Sequence Of Random Matrices Research Articles

Tight Bounds on the Convergence Rate of Generalized Ratio Consensus Algorithms

The problems discussed in this article are motivated by general ratio consensus algorithms, introduced by Kempe <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> in 2003 in a simple form as the push-sum algorithm, later extended by Bénézit <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> in 2010 under the name weighted gossip algorithm. We consider a communication protocol described by a strictly stationary, ergodic, sequentially primitive sequence of nonnegative matrices, applied iteratively to a pair of fixed initial vectors, the components of which are called values and weights defined at the nodes of a network. The subject of ratio consensus problems is to study the asymptotic properties of ratios of values and weights at each node, expecting convergence to the same limit for all nodes. The main results of this article provide upper bounds for the rate of the almost sure exponential convergence in terms of the spectral gap associated with the given sequence of random matrices. It will be shown that these upper bounds are sharp. Our results complement previous results of Picci and Taylor in 2013 and Iutzeler <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> in 2013.

Demographic Dynamics in Multitype Populations with Migrations

This research work deals with mathematical modeling in complex biological systems in which several types of individuals coexist in various populations. Migratory phenomena among the populations are allowed. We propose a class of mathematical models to describe the demographic dynamics of these type of complex systems. The probability model is defined through a sequence of random matrices in which rows and columns represent the various populations and the several types of individuals, respectively. We prove that this stochastic sequence can be studied under the general setting provided by the multitype branching process theory. Probabilistic properties and limiting results are then established. As application, we present an illustrative example about the population dynamics of biological systems formed by long-lived raptor colonies.

Regular variation and free regular infinitely divisible laws

In this article the relation between the tail behaviours of a free regular infinitely divisible probability measure and its Lévy measure is studied. An important example of such a measure is the compound free Poisson distribution, which often occurs as a limiting spectral distribution of certain sequences of random matrices. We also describe a connection between an analogous classical result of Embrechts et al. (1979) and our result using the Bercovici–Pata bijection.

Robust Stability Analysis and State Feedback Synthesis for Discrete-Time Systems Characterized by Random Polytopes

This paper deals with discrete-time linear systems whose state transition is determined by a sequence of random matrices having uncertain distributions. In representing random matrices with uncertain distributions, we use random polytopes whose vertices are random matrices with given fixed distributions. For such systems characterized by random polytopes, we first tackle a problem of analyzing robust second-moment exponential stability. In particular, we show a linear-matrix-inequality-based robust stability condition that can be solved through sample-based evaluation of the associated expectations. The confidence level for such sample-based analysis is ensured by the central limit theorem. Then, we extend the results about analysis toward robust stabilization state feedback synthesis in such a way that the confidence level arguments can be facilitated. We also provide numerical examples illustrating our analysis and synthesis framework.

Cycle-expansion method for the Lyapunov exponent, susceptibility, and higher moments.

Lyapunov exponents characterize the chaotic nature of dynamical systems by quantifying the growth rate of uncertainty associated with the imperfect measurement of initial conditions. Finite-time estimates of the exponent, however, experience fluctuations due to both the initial condition and the stochastic nature of the dynamical path. The scale of these fluctuations is governed by the Lyapunov susceptibility, the finiteness of which typically provides a sufficient condition for the law of large numbers to apply. Here, we obtain a formally exact expression for this susceptibility in terms of the Ruelle dynamical ζ function for one-dimensional systems. We further show that, for systems governed by sequences of random matrices, the cycle expansion of the ζ function enables systematic computations of the Lyapunov susceptibility and of its higher-moment generalizations. The method is here applied to a class of dynamical models that maps to static disordered spin chains with interactions stretching over a varying distance and is tested against Monte Carlo simulations.

Accessing the Appropriateness of a Spatial Regression Using Generalized (&lt;i&gt;h&lt;/i&gt;&lt;sub&gt;1&lt;/sub&gt;&lt;i&gt;h&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;-)Slepian Random Field

In this study we derived asymptotic goodness-of-fit test (model check) for spatial regression where the critical region as well as the p-value of the tests are approximated based on the distribution of a type of the integral functional of the generalized (h1h2-)-Slepian field and the set-indexed Gaussian white noise. Such random fields are obtained as the limit process of the moving and the cumulative sums processes of the sequence of random matrices consist of independent and identically distributed random variables indexed by the points of a design constructed by means of a given continuous probability measure. Although the common approach in model diagnostic for regression is based on the functional of the residuals, in this study a new different idea is proposed by directly investigating the moving and the cumulative sums of the array of the observations. It is shown that these approaches are mathematically tractable and practically more applicable. Simulation study is conducted for investigating the finite sample size behavior of the tests. An application of the procedure to a mining data is also discussed, where from the perspective of geology and geophysics, polynomial model is reasonable and suitable for the data.

