Large Deviation Probabilities for Square-Gaussian Stochastic Processes

We investigate properties of square-Gaussian stochastic processes. These processes are formed by quadratic forms of Gaussian processes or by limits in the mean square of quadratic forms of Gaussian processes. Special classes of these processes are determined and investigated. For processes from these classes estimates of large deviation probability are obtained. These estimates we use to estimate the probability that Gaussian vector-valued process leave some region on some interval of time. We construct asymptotic confidence regions for estimates of covariance functions of vector-valued Gaussian processes. Criterion of hypothesis testing on covariance functions of these processes is constructed.     

Doubly Feller Property of Brownian Motions with Robin Boundary Condition

In this paper, we consider first order Sobolev spaces with Robin boundary condition on unbounded Lipschitz domains. Hunt processes are associated with these spaces. We prove that the semigroup of these processes are doubly Feller. As a corollary, we provide a condition for semigroups generated by these processes being compact.

     

On a class of self-similar processes with stationary increments in higher order Wiener chaoses

We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local Hölder regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener–Itô integrals.     

Comparing equivalences on precubical sets and spaces

We study equivalences of concurrent processes represented by objects of algebraic topology. We use methods of category theory and consider precubical sets (analogs of semisimplicial sets) and precubical spaces (analogs of cell complexes). In particular, we consider categories of these objects and construct subcategories of path-objects. We define open morphisms with respect to these subcategories and formulate criteria for a morphism to be open. We prove that the equivalence of precubical sets (spaces) based on open morphisms coincides with a behavioral equivalence of concurrent processes.     

Approximations to permutation and exchangeable processes

We obtain weighted approximations by a Brownian bridge to permutation and exchangeable processes and to appropriately defined inverse processes. Our results provide as special cases useful weighted approximations to the uniform empirical and quantile processes and to generalized bootstrapped versions of these processes. A number of other applications are discussed. Our approach is based on the Skorokhod embedding for martingales.     

On the law of the iterated logarithm for Gaussian processes

We present some optimal conditions for the compact law of the iterated logarithm of a sequence of jointly Gaussian processes in different situations. We also discuss the local law of the iterated logarithm for Gaussian processes indexed by arbitrary index sets, in particular for self-similar Gaussian processes. We apply these results to obtain the law of the iterated logarithm for compositions of Gaussian processes. Research partially supported by NSF Grant DMS-93-02583.     

Forward Brownian motion

We consider processes which have the distribution of standard Brownian motion (in the forward direction of time) starting from random points on the trajectory which accumulate at \(-\infty \) . We show that these processes do not have to have the distribution of standard Brownian motion in the backward direction of time, no matter which random time we take as the origin. We study the maximum and minimum rates of growth for these processes in the backward direction. We also address the question of which extra assumptions make one of these processes a two-sided Brownian motion.     

Two-limit problems for almost semicontinuous processes defined on a Markov chain

We consider almost upper-semicontinuous processes defined on a finite Markov chain. The distributions of functionals associated with the exit of these processes from a finite interval are studied. We also consider some modifications of these processes. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 4, pp. 555–565, April, 2007.     

Limit theorems for path-functionals of regenerative processes

Regenerative processes were defined and investigated by Smith [12]. These processes have limiting distributions under very mild regularity conditions. In certain applications, such as shot-noise processes and some queueing problems, it is of interest to consider path-functionals of regenerative processes. We seek to extend the nice asymptotic properties of regenerative processes to path-functionals of regenerative processes. We show that these more general processes converge to a “steady-state” process in a certain weak sense. This is applied to show convergence of shot-noise processes. We also present a Blackwell theorem for path-functionals of regenerative processes.     

Goodness-of-fit test for switching diffusion

We consider two Cramér–von Mises goodness-of-fit tests for hypotheses that the observed diffusion process has sign-type trend coefficient based on empirical distribution function and empirical density function. It is shown that the limit distributions of the proposed tests statistics are defined by the integral type functionals of continuous Gaussian processes. We study the behavior of these statistics under the alternative hypothesis and we prove that the tests are consistent. We provide the Karhunen-Loève expansion on \mathbbR{\mathbb{R}} of the corresponding limiting processes and we show that the eigenfunctions in these expansions have expressions in term of Bessel functions.     

Reduced-load equivalence for Gaussian processes

We consider a fluid model fed by two Gaussian processes. We obtain necessary and sufficient conditions for the workload asymptotics to be completely determined by one of the two processes, and apply these results to the case of two fractional Brownian motions.     

