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This paper is about the rate of convergence of the Markov chain X n+1 = AX n +B n (mod p), where A is an integer matrix with nonzero eigenvalues and {B n } n is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of Q k invariant under A. If |λ i | = 1 for all eigenvalues λ i of A, then n = O (ln p) 2 steps are sufficient and n = O(ln p) steps are necessary to have X n sampling from a nearly uniform distribution. Conversely, if A has the eigenvalues λ i that are roots of positive integer numbers, |λ 1 | = 1 and |λ i | > 1 for all i = 1, then O p 2 steps are necessary and sufficient. Running head. Generating uniform random vectors. 1 1 MSC 2000 subject classifications. Primary 60B15; secondary 60J10.

In this paper an enlargement of the beta family of distributions on (0, 1) is presented. Distributions in this class are characterized as being the laws of certain random continued fractions associated with products of independent random matrices of order 2 whose entries are either constant or beta distributed. The result can be proved by a famous 1879 Thomae formula on generalized hypergeometric functions 3F2

Based on the recently-established Master kinetic equation and related Master constant H-theorem which describe the statistical behavior of the Boltzmann-Sinai classical dynamical system for smooth and hard spherical particles, the problem is posed of determining a microscopic statistical description holding for an incompressible Navier-Stokes fluid. The goal is reached by introducing a suitable mean-field interaction in the Master kinetic equation. The resulting Modified Master Kinetic Equation (MMKE) is proved to warrant at the same time the condition of mass-density incompressibility and the validity of the Navier-Stokes fluid equation. In addition, it is shown that the conservation of the Boltzmann-Shannon entropy can similarly be warranted. Applications to the plane Couette and Poiseuille flows are considered showing that they can be regarded as final decaying states for suitable non-stationary flows. As a result, it is shown that an arbitrary initial stochastic 1−body PDF evolving in time by means of MMKE necessarily exhibits the phenomenon of Decay to Kinetic Equilibrium (DKE), whereby the 1−body PDF asymptotically relaxes to a stationary and spatially-uniform Maxwellian PDF.

The local specification of priors in non-decomposable graphical models does not necessarily yield a proper joint prior for all the parameters of the model. Using results concerning general exponential families with cuts, we derive specific results for the multivariate Gamma distribution (conjugate prior for Poisson counts) and the Wishart distribution (conjugate prior for Gaussian models). These results link the existence of a locally specified joint prior to the solvability of a related marginal problem over the cliques of the graph. Copyright 2007 Board of the Foundation of the Scandinavian Journal of Statistics..