2,140 research outputs found
Spherical completeness of the non-archimedean ring of Colombeau generalized numbers
We show spherical completeness of the ring of Colombeau generalized real (or
complex) numbers endowed with the sharp norm. As an application, we establish a
Hahn Banach extension theorem for ultra pseudo normed modules (over the ring of
generalized numbers) of generalized functions in the sense of Colombeau.Comment: 11 pages, parts rewritten, notation improve
How singular are moment generating functions?
This short note concerns the possible singular behaviour of moment generating
functions of finite measures at the boundary of their domain of existence. We
look closer at Example 7.3 in O. Barndorff-Nielsen's book "Information and
Exponential Families in Statistical Theory (1978)" and elaborate on the type of
exhibited singularity. Finally, another regularity problem is discussed and it
is solved through tensorizing two Barndorff- Nielsen's distributions
Rational Decompositions of p-adic meromorphic functions
Let K be a non archimedean algebraically closed field of characteristic pi
complete for its ultrametric absolute value. In a recent paper by Escassut and
Yang, polynomial decompositions P(f)=Q(g) for meromorphic functions f, g on K
(resp. in a disk) have been considered, and for a class of polynomials P, Q,
estimates for the Nevanlinna function T(r,f) have been derived. In the present
paper we consider as a generalization rational decompositions of meromorphic
functions. In the case, where f, g are analytic functions, the Second
Nevanlinna Theorem yields an analogue result as in the mentioned paper.
However, if they are meromorphic, non trivial estimates for T(r,f) are more
sophisticated.Comment: 13 page
On Lorentz geometry in algebras of generalized functions
We introduce a concept of causality in the framework of generalized
pseudo-Riemannian Geometry in the sense of J.F. Colombeau and establish the
inverse Cauchy-Schwarz inequality in this context. As an application, we prove
a dominant energy condition for some energy tensors as put forward in Hawking
and Ellis's book "The large scale structure of space-time". Our work is based
on a new characterization of free elements in finite dimensional modules over
the ring of generalized numbers.Comment: 26 pages, reorganized, updated reference
On the existence of non-central Wishart distributions
This paper deals with the existence issue of non-central Wishart
distributions which is a research topic initiated by Wishart (1928), and with
important contributions by e.g., L\'evy (1937), Gindikin (1975), Shanbhag
(1988), Peddada and Richards (1991). We present a new method involving the
theory of affine Markov processes, which reveals joint necessary conditions on
shape and non-centrality parameter. While Eaton's conjecture concerning the
necessary range of the shape parameter is confirmed, we also observe that it is
not sufficient anymore that it only belongs to the Gindikin ensemble, as is in
the central case.Comment: This version contains an Appendix which explains the relation of my
definition of non-central Wishart distributions to alternative ones from the
standard literatur
On the parameter domain of Wishart distributions and their infinite divisibility
A complete characterization of Wishart distributions on the cones of positive
semi-definite matrices is provided in terms of a description of their maximal
parameter domain. This result is new in that also degenerate scale parameters
are included. For such cases, the standard constraints on the range of the
shape parameter may be relaxed. Furthermore, the infinitely divisible Wishart
distributions are revealed as suitable transformations and embeddings of one
dimensional gamma distributions. This note completes the findings of L\'evy
(1937) concerning infinite divisibility and Gindikin (1975) regarding the
existence issue
Reforming the Wishart characteristic function
The literature presents the characteristic function of the Wishart
distribution on m times m matrices as an inverse power of the determinant of
the Fourier variable, the exponent being the positive, real shape parameter.
I demonstrate that only for two times two matrices, this expression is
unambiguous -- in this case the complex range of the determinant excludes the
negative real line. When m greater or equals 3 the range of the determinant
contains closed lines around the origin, hence a single branch of the complex
logarithm does not suffice to define the determinant's power. To resolve this
issue, I give the correct analytic extension of the Laplace transform, by
exploiting the Fourier-Laplace transform of a Wishart process
Affine processes on positive semidefinite d x d matrices have jumps of finite variation in dimension d > 1
The theory of affine processes on the space of positive semidefinite d x d
matrices has been established in a joint work with Cuchiero, Filipovi\'c and
Teichmann (2011). We confirm the conjecture stated therein that in dimension d
greater than 1 this process class does not exhibit jumps of infinite total
variation. This constitutes a geometric phenomenon which is in contrast to the
situation on the positive real line (Kawazu and Watanabe, 1974). As an
application we prove that the exponentially affine property of the Laplace
transform carries over to the Fourier-Laplace transform if the diffusion
coefficient is zero or invertible.Comment: version to appear in Stochastic Processes and Their Application
Affine Processes
We put forward a complete theory on moment explosion for fairly general
state-spaces. This includes a characterization of the validity of the affine
transform formula in terms of minimal solutions of a system of generalized
Riccati differential equations.
Also, we characterize the class of positive semidefinite processes, and
provide existence of weak and strong solutions for Wishart SDEs. As an
application, we answer a conjecture of M.L. Eaton on the maximal parameter
domain of non-central Wishart distributions.
The last chapter of this thesis comprises three individual works on affine
models, such as a characterization of the martingale property of exponentially
affine processes, an investigation of the jump-behaviour of processes on
positive semidefinite cones, and an existence result for transition densities
of multivariate affine jump-diffusions and their approximation theory in
weighted Hilbert spaces
On the characterization of p-adic Colombeau-Egorov generalized functions by their point values
We show that contrary to recent papers by S. Albeverio, A. Yu. Khrennikov and
V. Shelkovich, point values do not determine elements of the so-called p-adic
Colombeau-Egorov algebra uniquely. We further show in a more general way that
for an Egorov algebra of generalized functions on a locally compact ultrametric
space (M,d) taking values in a non-trivial ring, a point value characterization
holds if and only if (M,d) is discrete. Finally, following an idea due to M.
Kunzinger and M. Oberguggenberger, a generalized point value characterization
of such an Egorov algebra is given. Elements of the latter are constructed
which differ from the p-adic delta-distribution considered as a generalized
function, yet coincide on point values with the latter.Comment: 5 pages, counterexampl
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