14,146 research outputs found
Invariance properties of random vectors and stochastic processes based on the zonoid concept
Two integrable random vectors and in are said
to be zonoid equivalent if, for each , the scalar products
and have the same first absolute
moments. The paper analyses stochastic processes whose finite-dimensional
distributions are zonoid equivalent with respect to time shift (zonoid
stationarity) and permutation of its components (swap invariance). While the
first concept is weaker than the stationarity, the second one is a weakening of
the exchangeability property. It is shown that nonetheless the ergodic theorem
holds for swap-invariant sequences and the limits are characterised.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ519 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
No-signaling, perfect bipartite dichotomic correlations and local randomness
The no-signaling constraint on bi-partite correlations is reviewed. It is
shown that in order to obtain non-trivial Bell-type inequalities that discern
no-signaling correlations from more general ones, one must go beyond
considering expectation values of products of observables only. A new set of
nontrivial no-signaling inequalities is derived which have a remarkably close
resemblance to the CHSH inequality, yet are fundamentally different. A set of
inequalities by Roy and Singh and Avis et al., which is claimed to be useful
for discerning no-signaling correlations, is shown to be trivially satisfied by
any correlation whatsoever. Finally, using the set of newly derived
no-signaling inequalities a result with potential cryptographic consequences is
proven: if different parties use identical devices, then, once they have
perfect correlations at spacelike separation between dichotomic observables,
they know that because of no-signaling the local marginals cannot but be
completely random.Comment: Published in 'Proceedings of the International Conference Advances in
Quantum Theory', AIP Conference Proceedings, vol. 1327, 2011. pp. 36-5
Bayesian inference for bivariate ranks
A recommender system based on ranks is proposed, where an expert's ranking of
a set of objects and a user's ranking of a subset of those objects are combined
to make a prediction of the user's ranking of all objects. The rankings are
assumed to be induced by latent continuous variables corresponding to the
grades assigned by the expert and the user to the objects. The dependence
between the expert and user grades is modelled by a copula in some parametric
family. Given a prior distribution on the copula parameter, the user's complete
ranking is predicted by the mode of the posterior predictive distribution of
the user's complete ranking conditional on the expert's complete and the user's
incomplete rankings. Various Markov chain Monte-Carlo algorithms are proposed
to approximate the predictive distribution or only its mode. The predictive
distribution can be obtained exactly for the Farlie-Gumbel-Morgenstern copula
family, providing a benchmark for the approximation accuracy of the algorithms.
The method is applied to the MovieLens 100k dataset with a Gaussian copula
modelling dependence between the expert's and user's grades.Comment: 21 page
Random Metric Spaces and Universality
WWe define the notion of a random metric space and prove that with
probability one such a space is isometricto the Urysohn universal metric space.
The main technique is the study of universal and random distance matrices; we
relate the properties of metric (in particulary universal) space to the
properties of distance matrices. We show the link between those questions and
classification of the Polish spaces with measure (Gromov or metric triples) and
with the problem about S_{\infty}-invariant measures in the space of symmetric
matrices. One of the new effects -exsitence in Urysohn space so called
anarchical uniformly distributed sequences. We give examples of other
categories in which the randomness and universality coincide (graph, etc.).Comment: 38 PAGE
Second-Order Weight Distributions
A fundamental property of codes, the second-order weight distribution, is
proposed to solve the problems such as computing second moments of weight
distributions of linear code ensembles. A series of results, parallel to those
for weight distributions, is established for second-order weight distributions.
In particular, an analogue of MacWilliams identities is proved. The
second-order weight distributions of regular LDPC code ensembles are then
computed. As easy consequences, the second moments of weight distributions of
regular LDPC code ensembles are obtained. Furthermore, the application of
second-order weight distributions in random coding approach is discussed. The
second-order weight distributions of the ensembles generated by a so-called
2-good random generator or parity-check matrix are computed, where a 2-good
random matrix is a kind of generalization of the uniformly distributed random
matrix over a finite filed and is very useful for solving problems that involve
pairwise or triple-wise properties of sequences. It is shown that the 2-good
property is reflected in the second-order weight distribution, which thus plays
a fundamental role in some well-known problems in coding theory and
combinatorics. An example of linear intersecting codes is finally provided to
illustrate this fact.Comment: 10 pages, accepted for publication in IEEE Transactions on
Information Theory, May 201
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