321 research outputs found
Nonclassical stochastic flows and continuous products
Contrary to the classical wisdom, processes with independent values (defined
properly) are much more diverse than white noise combined with Poisson point
processes, and product systems are much more diverse than Fock spaces.
This text is a survey of recent progress in constructing and investigating
nonclassical stochastic flows and continuous products of probability spaces and
Hilbert spaces.Comment: A survey, 126 pages. Version 3 (final): former Question 9d4 is
solved; 8a1 reformulated. Ref [41] added. For readability, sections are
reordered (123456..->142536..). Cosmetic changes, mostly in 1b, 2a, 3d, (4a7)
(v3 numbers) and Introductio
A Simple Proof of the Fundamental Theorem about Arveson Systems
With every Eo-semigroup (acting on the algebra of of bounded operators on a
separable infinite-dimensional Hilbert space) there is an associated Arveson
system. One of the most important results about Arveson systems is that every
Arveson system is the one associated with an Eo-semigroup. In these notes we
give a new proof of this result that is considerably simpler than the existing
ones and allows for a generalization to product systems of Hilbert module (to
be published elsewhere).Comment: Publication data added, acknowledgements and a note after acceptance
added, corrects a number of inconveniences that have been produced in the
published version during the publication proces
On continuity of measurable group representations and homomorphisms
Let G be a locally compact group, and let U be its unitary representation on
a Hilbert space H. Endow the space L(H) of linear bounded operators on H with
weak operator topology. We prove that if U is a measurable map from G to L(H)
then it is continuous. This result was known before for separable H. To prove
this, we generalize a known theorem on nonmeasuralbe unions of point finite
families of null sets. We prove also that the following statement is consistent
with ZFC: every measurable homomorphism from a locally compact group into any
topological group is continuous. This relies, in turn, on the following
theorem: it is consistent with ZFC that for every null set S in a locally
compact group there is a set A such that AS is non-measurable.Comment: The previous version was not final, I update it once notice
Identifying Demand with Multidimensional Unobservables: A Random Functions Approach
We explore the identification of nonseparable models without relying on the property that the model can be inverted in the econometric unobservables. In particular, we allow for infinite dimensional unobservables. In the context of a demand system, this allows each product to have multiple unobservables. We identify the distribution of demand both unconditional and conditional on market observables, which allows us to identify several quantities of economic interest such as the (conditional and unconditional) distributions of elasticities and the distribution of price effects following a merger. Our approach is based on a significant generalization of the linear in random coefficients model that only restricts the random functions to be analytic in the endogenous variables, which is satisfied by several standard demand models used in practice. We assume an (unknown) countable support for the the distribution of the infinite dimensional unobservables.
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