Showing papers on "Random element published in 2011"
TL;DR: A Chernoff inequality for matrices is used to show that the eigenvalues of the adjacency matrix and the normalized Laplacian of such a random graph can be approximated by those of the weighted expectation graph, with error bounds dependent upon the minimum and maximum expected degrees.
Abstract: We consider random graphs such that each edge is determined by an independent random variable, where the probability of each edge is not assumed to be equal. We use a Chernoff inequality for matrices to show that the eigenvalues of the adjacency matrix and the normalized Laplacian of such a random graph can be approximated by those of the weighted expectation graph, with error bounds dependent upon the minimum and maximum expected degrees. In particular, we use these results to bound the spectra of random graphs with given expected degree sequences, including random power law graphs. Moreover, we prove a similar result giving concentration of the spectrum of a matrix martingale on its expectation.
167 citations
TL;DR: In this article, the existence of global compact random attractors in weighted space l ρ 2 for stochastic lattice dynamical systems with random coupled coefficients and multiplicative/additive white noises was shown.
125 citations
TL;DR: In this paper, the conditional probability of max-stable random fields is derived for a large class of random fields and an algorithm for efficient and exact sampling from the conditional distributions is proposed.
Abstract: Max-stable random fields play a central role in modeling extreme value phenomena. We obtain an explicit formula for the conditional probability in general max-linear models, which include a large class of max-stable random fields. As a consequence, we develop an algorithm for efficient and exact sampling from the conditional distributions. Our method provides a computational solution to the prediction problem for spectrally discrete max-stable random fields. This work offers new tools and a new perspective to many statistical inference problems for spatial extremes, arising, for example, in meteorology, geology, and environmental applications.
79 citations
TL;DR: It is proved that for densities well below the satisfiability threshold, the solution space of random k‐SAT decomposes into an exponential number of connected components and give quantitative bounds for the diameter, volume, and number.
Abstract: For various random constraint satisfaction problems there is a significant gap between the largest constraint density for which solutions exist and the largest density for which any polynomial time algorithm is known to find solutions. Examples of this phenomenon include random k-SAT, random graph coloring, and a number of other random constraint satisfaction problems. To understand this gap, we study the structure of the solution space of random k-SAT (i.e., the set of all satisfying assignments viewed as a subgraph of the Hamming cube). We prove that for densities well below the satisfiability threshold, the solution space decomposes into an exponential number of connected components and give quantitative bounds for the diameter, volume, and number.© 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 251–268, 2011 © 2011 Wiley Periodicals, Inc.
77 citations
TL;DR: The distance function which appears in this estimate is not in general the graph metric, but a new metric which is adapted to the random walk, and long-range non-Gaussian bounds in this new metric are established.
Abstract: We consider continuous time simple random walks with arbitrary speed measure $\theta$ on infinite weighted graphs. Write $p_t(x,y)$ for the heat kernel of this process. Given on-diagonal upper bounds for the heat kernel at two points $x_1,x_2$, we obtain a Gaussian upper bound for $p_t(x_1,x_2)$ . The distance function which appears in this estimate is not in general the graph metric, but a new metric which is adapted to the random walk. Long-range non-Gaussian bounds in this new metric are also established. Applications to heat kernel bounds for various models of random walks in random environments are discussed.
72 citations
TL;DR: A theory for the eigenvalue density of arbitrary non-Hermitian Euclidean matrices is developed and applied to the random Green's matrix relevant to wave propagation in an ensemble of pointlike scattering centers.
Abstract: We develop a theory for the eigenvalue density of arbitrary non-Hermitian Euclidean matrices. Closed equations for the resolvent and the eigenvector correlator are derived. The theory is applied to the random Green's matrix relevant to wave propagation in an ensemble of pointlike scattering centers. This opens a new perspective in the study of wave diffusion, Anderson localization, and random lasing.
70 citations
TL;DR: In this paper, the authors develop approximation methods to achieve computationally tractable BN models of correlated random variables drawn from a Gaussian random field using several generic and systematic spatial configuration models, and numerical investigations are performed to compare the relative effectiveness of the proposed approximation methods.
57 citations
TL;DR: In this article, a class of one-dimensional interacting particle systems in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied/vacant sites has a local drift to the right/left.
