1,511 research outputs found
A class of scaled Bessel sampling theorems
Sampling theorems for a class of scaled Bessel unitary transforms are presented. The derivations are based on the properties of the generalized Laguerre functions. This class of scaled Bessel unitary transforms includes the classical sine and cosine transforms, but also novel chirp sine and modified Hankel transforms. The results for the sine and cosine transform can also be utilized to yield a sampling theorem, different from Shannon's, for the Fourier transform
Sampling and interpolation in de Branges spaces with doubling phase
The de Branges spaces of entire functions generalise the classical
Paley-Wiener space of square summable bandlimited functions. Specifically, the
square norm is computed on the real line with respect to weights given by the
values of certain entire functions. For the Paley-Wiener space, this can be
chosen to be an exponential function where the phase increases linearly. As our
main result, we establish a natural geometric characterisation, in terms of
densities, for real sampling and interpolating sequences in the case when the
derivative of the phase function merely gives a doubling measure on the real
line. Moreover, a consequence of this doubling condition, is that the spaces we
consider are one component model spaces. A novelty of our work is the
application to de Branges spaces of techniques developed by Marco, Massaneda
and Ortega-Cerd\'a for Fock spaces satisfying a doubling condition analogue to
ours.Comment: 31 pages, 1 figur
Large sample asymptotics for the two-parameter Poisson--Dirichlet process
This paper explores large sample properties of the two-parameter
Poisson--Dirichlet Process in two contexts. In a Bayesian
context of estimating an unknown probability measure, viewing this process as a
natural extension of the Dirichlet process, we explore the consistency and weak
convergence of the the two-parameter Poisson--Dirichlet posterior process. We
also establish the weak convergence of properly centered two-parameter
Poisson--Dirichlet processes for large This latter result
complements large results for the Dirichlet process and
Poisson--Dirichlet sequences, and complements a recent result on large
deviation principles for the two-parameter Poisson--Dirichlet process. A
crucial component of our results is the use of distributional identities that
may be useful in other contexts.Comment: Published in at http://dx.doi.org/10.1214/074921708000000147 the IMS
Collections (http://www.imstat.org/publications/imscollections.htm) by the
Institute of Mathematical Statistics (http://www.imstat.org
Analysis of continuous strict local martingales via h-transforms
We study strict local martingales via h-transforms, a method which first
appeared in Delbaen-Schachermayer. We show that strict local martingales arise
whenever there is a consistent family of change of measures where the two
measures are not equivalent to one another. Several old and new strict local
martingales are identified. We treat examples of diffusions with various
boundary behavior, size-bias sampling of diffusion paths, and non-colliding
diffusions. A multidimensional generalization to conformal strict local
martingales is achieved through Kelvin transform. As curious examples of
non-standard behavior, we show by various examples that strict local
martingales do not behave uniformly when the function (x-K)^+ is applied to
them. Implications to the recent literature on financial bubbles are discussed.Comment: Significantly revised version. 28 page
Approximation Error Bounds via Rademacher's Complexity
Approximation properties of some connectionistic models, commonly used to construct approximation schemes for optimization problems with multivariable functions as admissible solutions, are investigated. Such models are made up of linear combinations of computational units
with adjustable parameters. The relationship between model complexity (number of computational units) and approximation error is investigated using tools from Statistical Learning Theory, such as Talagrand's
inequality, fat-shattering dimension, and Rademacher's complexity. For some families of multivariable functions, estimates of the approximation accuracy of models with certain computational units are derived in dependence of the Rademacher's complexities of the families. The
estimates improve previously-available ones, which were expressed in terms of V C dimension and derived by exploiting union-bound techniques. The results are applied to approximation schemes with certain radial-basis-functions as computational units, for which it is shown that
the estimates do not exhibit the curse of dimensionality with respect to the number of variables
The falling appart of the tagged fragment and the asymptotic disintegration of the Brownian height fragmentation
We present a further analysis of the fragmentation at heights of the
normalized Brownian excursion. Specifically we study a representation for the
mass of a tagged fragment in terms of a Doob transformation of the 1/2-stable
subordinator and use it to study its jumps; this accounts for a description of
how a typical fragment falls apart. These results carry over to the height
fragmentation of the stable tree. Additionally, the sizes of the fragments in
the Brownian fragmentation when it is about to reduce to dust are described in
a limit theorem.Comment: 23 pages, 4 figures, AMSLaTeX (PDFLaTeX), accepted in Annales de
l'Institut Henri Poincar\'e (B
The falling appart of the tagged fragment and the asymptotic disintegration of the Brownian height fragmentation
We present a further analysis of the fragmentation at heights of the
normalized Brownian excursion. Specifically we study a representation for the
mass of a tagged fragment in terms of a Doob transformation of the 1/2-stable
subordinator and use it to study its jumps; this accounts for a description of
how a typical fragment falls apart. These results carry over to the height
fragmentation of the stable tree. Additionally, the sizes of the fragments in
the Brownian fragmentation when it is about to reduce to dust are described in
a limit theorem.Comment: 23 pages, 4 figures, AMSLaTeX (PDFLaTeX), accepted in Annales de
l'Institut Henri Poincar\'e (B
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