Closure of Linear Processes

We consider the sets of moving-average and autoregressive processes and study their closures under the Mallows metric and the total variation convergence on finite dimensional distributions. These closures are unexpectedly large, containing nonergodic processes which are Poisson sums of i.i.d. copies from a stationary process. The presence of these nonergodic Poisson sum processes has immediate implications. In particular, identifiability of the hypothesis of linearity of a process is in question. A discussion of some of these issues for the set of moving-average processes has already been given without proof in Bickel and Bühlmann.⁽²⁾ We establish here the precise mathematical arguments and present some additional extensions: results about the closure of autoregressive processes and natural sub-sets of moving-average and autoregressive processes which are closed.Research supported in part by grants NSA MDA 904-94-H-2020 and NSF DMS 95049555     

Non-extensive thermodynamics and stationary processes of localization

We focus our attention on dynamical processes characterized by an entropic index Q<1. According to the probabilistic arguments of Tsallis, C and Bukman, DJ [Phys Rev E 1996;54:R2197] these processes are subdiffusional in nature. The non-extensive generalization of the Kolmogorov–Sinai (KS) entropy yielding the same entropic index implies the stationary condition. We note, on the other hand, that enforcing the stationary property on subdiffusion has the effect of producing a localization process occurring within a finite time scale. We thus conclude that the stationary dynamic processes with Q<1 must undergo a localization process occurring at a finite time. We check the validity of this conclusion by means of a numerical treatment of the dynamics of the logistic map at the critical point.     

Small deviations of iterated processes in the space of trajectories

We derive logarithmic asymptotics of probabilities of small deviations for iterated processes in the space of trajectories. We find conditions under which these asymptotics coincide with those of processes generating iterated processes. When these conditions fail the asymptotics are quite different.     

Simulation Studies of Some Voronoi Point Processes

We introduce a new class of dynamic point process models with simple and intuitive dynamics that are based on the Voronoi tessellations generated by the processes. Under broad conditions, these processes prove to be ergodic and produce, on stabilisation, a wide range of clustering patterns. In the paper, we present results of simulation studies of three statistical measures (Thiel’s redundancy, van Lieshout and Baddeley’s J-function and the empirical distribution of the Voronoi nearest neighbours’ numbers) for inference on these models from the clustering behaviour in the stationary regime. In particular, we make comparisons with the area-interaction processes of Baddeley and van Lieshout.     

A new family of time-space harmonic polynomials with respect to Lévy processes

By means of a symbolic method, a new family of time-space harmonic polynomials with respect to Lévy processes is given. The coefficients of these polynomials involve a formal expression of Lévy processes by which many identities are stated. We show that this family includes classical families of polynomials such as Hermite polynomials. Poisson–Charlier polynomials result to be a linear combinations of these new polynomials, when they have the property to be time-space harmonic with respect to the compensated Poisson process. The more general class of Lévy–Sheffer polynomials is recovered as a linear combination of these new polynomials, when they are time-space harmonic with respect to Lévy processes of very general form. We show the role played by cumulants of Lévy processes, so that connections with boolean and free cumulants are also stated.     

Predator–Prey Dynamics on Infinite Trees: A Branching Random Walk Approach

We are interested in predator–prey dynamics on infinite trees, which can informally be seen as particular two-type branching processes where individuals may die (or be infected) only after their parent dies (or is infected). We study two types of such dynamics: the chase–escape process, introduced by Kordzakhia with a variant by Bordenave who sees it as a rumor propagation model, and the birth-and-assassination process, introduced by Aldous and Krebs. We exhibit a coupling between these processes and branching random walks killed at the origin. This sheds new light on the chase–escape and birth-and-assassination processes, which allows us to recover by probabilistic means previously known results and also to obtain new results. For instance, we find the asymptotic behavior of the tail of the number of infected individuals in both the subcritical and critical regimes for the chase–escape process and show that the birth-and-assassination process ends almost surely at criticality.     

Reduction principles for quantile and Bahadur–Kiefer processes of long-range dependent linear sequences

In this paper we consider quantile and Bahadur–Kiefer processes for long range dependent linear sequences. These processes, unlike in previous studies, are considered on the whole interval (0, 1). As it is well-known, quantile processes can have very erratic behavior on the tails. We overcome this problem by considering these processes with appropriate weight functions. In this way we conclude strong approximations that yield some remarkable phenomena that are not shared with i.i.d. sequences, including weak convergence of the Bahadur–Kiefer processes, a different pointwise behavior of the general and uniform Bahadur–Kiefer processes, and a somewhat “strange” behavior of the general quantile process.     

Periodically correlated autoregressive Hilbertian processes

We consider periodically correlated autoregressive processes in Hilbert spaces. Our studies on these processes involve existence, covariance structure, estimation of the covariance operators, strong law of large numbers and central limit theorem.