A note on the invariant distribution of a quasi-birth-and-death process

The aim of this paper is to give an explicit formula of the invariant distribution of a quasi-birth-and-death process in terms of the block entries of the transition probability matrix using a matrix-valued orthogonal polynomials approach. We will show that the invariant distribution can be computed using the squared norms of the corresponding matrix-valued orthogonal polynomials, no matter if they are or not diagonal matrices. We will give an example where the squared norms are not diagonal matrices, but nevertheless we can compute its invariant distribution

The performance of the MROI fast tip-tilt correction system

The fast tip-tilt (FTT) correction system for the Magdalena Ridge Observatory Interferometer (MROI) is being developed by the University of Cambridge. The design incorporates an EMCCD camera protected by a thermal enclosure, optical mounts with passive thermal compensation, and control software running under Xenomai real-time Linux. The complete FTT system is now undergoing laboratory testing prior to being installed on the first MROI unit telescope in the fall of 2014. We are following a twin-track approach to testing the closed-loop performance: tracking tip-tilt perturbations introduced by an actuated flat mirror in the laboratory, and undertaking end-to-end simulations that incorporate realistic higher-order atmospheric perturbations. We report test results that demonstrate (a) the high stability of the entire opto-mechanical system, realized with a completely passive design; and (b) the fast tip-tilt correction performance and limiting sensitivity. Our preliminary results in both areas are close to those needed to realise the ambitious stability and sensitivity goals of the MROI which aims to match the performance of current natural guide star adaptive optics systems.Previously funded by the Naval Research Laboratory (under Agreement No. N00173-01-2-C902), MROI is currently funded by the US Department of Transportation, the State of New Mexico and by New Mexico Tech

Finite type approximations of Gibbs measures on sofic subshifts

Consider a H\"older continuous potential $\phi$ defined on the full shift A^\nn, where $A$ is a finite alphabet. Let X\subset A^\nn be a specified sofic subshift. It is well-known that there is a unique Gibbs measure $\mu_\phi$ on $X$ associated to $\phi$. Besides, there is a natural nested sequence of subshifts of finite type $(X_m)$ converging to the sofic subshift $X$. To this sequence we can associate a sequence of Gibbs measures $(\mu_{\phi}^m)$. In this paper, we prove that these measures weakly converge at exponential speed to $\mu_\phi$ (in the classical distance metrizing weak topology). We also establish a strong mixing property (ensuring weak Bernoullicity) of $\mu_\phi$. Finally, we prove that the measure-theoretic entropy of $\mu_\phi^m$ converges to the one of $\mu_\phi$ exponentially fast. We indicate how to extend our results to more general subshifts and potentials. We stress that we use basic algebraic tools (contractive properties of iterated matrices) and symbolic dynamics.Comment: 18 pages, no figure

Martin boundary of a reflected random walk on a half-space

The complete representation of the Martin compactification for reflected random walks on a half-space $\Z^d\times\N$ is obtained. It is shown that the full Martin compactification is in general not homeomorphic to the ``radial'' compactification obtained by Ney and Spitzer for the homogeneous random walks in $\Z^d$ : convergence of a sequence of points $z_n\in\Z^{d-1}\times\N$ to a point of on the Martin boundary does not imply convergence of the sequence $z_n/|z_n|$ on the unit sphere $S^d$. Our approach relies on the large deviation properties of the scaled processes and uses Pascal's method combined with the ratio limit theorem. The existence of non-radial limits is related to non-linear optimal large deviation trajectories.Comment: 42 pages, preprint, CNRS UMR 808

Entropy: The Markov Ordering Approach

The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and analyze two invariant "additivity" properties: (i) existence of a monotonic transformation which makes the functional additive with respect to the joining of independent systems and (ii) existence of a monotonic transformation which makes the functional additive with respect to the partitioning of the space of states. All Lyapunov functionals for Markov chains which have properties (i) and (ii) are derived. We describe the most general ordering of the distribution space, with respect to which all continuous-time Markov processes are monotonic (the {\em Markov order}). The solution differs significantly from the ordering given by the inequality of entropy growth. For inference, this approach results in a convex compact set of conditionally "most random" distributions.Comment: 50 pages, 4 figures, Postprint version. More detailed discussion of the various entropy additivity properties and separation of variables for independent subsystems in MaxEnt problem is added in Section 4.2. Bibliography is extende

