Discrete Schur-constant models

This paper introduces a class of Schur-constant survival models, of dimension n, for arithmetic non-negative random variables. Such a model is defined through a univariate survival function that is shown to be n-monotone. Two general representations are obtained, by conditioning on the sum of the n variables or through a doubly mixed multinomial distribution. Several other properties including correlation measures are derived. Three processes in insurance theory are discussed for which the claim interarrival periods form a Schur-constant model

Equilibrium distributions and discrete Schur-constant models

This paper introduces Schur-constant equilibrium distribution models of dimension n for arithmetic non-negative random variables. Such a model is defined through the (several orders) equilibrium distributions of a univariate survival function. First, the bivariate case is considered and analyzed in depth, stressing the main characteristics of the Poisson case. The analysis is then extended to the multivariate case. Several properties are derived, including the implicit correlation and the distribution of the sum

Variational Theory and Domain Decomposition for Nonlocal Problems

In this article we present the first results on domain decomposition methods for nonlocal operators. We present a nonlocal variational formulation for these operators and establish the well-posedness of associated boundary value problems, proving a nonlocal Poincar\'{e} inequality. To determine the conditioning of the discretized operator, we prove a spectral equivalence which leads to a mesh size independent upper bound for the condition number of the stiffness matrix. We then introduce a nonlocal two-domain variational formulation utilizing nonlocal transmission conditions, and prove equivalence with the single-domain formulation. A nonlocal Schur complement is introduced. We establish condition number bounds for the nonlocal stiffness and Schur complement matrices. Supporting numerical experiments demonstrating the conditioning of the nonlocal one- and two-domain problems are presented.Comment: Updated the technical part. In press in Applied Mathematics and Computatio

Brownian Motion in a Weyl Chamber, Non-Colliding Particles, and Random Matrices

Let $n$ particles move in standard Brownian motion in one dimension, with the process terminating if two particles collide. This is a specific case of Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for this chamber is $A_{n-1}$, the symmetric group. For any starting positions, we compute a determinant formula for the density function for the particles to be at specified positions at time $t$ without having collided by time $t$. We show that the probability that there will be no collision up to time $t$ is asymptotic to a constant multiple of $t^{-n(n-1)/4}$ as $t$ goes to infinity, and compute the constant as a polynomial of the starting positions. We have analogous results for the other classical Weyl groups; for example, the hyperoctahedral group $B_n$ gives a model of $n$ independent particles with a wall at $x=0$. We can define Brownian motion on a Lie algebra, viewing it as a vector space; the eigenvalues of a point in the Lie algebra correspond to a point in the Weyl chamber, giving a Brownian motion conditioned never to exit the chamber. If there are $m$ roots in $n$ dimensions, this shows that the radial part of the conditioned process is the same as the $n+2m$-dimensional Bessel process. The conditioned process also gives physical models, generalizing Dyson's model for $A_{n-1}$ corresponding to ${\mathfrak s}{\mathfrak u}_n$ of $n$ particles moving in a diffusion with a repelling force between two particles proportional to the inverse of the distance between them

Robust filtering for gene expression time series data with variance constraints

This is the post print version of the article. The official published version can be obtained from the link below - Copyright 2007 Taylor & Francis Ltd.In this paper, an uncertain discrete-time stochastic system is employed to represent a model for gene regulatory networks from time series data. A robust variance-constrained filtering problem is investigated for a gene expression model with stochastic disturbances and norm-bounded parameter uncertainties, where the stochastic perturbation is in the form of a scalar Gaussian white noise with constant variance and the parameter uncertainties enter both the system matrix and the output matrix. The purpose of the addressed robust filtering problem is to design a linear filter such that, for the admissible bounded uncertainties, the filtering error system is Schur stable and the individual error variance is less than a prespecified upper bound. By using the linear matrix inequality (LMI) technique, sufficient conditions are first derived for ensuring the desired filtering performance for the gene expression model. Then the filter gain is characterized in terms of the solution to a set of LMIs, which can easily be solved by using available software packages. A simulation example is exploited for a gene expression model in order to demonstrate the effectiveness of the proposed design procedures.This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grants GR/S27658/01 and EP/C524586/1, the Biotechnology and Biological Sciences Research Council (BBSRC) of the UK under Grants BB/C506264/1 and 100/EGM17735, the Nuffield Foundation of the UK under Grant NAL/00630/G, and the Alexander von Humboldt Foundation of Germany

Fermionic construction of tau functions and random processes

Tau functions expressed as fermionic expectation values are shown to provide a natural and straightforward description of a number of random processes and statistical models involving hard core configurations of identical particles on the integer lattice, like a discrete version simple exclusion processes (ASEP), nonintersecting random walkers, lattice Coulomb gas models and others, as well as providing a powerful tool for combinatorial calculations involving paths between pairs of partitions. We study the decay of the initial step function within the discrete ASEP (d-ASEP) model as an example.Comment: 53 pages, 13 figures, a contribution to Proc. "Mathematics and Physics of Growing Interfaces