Abstract: In this paper we consider a class of one-dimensional interacting particle systems in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied/vacant sites has a local drift to the right/left. We adapt a regeneration-time argument originally developed by Comets and Zeitouni for static random environments to prove that, under a space-time mixing property for the dynamic random environment called cone-mixing, the random walk has an a.s. constant global speed. In addition, we show that if the dynamic random environment is exponentially mixing in space-time and the local drifts are small, then the global speed can be written as a power series in the size of the local drifts. From the first term in this series the sign of the global speed can be read off. The results can be easily extended to higher dimensions.
54 citations
TL;DR: In this article, a class of stable random fields, called operator-self-similar random fields (OSRFs), whose distributions are invariant under operator-scalings in both the time domain and the state space are studied.
52 citations
TL;DR: In this paper, the authors apply the concept of free random variables to doubly correlated Wishart random matrix models, appearing, for example, in a multivariate analysis of financial time series, and displaying both inter-asset cross-covariances and temporal auto covariances.
Abstract: We apply the concept of free random variables to doubly correlated (Gaussian) Wishart random matrix models, appearing, for example, in a multivariate analysis of financial time series, and displaying both inter-asset cross-covariances and temporal auto-covariances. We give a comprehensive introduction to the rich financial reality behind such models. We explain in an elementary way the main techniques of free random variables calculus, with a view to promoting them in the quantitative finance community. We apply our findings to tackle several financially relevant problems, such as a universe of assets displaying exponentially decaying temporal covariances, or the exponentially weighted moving average, both with an arbitrary structure of cross-covariances.
TL;DR: It is shown the existence and uniqueness of a solution that generates a cocycle and the existence of an attractor is proved, allowing us to prove that the solution to the random differential equation generates a random dynamical system.
Abstract: We investigate a random differential equation with random delay. First the non-autonomous case is considered. We show the existence and uniqueness of a solution that generates a cocycle. In particular, the existence of an attractor is proved. Secondly we look at the random case. We pay special attention to the measurability. This allows us to prove that the solution to the random differential equation generates a random dynamical system. The existence result of the attractor can be carried over to the random case.
TL;DR: In this paper, a model of random walk on random marked trees, following a model introduced by R. Lyons and R. Pemantle (1992), is presented, where the authors consider a very general way of constructing random trees with random transition probabilities.
Abstract: Models of random walks in a random environment were introduced at first by Chernoff in 1967 in order to study biological mechanisms. The original model has been intensively studied since then and is now well understood. In parallel, similar models of random processes in a random environment have been studied. In this article we focus on a model of random walk on random marked trees, following a model introduced by R. Lyons and R. Pemantle (1992). Our point of view is a bit different yet, as we consider a very general way of constructing random trees with random transition probabilities on them. We prove an analogue of R. Lyons and R. Pemantle's recurrence criterion in this setting, and we study precisely the asymptotic behavior, under restrictive assumptions. Our last result is a generalization of a result of Y. Peres and O. Zeitouni (2006) concerning biased random walks on Galton-Watson trees.
TL;DR: In this article, the dynamical behavior of a stochastic wave equation on a bounded domain in R 3 with smooth boundary, for which the nonlinear damping has a critical cubic growth rate, is investigated.
Abstract: The dynamical behavior of a stochastic wave equation on a bounded domain in R 3 with smooth boundary, for which the nonlinear damping has a critical cubic growth rate, is investigated Its dissipativeness in higher-energy spaces W s , 2 × W s − 1 , 2 for 1 s ⩽ 2 is established This implies that the random dynamical system (RDS) generated by the equation has a random attractor in H 0 1 × L 2 , which is a tempered random set in the space H 2 × H 0 1
TL;DR: A full and rigorous proof of a theorem, stating that random groups in the Gromov density model for d > 1/3 have property (T) with high probability, is provided.
Abstract: We provide a full and rigorous proof of a theorem attributed to Żuk, stating that random groups in the Gromov density model for d > 1/3 have property (T) with high probability. The original paper had numerous gaps, in particular, crucial steps involving passing between different models of random groups were not described. We fix the gaps using combinatorial arguments and a recent result concerning perfect matchings in random hypergraphs. We also provide an alternative proof, avoiding combinatorial difficulties and relying solely on spectral properties of random graphs in G(n, p) model.
TL;DR: In this paper, a bounded step size random walk in an ergodic random environment with some ellipticity, on an integer lattice of arbitrary dimension, was considered and the level 3 large deviation principle was proved under almost every environment, with rate function related to a relative entropy.
