2,484 research outputs found
A de Bruijn identity for discrete random variables
We discuss properties of the "beamsplitter addition" operation, which
provides a non-standard scaled convolution of random variables supported on the
non-negative integers. We give a simple expression for the action of
beamsplitter addition using generating functions. We use this to give a
self-contained and purely classical proof of a heat equation and de Bruijn
identity, satisfied when one of the variables is geometric.Comment: 9 pages, shorter version submitted to ISIT 201
Concave Renewal Functions Do Not Imply DFR Inter-Renewal Times
Brown (1980, 1981) proved that the renewal function is concave if the
inter-renewal distribution is DFR (decreasing failure rate), and conjectured
the converse. This note settles Brown's conjecture with a class of
counter-examples. We also give a short proof of Shanthikumar's (1988) result
that the DFR property is closed under geometric compounding.Comment: 8 pages, 1 figur
A de Bruijn identity for symmetric stable laws
We show how some attractive information--theoretic properties of Gaussians
pass over to more general families of stable densities. We define a new score
function for symmetric stable laws, and use it to give a stable version of the
heat equation. Using this, we derive a version of the de Bruijn identity,
allowing us to write the derivative of relative entropy as an inner product of
score functions. We discuss maximum entropy properties of symmetric stable
densities
Cellular Probabilistic Automata - A Novel Method for Uncertainty Propagation
We propose a novel density based numerical method for uncertainty propagation
under certain partial differential equation dynamics. The main idea is to
translate them into objects that we call cellular probabilistic automata and to
evolve the latter. The translation is achieved by state discretization as in
set oriented numerics and the use of the locality concept from cellular
automata theory. We develop the method at the example of initial value
uncertainties under deterministic dynamics and prove a consistency result. As
an application we discuss arsenate transportation and adsorption in drinking
water pipes and compare our results to Monte Carlo computations
On mutual information, likelihood-ratios and estimation error for the additive Gaussian channel
This paper considers the model of an arbitrary distributed signal x observed
through an added independent white Gaussian noise w, y=x+w. New relations
between the minimal mean square error of the non-causal estimator and the
likelihood ratio between y and \omega are derived. This is followed by an
extended version of a recently derived relation between the mutual information
I(x;y) and the minimal mean square error. These results are applied to derive
infinite dimensional versions of the Fisher information and the de Bruijn
identity. The derivation of the results is based on the Malliavin calculus.Comment: 21 pages, to appear in the IEEE Transactions on Information Theor
Perfect Necklaces
We introduce a variant of de Bruijn words that we call perfect necklaces. Fix
a finite alphabet. Recall that a word is a finite sequence of symbols in the
alphabet and a circular word, or necklace, is the equivalence class of a word
under rotations. For positive integers k and n, we call a necklace
(k,n)-perfect if each word of length k occurs exactly n times at positions
which are different modulo n for any convention on the starting point. We call
a necklace perfect if it is (k,k)-perfect for some k. We prove that every
arithmetic sequence with difference coprime with the alphabet size induces a
perfect necklace. In particular, the concatenation of all words of the same
length in lexicographic order yields a perfect necklace. For each k and n, we
give a closed formula for the number of (k,n)-perfect necklaces. Finally, we
prove that every infinite periodic sequence whose period coincides with some
(k,n)-perfect necklace for any n, passes all statistical tests of size up to k,
but not all larger tests. This last theorem motivated this work
Flip dynamics in octagonal rhombus tiling sets
We investigate the properties of classical single flip dynamics in sets of
two-dimensional random rhombus tilings. Single flips are local moves involving
3 tiles which sample the tiling sets {\em via} Monte Carlo Markov chains. We
determine the ergodic times of these dynamical systems (at infinite
temperature): they grow with the system size like ;
these dynamics are rapidly mixing. We use an inherent symmetry of tiling sets
and a powerful tool from probability theory, the coupling technique. We also
point out the interesting occurrence of Gumbel distributions.Comment: 5 Revtex pages, 4 figures; definitive versio
Stationary Distribution and Eigenvalues for a de Bruijn Process
We define a de Bruijn process with parameters n and L as a certain
continuous-time Markov chain on the de Bruijn graph with words of length L over
an n-letter alphabet as vertices. We determine explicitly its steady state
distribution and its characteristic polynomial, which turns out to decompose
into linear factors. In addition, we examine the stationary state of two
specializations in detail. In the first one, the de Bruijn-Bernoulli process,
this is a product measure. In the second one, the Skin-deep de Bruin process,
the distribution has constant density but nontrivial correlation functions. The
two point correlation function is determined using generating function
techniques.Comment: Dedicated to Herb Wilf on the occasion of his 80th birthda
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