Trajectory composition of Poisson time changes and Markov counting systems

Changing time of simple continuous-time Markov counting processes by independent unit-rate Poisson processes results in Markov counting processes for which we provide closed-form transition rates via composition of trajectories and with which we construct novel, simpler infinitesimally over-dispersed processes.     

On the infinitesimal dispersion of multivariate Markov counting systems

We provide a multivariate extension of a recent result for univariate Markov counting processes: necessity and sufficiency of compoundness for infinitesimal over-dispersion. As an illustration, we show that infinitesimally over-dispersed epidemiological SIR-type compartment models must rely on compound counting processes.     

Integer Valued Autoegressive Process with Katz Arrivals and Its Application in Predicting the Count of the Cases of Respiratory Disease

The traditional PAR process (Poisson autoregressive process) assumes that the arrival process is the equi-dispersed Poisson process, with its mean being equal to its variance. Whereas the arrival process in the real DGP (data generating process) could either be over-dispersed, with variance being greater than the mean, or under-dispersed, with variance being less than the mean. This paper proposes using the Katz family distributions to model the arrival process in the INAR process (integer valued autoregressive process with Katz arrivals) and deploying Monte Carlo simulations to examine the performance of maximum likelihood (ML) and method of moments (MM) estimators of INAR-Katz model. Finally, we used the INAR-Katz process to model count data of hospital emergency room visits for respiratory disease. The results show that the INAR-Katz model outperforms the Poisson model, PAR(1) model, and has great potential in empirical application.     

Infinitesimally flexible meshes and discrete minimal surfaces

We explore the geometry of isothermic meshes, conical meshes, and asymptotic meshes around the Christoffel dual construction of a discrete minimal surface. We present a discrete Legendre transform which realizes discrete minimal surfaces as conical meshes. Conical meshes turn out to be infinitesimally flexible if and only if their spherical image is isothermic, which implies that discrete minimal surfaces constructed in this way are infinitesimally flexible, and therefore possess reciprocal-parallel meshes. These are discrete minimal surfaces in their own right. In our study of relative kinematics of infinitesimally flexible meshes, we encounter characterizations of flexibility and isothermicity which are of incidence-geometric nature and are related to the classical Desargues configuration. The Lelieuvre formula for asymptotic meshes leads to another characterization of isothermic meshes in the sphere which is based on triangle areas. Authors’ addresses: Johannes Wallner (corresponding author), Institut für Geometrie, TU Graz, Kopernikusgasse 24, A 8010 Graz, Austria; Helmut Pottmann, Institut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner Hauptstr. 8-10/104, A 1040 Wien, Austria     

Strongly stable real infinitesimally symplectic mappings

We prove that a mapA εsp(σ,R), the set of infinitesimally symplectic maps, is strongly stable if and only if its centralizerC(A) insp(σ,R) contains only semisimple elements. Using the theorem that everyB insp(σ,R) close toA is conjugate by a real symplectic map to an element ofC(A), we give a new proof of the openness of the set of strongly stable maps. Then we prove that the set of strongly stable maps is the interior of the set of all infinitesimally symplectic maps with purely imaginary or zero eigenvalues, and the connected components of this set are described. Finally, we give a new proof of the analytic conjugacy theorem for an analytic curve through a given strongly stable map.     

Critical Bellman-Harris branching processes with long-living particles

A critical indecomposable two-type Bellman-Harris branching process is considered in which the life-length of the first-type particles has finite variance while the tail of the life-length distribution of the second-type particles is regularly varying at infinity with parameter β ∈ (0, 1]. It is shown that, contrary to the critical indecomposable Bellman-Harris branching processes with finite variances of the life-lengths of particles of both types, the probability of observing first-type particles at a distant moment t is infinitesimally less than the survival probability of the whole process. In addition, a Yaglom-type limit theorem is proved for the distribution of the number of the first-type particles at moment t given that the population contains particles of the first type at this moment.     

Generation of Over-Dispersed and Under-Dispersed Binomial Variates

Abstract

This article proposes an algorithm for generating over-dispersed and under-dispersed binomial variates with specified mean and variance. The over-dispersed/under-dispersed distributions are derived from correlated binary variables with an underlying continuous multivariate distribution. Different multivariate distributions or different correlation matrices result in different over-dispersed (or under-dispersed) distributions. The over-dispersed binomial distributions that are generated from three different correlation matrices of a multivariate normal are compared with the beta-binomial distribution for various mean and over-dispersion parameters by quantile-quantile (Q-Q) plots. The two distributions appear to be similar. The under-dispersed binomial distribution is simulated to model an example data set that exhibits under-dispersed binomial variation.     

