3,665 research outputs found
Dirichlet random walks
This article provides tools for the study of the Dirichlet random walk in
. By this we mean the random variable
where is Dirichlet distributed and where
are iid, uniformly distributed on the unit sphere of
and independent of In particular we compute explicitely in
a number of cases the distribution of Some of our results appear already
in the literature, in particular in the papers by G\'erard Le Ca\"{e}r (2010,
2011). In these cases, our proofs are much simpler from the original ones,
since we use a kind of Stieltjes transform of instead of the Laplace
transform: as a consequence the hypergeometric functions replace the Bessel
functions. A crucial ingredient is a particular case of the classical and non
trivial identity, true for : We extend these results to a study of
the limits of the Dirichlet random walks when the number of added terms goes to
infinity, interpreting the results in terms of an integral by a Dirichlet
process. We introduce the ideas of Dirichlet semigroups and of Dirichlet
infinite divisibility and characterize these infinite divisible distributions
in the sense of Dirichlet when they are concentrated on the unit ball of
{4mm}\noindent \textsc{Keywords:} Dirichlet processes, Stieltjes transforms,
random flight, distributions in a ball, hyperuniformity, infinite divisibility
in the sense of Dirichlet.
{4mm}\noindent \textsc{AMS classification}: 60D99, 60F99
One-dimensional infinite memory imitation models with noise
In this paper we study stochastic process indexed by
constructed from certain transition kernels depending on the whole past. These
kernels prescribe that, at any time, the current state is selected by looking
only at a previous random instant. We characterize uniqueness in terms of
simple concepts concerning families of stochastic matrices, generalizing the
results previously obtained in De Santis and Piccioni (J. Stat. Phys.,
150(6):1017--1029, 2013).Comment: 22 pages, 3 figure
Perfect simulation of autoregressive models with infinite memory
In this paper we consider the problem of determining the law of binary
stochastic processes from transition kernels depending on the whole past. These
kernels are linear in the past values of the process. They are allowed to
assume values close to both 0 and 1, preventing the application of usual
results on uniqueness. More precisely we give sufficient conditions for
uniqueness and non-uniqueness. In the former case a perfect simulation
algorithm is also given.Comment: 12 page
Criticality of the "critical state" of granular media: Dilatancy angle in the tetris model
The dilatancy angle describes the propensity of a granular medium to dilate
under an applied shear. Using a simple spin model (the ``tetris'' model) which
accounts for geometrical ``frustration'' effects, we study such a dilatancy
angle as a function of density. An exact mapping can be drawn with a directed
percolation process which proves that there exists a critical density
above which the system expands and below which it contracts under shear. When
applied to packings constructed by a random deposition under gravity, the
dilatancy angle is shown to be strongly anisotropic, and it constitutes an
efficient tool to characterize the texture of the medium.Comment: 7 pages RevTex, 8eps figure, to appear in Phys. Rev.
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