Search results for "Combinatorics"
showing 10 items of 1770 documents
The smallest singular value of a shifted $d$-regular random square matrix
2017
We derive a lower bound on the smallest singular value of a random d-regular matrix, that is, the adjacency matrix of a random d-regular directed graph. Specifically, let $$C_1<d< c n/\log ^2 n$$ and let $$\mathcal {M}_{n,d}$$ be the set of all $$n\times n$$ square matrices with 0 / 1 entries, such that each row and each column of every matrix in $$\mathcal {M}_{n,d}$$ has exactly d ones. Let M be a random matrix uniformly distributed on $$\mathcal {M}_{n,d}$$ . Then the smallest singular value $$s_{n} (M)$$ of M is greater than $$n^{-6}$$ with probability at least $$1-C_2\log ^2 d/\sqrt{d}$$ , where c, $$C_1$$ , and $$C_2$$ are absolute positive constants independent of any other parameter…
Hitting straight lines by compound Poisson process paths
1990
In a recent article Mallows and Nair (1989,Ann. Inst. Statist. Math.,41, 1–8) determined the probability of intersectionP{X(t)=αt for somet≥0} between a compound Poisson process {X(t), t≥0} and a straight line through the origin. Using four different approaches (direct probabilistic, via differential equations and via Laplace transforms) we extend their results to obtain the probability of intersection between {X(t), t≥0} and arbitrary lines. Also, we display a relationship with the theory of Galton-Watson processes. Additional results concern the intersections with two (or more) parallel lines.
Decomposable multiphase entropic descriptor
2013
To quantify degree of spatial inhomogeneity for multiphase materials we adapt the entropic descriptor (ED) of a pillar model developed to greyscale images. To uncover the contribution of each phase we introduce the suitable 'phase splitting' of the adapted descriptor. As a result, each of the phase descriptors (PDs) describes the spatial inhomogeneity attributed to each phase-component. Obviously, their sum equals to the value of the overall spatial inhomogeneity. We apply this approach to three-phase synthetic patterns. The black and grey components are aggregated or clustered while the white phase is the background one. The examples show how the valuable microstuctural information related…
Multitype spatial point patterns with hierarchical interactions.
2001
Multitype spatial point patterns with hierarchical interactions are considered. Here hierarchical interaction means directionality: points on a higher level of hierarchy affect the locations of points on the lower levels, but not vice versa. Such relations are common, for example, in ecological communities. Interacting point patterns are often modeled by Gibbs processes with pairwise interactions. However, these models are inherently symmetric, and the hierarchy can be acknowledged only when interpreting the results. We suggest the following in allowing the inclusion of the hierarchical structure in the model. Instead of regarding the pattern as a realization of a stationary multivariate po…
Pairwise Markov properties for regression graphs
2016
With a sequence of regressions, one may generate joint probability distributions. One starts with a joint, marginal distribution of context variables having possibly a concentration graph structure and continues with an ordered sequence of conditional distributions, named regressions in joint responses. The involved random variables may be discrete, continuous or of both types. Such a generating process specifies for each response a conditioning set that contains just its regressor variables, and it leads to at least one valid ordering of all nodes in the corresponding regression graph that has three types of edge: one for undirected dependences among context variables, another for undirect…
Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers
2018
We establish quantitative bounds for rates of convergence and asymptotic variances for iterated conditional sequential Monte Carlo (i-cSMC) Markov chains and associated particle Gibbs samplers. Our main findings are that the essential boundedness of potential functions associated with the i-cSMC algorithm provide necessary and sufficient conditions for the uniform ergodicity of the i-cSMC Markov chain, as well as quantitative bounds on its (uniformly geometric) rate of convergence. Furthermore, we show that the i-cSMC Markov chain cannot even be geometrically ergodic if this essential boundedness does not hold in many applications of interest. Our sufficiency and quantitative bounds rely on…
Affine-invariant rank tests for multivariate independence in independent component models
2016
We consider the problem of testing for multivariate independence in independent component (IC) models. Under a symmetry assumption, we develop parametric and nonparametric (signed-rank) tests. Unlike in independent component analysis (ICA), we allow for the singular cases involving more than one Gaussian independent component. The proposed rank tests are based on componentwise signed ranks, à la Puri and Sen. Unlike the Puri and Sen tests, however, our tests (i) are affine-invariant and (ii) are, for adequately chosen scores, locally and asymptotically optimal (in the Le Cam sense) at prespecified densities. Asymptotic local powers and asymptotic relative efficiencies with respect to Wilks’…
On (n-l)-wise and joint independence and normality of n Random variables: an example
1981
An example is given of a vector of n random variables such that any (n-1)-dimensional subvector consists of n-1 independent standard normal variables. The whole vector however is neither independent nor normal.
The Concept of Duality and Applications to Markov Processes Arising in Neutral Population Genetics Models
1999
One possible and widely used definition of the duality of Markov processes employs functions H relating one process to another in a certain way. For given processes X and Y the space U of all such functions H, called the duality space of X and Y, is studied in this paper. The algebraic structure of U is closely related to the eigenvalues and eigenvectors of the transition matrices of X and Y. Often as for example in physics (interacting particle systems) and in biology (population genetics models) dual processes arise naturally by looking forwards and backwards in time. In particular, time-reversible Markov processes are self-dual. In this paper, results on the duality space are presented f…
On the empirical spectral distribution for certain models related to sample covariance matrices with different correlations
2021
Given [Formula: see text], we study two classes of large random matrices of the form [Formula: see text] where for every [Formula: see text], [Formula: see text] are iid copies of a random variable [Formula: see text], [Formula: see text], [Formula: see text] are two (not necessarily independent) sets of independent random vectors having different covariance matrices and generating well concentrated bilinear forms. We consider two main asymptotic regimes as [Formula: see text]: a standard one, where [Formula: see text], and a slightly modified one, where [Formula: see text] and [Formula: see text] while [Formula: see text] for some [Formula: see text]. Assuming that vectors [Formula: see t…