lindel\"of Hypothesis. We review some typical properties of their
means and continue with some investigations concerning their supremum
properties. Their random counterpart is considered in the last part of the
paper, where a analysis of their supremum properties, based on methods of
stochastic processes, is developed.Comment: 29 page
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