Random Time-Dependent Quantum Walks

We consider the discrete time unitary dynamics given by a quantum walk on the lattice $\Z^d$ performed by a quantum particle with internal degree of freedom, called coin state, according to the following iterated rule: a unitary update of the coin state takes place, followed by a shift on the lattice, conditioned on the coin state of the particle. We study the large time behavior of the quantum mechanical probability distribution of the position observable in $\Z^d$ when the sequence of unitary updates is given by an i.i.d. sequence of random matrices. When averaged over the randomness, this distribution is shown to display a drift proportional to the time and its centered counterpart is shown to display a diffusive behavior with a diffusion matrix we compute. A moderate deviation principle is also proven to hold for the averaged distribution and the limit of the suitably rescaled corresponding characteristic function is shown to satisfy a diffusion equation. A generalization to unitary updates distributed according to a Markov process is also provided. An example of i.i.d. random updates for which the analysis of the distribution can be performed without averaging is worked out. The distribution also displays a deterministic drift proportional to time and its centered counterpart gives rise to a random diffusion matrix whose law we compute. A large deviation principle is shown to hold for this example. We finally show that, in general, the expectation of the random diffusion matrix equals the diffusion matrix of the averaged distribution.

A finitization of the bead process

The bead process is the particle system defined on parallel lines, with underlying measure giving constant weight to all configurations in which particles on neighbouring lines interlace, and zero weight otherwise. Motivated by the statistical mechanical model of the tiling of an abc-hexagon by three species of rhombi, a finitized version of the bead process is defined. The corresponding joint distribution can be realized as an eigenvalue probability density function for a sequence of random matrices. The finitized bead process is determinantal, and we give the correlation kernel in terms of Jacobi polynomials. Two scaling limits are considered: a global limit in which the spacing between lines goes to zero, and a certain bulk scaling limit. In the global limit the shape of the support of the particles is determined, while in the bulk scaling limit the bead process kernel of Boutillier is reclaimed, after appropriate identification of the anisotropy parameter therein.

Tail probabilities for infinite series of regularly varying random vectors

A random vector $X$ with representation $X=\sum_{j\geq0}A_jZ_j$ is considered. Here, $(Z_j)$ is a sequence of independent and identically distributed random vectors and $(A_j)$ is a sequence of random matrices, `predictable' with respect to the sequence $(Z_j)$. The distribution of $Z_1$ is assumed to be multivariate regular varying. Moment conditions on the matrices $(A_j)$ are determined under which the distribution of $X$ is regularly varying and, in fact, `inherits' its regular variation from that of the $(Z_j)$'s. We compute the associated limiting measure. Examples include linear processes, random coefficient linear processes such as stochastic recurrence equations, random sums and stochastic integrals.

Self-organizing algorithms for generalized eigen-decomposition

We discuss a new approach to self-organization that leads to novel adaptive algorithms for generalized eigen-decomposition and its variance for a single-layer linear feedforward neural network. First, we derive two novel iterative algorithms for linear discriminant analysis (LDA) and generalized eigen-decomposition by utilizing a constrained least-mean-squared classification error cost function, and the framework of a two-layer linear heteroassociative network performing a one-of-m classification. By using the concept of deflation, we are able to find sequential versions of these algorithms which extract the LDA components and generalized eigenvectors in a decreasing order of significance. Next, two new adaptive algorithms are described to compute the principal generalized eigenvectors of two matrices (as well as LDA) from two sequences of random matrices. We give a rigorous convergence analysis of our adaptive algorithms by using stochastic approximation theory, and prove that our algorithms converge with probability one.

An adaptive stochastic approximation algorithm for simultaneous diagonalization of matrix sequences with applications

We describe an adaptive algorithm based on stochastic approximation theory for the simultaneous diagonalization of the expectations of two random matrix sequences. Although there are several conventional approaches to solving this problem, there are many applications in pattern analysis and signal detection that require an online (i.e., real-time) procedure for this computation. In these applications, we are given two sequences of random matrices {A/sub k/} and {B/sub k/} as online observations, with lim/sub k/spl rarr//spl infin//E[A/sub k/]=A and lim/sub k/spl rarr//spl infin//E[B/sub k/]=B, where A and B are real, symmetric and positive definite. For every sample (A/sub k/, B/sub k/), we need the current estimates /spl Phi//sub k/ and /spl Lambda//sub k/, respectively of the eigenvectors /spl Phi/ and eigenvalues /spl Lambda/ of A/sup -1/ B. We have described such an algorithm where /spl Phi//sub k/ and /spl Lambda//sub k/ converge provably with probability one to /spl Phi/ and /spl Lambda/ respectively. A novel computational procedure used in the algorithm is the adaptive computation of A/sup - 1/2 /. Besides its use in the generalized eigen-decomposition problem, this procedure can be used on its own in several feature extraction problems. The performance of the algorithm is demonstrated with an example of detecting a high-dimensional signal in the presence of interference and noise, in a digital mobile communications problem. Experiments comparing computational complexity and performance demonstrate the effectiveness of the algorithm in this real-time application.

Weak ergodicity and products of random matrices

A representation for a weakly ergodic sequence of (nonstochastic) matrices allows products of nonnegative matrices which eventually become strictly positive to be expressed via products of some associated stochastic matrices and ratios of values of a certain function. This formula used in a random setup leads to a representation for the logarithm of a random matrix product. If the sequence of random matrices is in addition stationary then automatically almost all sequences are weakly ergodic, and the representation is expressed in terms of an one-dimensional stationary process. This permits properties of products of random matrices to be deduced from the latter. Second moment assumptions guarantee that central limit theorems and laws of the iterated logarithm hold for the random matrix products if and only if they hold for the corresponding stationary process. Finally, a central limit theorem for some classes of weakly dependent stationary random matrices is derived doing away with the restriction of boundedness of the ratios of colum entries assumed by previous studies. Extensions beyond stationarity are discussed.