Infinitesimally flexible meshes and discrete minimal surfaces

We explore the geometry of isothermic meshes, conical meshes, and asymptotic meshes around the Christoffel dual construction of a discrete minimal surface. We present a discrete Legendre transform which realizes discrete minimal surfaces as conical meshes. Conical meshes turn out to be infinitesimally flexible if and only if their spherical image is isothermic, which implies that discrete minimal surfaces constructed in this way are infinitesimally flexible, and therefore possess reciprocal-parallel meshes. These are discrete minimal surfaces in their own right. In our study of relative kinematics of infinitesimally flexible meshes, we encounter characterizations of flexibility and isothermicity which are of incidence-geometric nature and are related to the classical Desargues configuration. The Lelieuvre formula for asymptotic meshes leads to another characterization of isothermic meshes in the sphere which is based on triangle areas.     

On the existence of moments of solutions to fragmentation equations

We consider the multiple fragmentation equations with polynomially bounded fragmentation rates, both in the discrete and continuous cases. The theory of semigroups of operators on Fréchet spaces is used to produce a simple proof that if moments of all non-negative orders of solutions are initially finite then they remain finite for all future times. Moreover, a class of fragmentation processes is identified in which the existence of the first moment of the initial distribution suffices for the existence of all other moments for positive times.     

Generalized Nonparametric Mixed Effects Models

Generalized linear mixed effects models (GLMM) provide useful tools for correlated and/or over-dispersed non-Gaussian data. This article considers generalized nonparametric mixed effects models (GNMM), which relax the rigid linear assumption on the conditional predictor in a GLMM. We use smoothing splines to model fixed effects. The random effects are general and may also contain stochastic processes corresponding to smoothing splines. We show how to construct smoothing spline ANOVA (SS ANOVA) decompositions for the predictor function. Components in a SS ANOVA decomposition have nice interpretations as main effects and interactions. Experimental design considerations help determine which components are fixed or random. We estimate all parameters and spline functions using stochastic approximation with Markov chain Monte Carlo (MCMC). As iteration increases we increase the MCMC sample size and decrease the step-size of the parameter update. This approach guarantees convergence of the estimates to the expected fixed points. We evaluate our methods through a simulation study.     

α-TRANSIENCE AND α-RECURRENCE FOR RANDOM WALKS AND LEVY PROCESSES

The authors investigate the α-transience and α-recurrence for random walks and Levy processes by means of the associated moment generating function, give a dichotomy theorem for not one-sided processes and prove that the process X is quasi-symmetric if and only if X is not α-recurrent for all α< 0 which gives a probabilistic explanation of quasi-symmetry, a concept originated from C. J. Stone.     

On the stability of open networks: A unified approach by stochastic dominance

Using stochastic dominance, in this paper we provide a new characterization of point processes. This characterization leads to a unified proof for various stability results of open Jackson networks where service times are i.i.d. with a general distribution, external interarrivai times are i.i.d. with a general distribution and the routing is Bernoulli. We show that if the traffic condition is satisfied, i.e., the input rate is smaller than the service rate at each queue, then the queue length process (the number of customers at each queue) is tight. Under the traffic condition, the pth moment of the queue length process is bounded for allt if the p+lth moment of the service times at all queues are finite. If, furthermore, the moment generating functions of the service times at all queues exist, then all the moments of the queue length process are bounded for allt. When the interarrivai times are unbounded and non-lattice (resp. spreadout), the queue lengths and the remaining service times converge in distribution (resp. in total variation) to a steady state. Also, the moments converge if the corresponding moment conditions are satisfied.     

Strong Martingales: Their Decompositions and Quadratic Variation

Set-indexed strong martingales and a form of predictability for set-indexed processes are defined. Under a natural integrability condition, we show that any set-indexed strong submartingale can be decomposed in the Doob–Meyer sense. A form of predictable quadratic variation for square-integrable set-indexed strong martingales is defined and sufficient conditions for its existence are given. Under a conditional independence assumption, these reduce to a simple moment condition and, if the strong martingale has continuous sample paths, the resulting quadratic variation can be approximated in the L ₂-sense by sums of conditional expectations of squared increments.     