Abstract: We consider a bounded step size random walk in an ergodic random environment with some ellipticity, on an integer lattice of arbitrary dimension. We prove a level 3 large deviation principle, under almost every environment, with rate function related to a relative entropy.
TL;DR: The use of upper-lower models instead of second-order models will enable us in the future to reach crisp decisions in some specific statistical problems, without adding any arbitrary information, and taking into account the imprecision in data.
TL;DR: An interesting hierarchy of random number generators is introduced in this paper based on the review of random numbers characteristics and chaotic functions theory, and offers a great capability to be employed in a pseudo random number generator.
TL;DR: In this article, the total macroscopic cross section is taken to be a spatially continuous log-normal random process with known covariance function and expressed as a memoryless transformation of a Gaussian random process.
Abstract: Stochastic spectral expansions are used to represent random input parameters and the random unknown solution to describe radiation transport in random media. The total macroscopic cross section is taken to be a spatially continuous log-normal random process with known covariance function and expressed as a memoryless transformation of a Gaussian random process. The Karhunen–Loeve expansion is applied to represent the spatially continuous random cross section in terms of a finite number of discrete Gaussian random variables. The angular flux is then expanded in terms of Hermite polynomials and, using a quadrature-based stochastic collocation method, the expansion coefficients are shown to satisfy uncoupled deterministic transport equations. Sparse grid Gauss quadrature rules are investigated to establish the efficacy of the polynomial chaos-collocation scheme. Numerical results for the mean and standard deviation of the scalar flux as well as probability density functions of the scalar flux and transmission function are obtained for a deterministic incident source, contrasting between absorbing and diffusive media.
TL;DR: In this article, the authors consider random walks in random Dirichlet environment (RWDE) which is a special type of random walk in random environment where the exit probabilities at each site are i.i.d.
Abstract: We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On $\Z^d$, RWDE are parameterized by a $2d$-uplet of positive reals. We prove that for all values of the parameters, RWDE are transient in dimension $d\ge 3$. We also prove that the Green function has some finite moments and we characterize the finite moments. Our result is more general and applies for example to finitely generated symmetric transient Cayley graphs. In terms of reinforced random walks it implies that directed edge reinforced random walks are transient for $d\ge 3$.
TL;DR: The distribution functions of the two pivotal quantities, when the sample size is assumed to be integer-valued random variable, are derived and three important special cases for the random sample size are presented.
Abstract: In this paper we develop two pivotal quantities to construct exact predication intervals for future exponential lifetime based on a random number of lower generalized order statistics. The distribution functions of the two pivotal quantities, when the sample size is assumed to be integer-valued random variable, are derived. Three important special cases for the random sample size are presented. A simulation study is conducted for illustrative purposes.
TL;DR: In this paper, the authors study right tail large deviations of the logarithm of the partition function for directed lattice paths in i.i.d. random potentials.
Abstract: We study right tail large deviations of the logarithm of the partition function for directed lattice paths in i.i.d. random potentials. The main purpose is the derivation of explicit formulas for the $1+1$-dimensional exactly solvable case with log-gamma distributed random weights. Along the way we establish some regularity results for this rate function for general distributions in arbitrary dimensions.
TL;DR: In this paper, the authors considered the branching random walks in d-dimensional integer lattice with time-space i.i.d. offspring distributions and proved a central limit theorem for the density of the population in terms of almost sure convergence.
Abstract: We consider the branching random walks in d-dimensional integer lattice with time–space iid offspring distributions Then the normalization of the total population is a nonnegative martingale and it almost surely converges to a certain random variable When d≥3 and the fluctuation of environment satisfies a certain uniform square integrability then it is nondegenerate and we prove a central limit theorem for the density of the population in terms of almost sure convergence
TL;DR: An approach to modeling dependent nonparametric random density functions is presented, based on the well known mixture of Dirichlet process model.
05 Jun 2011
TL;DR: It is found that for a sufficiently large field size, whether the transfer matrix is singular or not is determined with probability one by the zero pattern of the matrix, i.e., where the zeroes are located in the matrix.