Nonequilibrium stationary states and equilibrium models with long range interactions

It was recently suggested by Blythe and Evans that a properly defined steady state normalisation factor can be seen as a partition function of a fictitious statistical ensemble in which the transition rates of the stochastic process play the role of fugacities. In analogy with the Lee-Yang description of phase transition of equilibrium systems, they studied the zeroes in the complex plane of the normalisation factor in order to find phase transitions in nonequilibrium steady states. We show that like for equilibrium systems, the ``densities'' associated to the rates are non-decreasing functions of the rates and therefore one can obtain the location and nature of phase transitions directly from the analytical properties of the ``densities''. We illustrate this phenomenon for the asymmetric exclusion process. We actually show that its normalisation factor coincides with an equilibrium partition function of a walk model in which the ``densities'' have a simple physical interpretation.Comment: LaTeX, 23 pages, 3 EPS figure

Survival, extinction and approximation of discrete-time branching random walks

We consider a general discrete-time branching random walk on a countable set X. We relate local, strong local and global survival with suitable inequalities involving the first-moment matrix M of the process. In particular we prove that, while the local behavior is characterized by M, the global behavior cannot be completely described in terms of properties involving M alone. Moreover we show that locally surviving branching random walks can be approximated by sequences of spatially confined and stochastically dominated branching random walks which eventually survive locally if the (possibly finite) state space is large enough. An analogous result can be achieved by approximating a branching random walk by a sequence of multitype contact processes and allowing a sufficiently large number of particles per site. We compare these results with the ones obtained in the continuous-time case and we give some examples and counterexamples.Comment: 32 pages, a few misprints have been correcte

Orbital elements, masses and distance of lambda Scorpii A and B determined with the Sydney University Stellar Interferometer and high resolution spectroscopy

The triple system HD158926 (lambda Sco) has been observed interferometrically with the Sydney University Stellar Interferometer and the elements of the wide orbit have been determined. These are significantly more accurate than the previous elements found spectroscopically. The inclination of the wide orbit is consistent with the inclination previously found for the orbit of the close companion. The wide orbit also has low eccentricity, suggesting that the three stars were formed at the same time. The brightness ratio between the two B stars was also measured at lambda = 442nm and 700nm. The brightness ratio and colour index are consistent with the previous classification of lambda Sco A as B1.5 and lambda Sco B as B2. Evolutionary models show that the two stars lie on the main sequence. Since they have have the same age and luminosity class (IV) the mass-luminosity relation can be used to determine the mass ratio of the two stars: M_B/M_A = 0.76+/-0.04. The spectroscopic data have been reanalyzed using the interferometric values for P, T, e and omega, leading to revised values for a_1sin i and the mass function. The individual masses can be found from the mass ratio, the mass function, spectrum synthesis and the requirement that the age of both components must be the same: M_A = 10.4+/-1.3 Msun and M_B = 8.1+/-1.0 Msun. The masses, angular semimajor axis and the period of the system can be used to determine the dynamical parallax. We find the distance to lambda Sco to be 112+/-5 pc, which is approximately a factor of two closer than the HIPPARCOS value of 216+/-42 pc.Comment: 8 pages, 4 figures. Accepted for publication by Monthly Notices of the Royal Astronomical Societ

Classification and realizations of type III factor representations of Cuntz-Krieger algebras associated with quasi-free states

We completely classify type III factor representations of Cuntz-Krieger algebras associated with quasi-free states up to unitary equivalence. Furthermore, we realize these representations on concrete Hilbert spaces without using GNS construction. Free groups and their type ${\rm II}_{1}$ factor representations are used in these realizations.Comment: 11 page