Lévy-Driven Carma Processes

Properties and examples of continuous-time ARMA (CARMA) processes driven by Lévy processes are examined. By allowing Lévy processes to replace Brownian motion in the definition of a Gaussian CARMA process, we obtain a much richer class of possibly heavy-tailed continuous-time stationary processes with many potential applications in finance, where such heavy tails are frequently observed in practice. If the Lévy process has finite second moments, the correlation structure of the CARMA process is the same as that of a corresponding Gaussian CARMA process. In this paper we make use of the properties of general Lévy processes to investigate CARMA processes driven by Lévy processes {W(t)} without the restriction to finite second moments. We assume only that W (1) has finite r-th absolute moment for some strictly positive r. The processes so obtained include CARMA processes with marginal symmetric stable distributions.     

On left regular and intra-regular ordered semigroups

We study the decomposition of left regular ordered semigroups into left regular components and the decomposition of intra-regular ordered semigroups into simple or intra-regular components, adding some additional information to the results considered in [KEHAYOPULU, N.: On left regular ordered semigroups, Math. Japon. 35 (1990), 1057–1060] and [KEHAYOPULU, N.: On intra-regular ordered semigroups, Semigroup Forum 46 (1993), 271–278]. We prove that an ordered semigroup S is left regular if and only if it is a semilattice (or a complete semilattice) of left regular semigroups, equivalently, it is a union of left regular subsemigroups of S. Moreover, S is left regular if and only if it is a union of pairwise disjoint left regular subsemigroups of S. The right analog also holds. The same result is true if we replace the words “left regular” by “intraregular”. Moreover, an ordered semigroup is intra-regular if and only if it is a semilattice (or a complete semilattice) of simple semigroups. On the other hand, if an ordered semigroup is a semilattice (or a complete semilattice) of left simple semigroups, then it is left regular, but the converse statement does not hold in general. Illustrative examples are given.     

α-TRANSIENCE AND α-RECURRENCE FOR RANDOM WALKS AND L(E)VY PROCESSES

The authors investigate the α-transience and α-recurrence for random walks and Levy processes by means of the associated moment generating function, give a dichotomy theorem for not one-sided processes and prove that the process X is quasisymmetric if and only if X is not α-recurrent for all α＜ 0 which gives a probabilistic explanation of quasi-symmetry, a concept originated from C. J. Stone.     

Convergence rates for probabilities of moderate deviations for moving average processes

The present paper first shows that, without any dependent structure assumptions for a sequence of random variables, the refined results of the complete convergence for the sequence is equivalent to the corresponding complete moment convergence of the sequence. Then this paper investigates the convergence rates and refined convergence rates （or complete moment convergence） for probabilities of moderate deviations of moving average processes. The results in this paper extend and generalize some well-known results.     

Global Rigidity: The Effect of Coning

Recent results have confirmed that the global rigidity of bar-and-joint frameworks on a graph G is a generic property in Euclidean spaces of all dimensions. Although it is not known if there is a deterministic algorithm that runs in polynomial time and space, to decide if a graph is generically globally rigid, there is an algorithm (Gortler et al. in Characterizing generic global rigidity, arXiv:, 2007) running in polynomial time and space that will decide with no false positives and only has false negatives with low probability. When there is a framework that is infinitesimally rigid with a stress matrix of maximal rank, we describe it as a certificate which guarantees that the graph is generically globally rigid, although this framework, itself, may not be globally rigid. We present a set of examples which clarify a number of aspects of global rigidity.     

On Sobolev infinitesimal rigidity of linear hyperbolic actions on the 2-torus*

Let A be a symmetric hyperbolic matrix in SL(2, ℤ) and Γ the subgroup of SL(2, ℤ) generated by A. We aim to study the infinitesimal rigidity of the standard action of Γ on the torus . More precisely, we will consider the Sobolev W^s–infinitesimal rigidity of this action (that is to determine if the cohomology space H¹(Γ,W^s (T M)) is trivial or not), and show that it is W^s–infinitesimally rigid only if 0 ≤ s < 1. A consequence will be that this action is not C^∞–infinitesimally rigid. *I would like to thank A. El Kacimi for introducing me this problem about which we had many fruitful discussions.     

Infinitesimal Rigidity of Convex Polytopes

Aleksandrov [1] proved that a simple convex d -dimensional polytope, d ≥ 3 , is infinitesimally rigid if the volumes of its facets satisfy a certain assumption of stationarity. We extend this result by proving that this assumption can be replaced by a stationarity assumption on the k -dimensional volumes of the polytope's k -dimensional faces, where k ∈{1,. . .,d-1} . Received November 20, 1997.