Abstract: Motivated by the noncoherent subspace coding approach and the low-complexity sparse coding approach to realize random linear network coding, we consider the problem of characterizing the probability of having a full rank (or nonsingular) square transfer matrix over a finite field, for which the probability of choosing the zero element is different from that of choosing a nonzero element. We found that for a sufficiently large field size, whether the transfer matrix is singular or not is determined with probability one by the zero pattern of the matrix, i.e., where the zeroes are located in the matrix. This result provides insight for optimizing sparse random linear network coding schemes and allows the problem of determining the probability of having a nonsingular transfer matrix over a large field size to be transformed into a combinatorial problem. By using some combinatorial arguments, useful upper and lower bounds on the singularity probability of the random transfer matrix are derived.
TL;DR: In this article, the authors studied the recurrence behavior of random dynamical systems and randomly perturbed dynamical system generated by the composition of random maps and proved that the random recurrence rate is equal to the local dimension of the stationary measure.
Abstract: This paper is a first step in the study of the recurrence behaviour in random dynamical systems and randomly perturbed dynamical systems. In particular we define a concept of quenched and annealed return times for systems generated by the composition of random maps. We moreover prove that for super-polynomially mixing systems, the random recurrence rate is equal to the local dimension of the stationary measure.
Posted Content•
TL;DR: In this paper, the authors considered a simple symmetric exclusion process in equilibrium, constituting a dynamic random environment for a nearest-neighbor random walk that on occupied/vacant sites has two different local drifts to the right.
Abstract: We consider a one-dimensional simple symmetric exclusion process in equilibrium, constituting a dynamic random environment for a nearest-neighbor random walk that on occupied/vacant sites has two different local drifts to the right. We construct a renewal structure from which a LLN, a functional CLT and large deviation bounds for the random walk under the annealed measure follow. We further prove an Einstein relation under a suitable perturbation. A brief discussion on the topic of random walks in slowly mixing dynamic random environments is presented.
Journal Article•
TL;DR: In this article, the authors extend Fulman's result to the multivariate central limit theorem for the full vector of traces of powers for a single power j (which may grow with n), using Stein's method of exchangeable pairs.
Abstract: Let Mn be a random element of the unitary, special orthogonal, or unitary symplectic groups, distributed according to Haar measure. By a classical result of Diaconis and Shahshahani, for large matrix size n, the vector(Tr(Mn), Tr(M 2 ), . . . , Tr(M d n )) tends to a vector of independent (real or complex) Gaussian random variables. Recently, Jason Fulman has demonstrated that for a single power j (which may grow with n), a speed of convergence result may be obtained via Stein’s method of exchangeable pairs. In this note, we extend Fulman’s result to the multivariate central limit theorem for the full vector of traces of powers. .
TL;DR: In this paper, the authors consider random dissections of a regular polygon with $n$ sides, which are chosen according to Boltzmann weights in the domain of attraction of a stable law of index $\theta\in(1,2]$.
Abstract: We study large random dissections of polygons. We consider random dissections of a regular polygon with $n$ sides, which are chosen according to Boltzmann weights in the domain of attraction of a stable law of index $\theta\in(1,2]$. As $n$ goes to infinity, we prove that these random dissections converge in distribution toward a random compact set, called the random stable lamination. If $\theta=2$, we recover Aldous' Brownian triangulation. However, if $\theta\in(1,2)$, large faces remain in the limit and a different random compact set appears. We show that the random stable lamination can be coded by the continuous-time height function associated to the normalized excursion of a strictly stable spectrally positive Levy process of index $\theta$. Using this coding, we establish that the Hausdorff dimension of the stable random lamination is almost surely $2-1/\theta$.
TL;DR: In this article, it was shown that the weak convergence of a function of the data to a limiting random element is the starting point for all tests that remain asymptotically valid for all models that induce a weak limit.
Abstract: The asymptotic validity of tests is usually established by making appropriate primitive assumptions, which imply the weak convergence of a specific function of the data, and an appeal to the continuous mapping theorem. This paper, instead, takes the weak convergence of some function of the data to a limiting random element as the starting point and studies efficiency in the class of tests that remain asymptotically valid for all models that induce the same weak limit. It is found that efficient tests in this class are simply given by efficient tests in the limiting problem—that is, with the limiting random element assumed observed—evaluated at sample analogues. Efficient tests in the limiting problem are usually straightforward to derive, even in nonstandard testing problems. What is more, their evaluation at sample analogues typically yields tests that coincide with suitably robustified versions of optimal tests in canonical parametric versions of the model. This paper thus establishes an alternative and broader sense of asymptotic efficiency for many previously derived tests in econometrics, such as tests for unit roots, parameter stability tests, and tests about regression coefficients under weak instruments.