26,552 research outputs found
Uniform test of algorithmic randomness over a general space
The algorithmic theory of randomness is well developed when the underlying
space is the set of finite or infinite sequences and the underlying probability
distribution is the uniform distribution or a computable distribution. These
restrictions seem artificial. Some progress has been made to extend the theory
to arbitrary Bernoulli distributions (by Martin-Loef), and to arbitrary
distributions (by Levin). We recall the main ideas and problems of Levin's
theory, and report further progress in the same framework.
- We allow non-compact spaces (like the space of continuous functions,
underlying the Brownian motion).
- The uniform test (deficiency of randomness) d_P(x) (depending both on the
outcome x and the measure P should be defined in a general and natural way.
- We see which of the old results survive: existence of universal tests,
conservation of randomness, expression of tests in terms of description
complexity, existence of a universal measure, expression of mutual information
as "deficiency of independence.
- The negative of the new randomness test is shown to be a generalization of
complexity in continuous spaces; we show that the addition theorem survives.
The paper's main contribution is introducing an appropriate framework for
studying these questions and related ones (like statistics for a general family
of distributions).Comment: 40 pages. Journal reference and a slight correction in the proof of
Theorem 7 adde
Computable de Finetti measures
We prove a computable version of de Finetti's theorem on exchangeable
sequences of real random variables. As a consequence, exchangeable stochastic
processes expressed in probabilistic functional programming languages can be
automatically rewritten as procedures that do not modify non-local state. Along
the way, we prove that a distribution on the unit interval is computable if and
only if its moments are uniformly computable.Comment: 32 pages. Final journal version; expanded somewhat, with minor
corrections. To appear in Annals of Pure and Applied Logic. Extended abstract
appeared in Proceedings of CiE '09, LNCS 5635, pp. 218-23
Computability of probability measures and Martin-Lof randomness over metric spaces
In this paper we investigate algorithmic randomness on more general spaces
than the Cantor space, namely computable metric spaces. To do this, we first
develop a unified framework allowing computations with probability measures. We
show that any computable metric space with a computable probability measure is
isomorphic to the Cantor space in a computable and measure-theoretic sense. We
show that any computable metric space admits a universal uniform randomness
test (without further assumption).Comment: 29 page
Beyond cash-additive risk measures: when changing the num\'{e}raire fails
We discuss risk measures representing the minimum amount of capital a
financial institution needs to raise and invest in a pre-specified eligible
asset to ensure it is adequately capitalized. Most of the literature has
focused on cash-additive risk measures, for which the eligible asset is a
risk-free bond, on the grounds that the general case can be reduced to the
cash-additive case by a change of numeraire. However, discounting does not work
in all financially relevant situations, typically when the eligible asset is a
defaultable bond. In this paper we fill this gap allowing for general eligible
assets. We provide a variety of finiteness and continuity results for the
corresponding risk measures and apply them to risk measures based on
Value-at-Risk and Tail Value-at-Risk on spaces, as well as to shortfall
risk measures on Orlicz spaces. We pay special attention to the property of
cash subadditivity, which has been recently proposed as an alternative to cash
additivity to deal with defaultable bonds. For important examples, we provide
characterizations of cash subadditivity and show that, when the eligible asset
is a defaultable bond, cash subadditivity is the exception rather than the
rule. Finally, we consider the situation where the eligible asset is not
liquidly traded and the pricing rule is no longer linear. We establish when the
resulting risk measures are quasiconvex and show that cash subadditivity is
only compatible with continuous pricing rules
Measuring risk with multiple eligible assets
The risk of financial positions is measured by the minimum amount of capital
to raise and invest in eligible portfolios of traded assets in order to meet a
prescribed acceptability constraint. We investigate nondegeneracy, finiteness
and continuity properties of these risk measures with respect to multiple
eligible assets. Our finiteness and continuity results highlight the interplay
between the acceptance set and the class of eligible portfolios. We present a
simple, alternative approach to the dual representation of convex risk measures
by directly applying to the acceptance set the external characterization of
closed, convex sets. We prove that risk measures are nondegenerate if and only
if the pricing functional admits a positive extension which is a supporting
functional for the underlying acceptance set, and provide a characterization of
when such extensions exist. Finally, we discuss applications to set-valued risk
measures, superhedging with shortfall risk, and optimal risk sharing
Persistent Homology and String Vacua
We use methods from topological data analysis to study the topological
features of certain distributions of string vacua. Topological data analysis is
a multi-scale approach used to analyze the topological features of a dataset by
identifying which homological characteristics persist over a long range of
scales. We apply these techniques in several contexts. We analyze N=2 vacua by
focusing on certain distributions of Calabi-Yau varieties and Landau-Ginzburg
models. We then turn to flux compactifications and discuss how we can use
topological data analysis to extract physical informations. Finally we apply
these techniques to certain phenomenologically realistic heterotic models. We
discuss the possibility of characterizing string vacua using the topological
properties of their distributions.Comment: 32 pages, 12 pdf figure
- …