429 research outputs found
On a class of distributions stable under random summation
We investigate a family of distributions having a property of
stability-under-addition, provided that the number of added-up random
variables in the random sum is also a random variable. We call the
corresponding property a \,-stability and investigate the situation with
the semigroup generated by the generating function of is commutative.
Using results from the theory of iterations of analytic functions, we show that
the characteristic function of such a -stable distribution can be
represented in terms of Chebyshev polynomials, and for the case of -normal
distribution, the resulting characteristic function corresponds to the
hyperbolic secant distribution. We discuss some specific properties of the
class and present particular examples.Comment: 12 pages, 1 figur
Flows on Graphs with Random Capacities
We investigate flows on graphs whose links have random capacities. For binary
trees we derive the probability distribution for the maximal flow from the root
to a leaf, and show that for infinite trees it vanishes beyond a certain
threshold that depends on the distribution of capacities. We then examine the
maximal total flux from the root to the leaves. Our methods generalize to
simple graphs with loops, e.g., to hierarchical lattices and to complete
graphs.Comment: 8 pages, 6 figure
Multivariate Copula Analysis Toolbox (MvCAT): Describing Dependence and Underlying Uncertainty Using a Bayesian Framework
We present a newly developed Multivariate Copula Analysis Toolbox (MvCAT) which includes a wide range of copula families with different levels of complexity. MvCAT employs a Bayesian framework with a residual-based Gaussian likelihood function for inferring copula parameters and estimating the underlying uncertainties. The contribution of this paper is threefold: (a) providing a Bayesian framework to approximate the predictive uncertainties of fitted copulas, (b) introducing a hybrid-evolution Markov Chain Monte Carlo (MCMC) approach designed for numerical estimation of the posterior distribution of copula parameters, and (c) enabling the community to explore a wide range of copulas and evaluate them relative to the fitting uncertainties. We show that the commonly used local optimization methods for copula parameter estimation often get trapped in local minima. The proposed method, however, addresses this limitation and improves describing the dependence structure. MvCAT also enables evaluation of uncertainties relative to the length of record, which is fundamental to a wide range of applications such as multivariate frequency analysis
Statistical Consequences of Devroye Inequality for Processes. Applications to a Class of Non-Uniformly Hyperbolic Dynamical Systems
In this paper, we apply Devroye inequality to study various statistical
estimators and fluctuations of observables for processes. Most of these
observables are suggested by dynamical systems. These applications concern the
co-variance function, the integrated periodogram, the correlation dimension,
the kernel density estimator, the speed of convergence of empirical measure,
the shadowing property and the almost-sure central limit theorem. We proved in
\cite{CCS} that Devroye inequality holds for a class of non-uniformly
hyperbolic dynamical systems introduced in \cite{young}. In the second appendix
we prove that, if the decay of correlations holds with a common rate for all
pairs of functions, then it holds uniformly in the function spaces. In the last
appendix we prove that for the subclass of one-dimensional systems studied in
\cite{young} the density of the absolutely continuous invariant measure belongs
to a Besov space.Comment: 33 pages; companion of the paper math.DS/0412166; corrected version;
to appear in Nonlinearit
Monge Distance between Quantum States
We define a metric in the space of quantum states taking the Monge distance
between corresponding Husimi distributions (Q--functions). This quantity
fulfills the axioms of a metric and satisfies the following semiclassical
property: the distance between two coherent states is equal to the Euclidean
distance between corresponding points in the classical phase space. We compute
analytically distances between certain states (coherent, squeezed, Fock and
thermal) and discuss a scheme for numerical computation of Monge distance for
two arbitrary quantum states.Comment: 9 pages in LaTex - RevTex + 2 figures in ps. submitted to Phys. Rev.
Parameters of the fractional Fokker-Planck equation
We study the connection between the parameters of the fractional
Fokker-Planck equation, which is associated with the overdamped Langevin
equation driven by noise with heavy-tailed increments, and the transition
probability density of the noise generating process. Explicit expressions for
these parameters are derived both for finite and infinite variance of the
rescaled transition probability density.Comment: 5 page
A multitype sticky particle construction of Wasserstein stable semigroups solving one-dimensional diagonal hyperbolic systems with large monotonic data
International audienceThis article is dedicated to the study of diagonal hyperbolic systems in one space dimension, with cumulative distribution functions, or more generally nonconstant monotonic bounded functions, as initial data. Under a uniform strict hyperbolicity assumption on the characteristic fields, we construct a multitype version of the sticky particle dynamics and obtain existence of global weak solutions by compactness. We then derive a stability estimate on the particle system uniform in the number of particles. This allows to construct nonlinear semigroups solving the system in the sense of Bianchini and Bressan [Ann. of Math. (2), 2005]. We also obtain that these semigroup solutions satisfy a stability estimate in Wasserstein distances of all orders, which encompasses the classical estimate and generalises to diagonal systems the results by Bolley, Brenier and Loeper [J. Hyperbolic Differ. Equ., 2005] in the scalar case. Our results are obtained without any smallness assumption on the variation of the data, and only require the characteristic fields to be Lipschitz continuous and the system to be uniformly strictly hyperbolic
Brownian markets
Financial market dynamics is rigorously studied via the exact generalized
Langevin equation. Assuming market Brownian self-similarity, the market return
rate memory and autocorrelation functions are derived, which exhibit an
oscillatory-decaying behavior with a long-time tail, similar to empirical
observations. Individual stocks are also described via the generalized Langevin
equation. They are classified by their relation to the market memory as heavy,
neutral and light stocks, possessing different kinds of autocorrelation
functions
Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model
This paper deals with a one--dimensional model for granular materials, which
boils down to an inelastic version of the Kac kinetic equation, with
inelasticity parameter . In particular, the paper provides bounds for
certain distances -- such as specific weighted --distances and the
Kolmogorov distance -- between the solution of that equation and the limit. It
is assumed that the even part of the initial datum (which determines the
asymptotic properties of the solution) belongs to the domain of normal
attraction of a symmetric stable distribution with characteristic exponent
\a=2/(1+p). With such initial data, it turns out that the limit exists and is
just the aforementioned stable distribution. A necessary condition for the
relaxation to equilibrium is also proved. Some bounds are obtained without
introducing any extra--condition. Sharper bounds, of an exponential type, are
exhibited in the presence of additional assumptions concerning either the
behaviour, near to the origin, of the initial characteristic function, or the
behaviour, at infinity, of the initial probability distribution function
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