Ergodicity of the Finite and Infinite Dimensional α-Stable Systems

Abstract

Some finite and infinite dimensional perturbed α-stable dynamics are constructed and studied in this article. We prove that the finite dimensional system is strongly mixing, while in the infinite dimensional case that the functional coercive inequalities are not available, we develop and apply a technique to prove the point-wise ergodicity for systems with sufficiently small interaction in a large subspace of Ω = R ^{Z d} .     

Central Limit Theorem for Nonlinear Semi-Markov Reward Processes

Abstract

In this work, we obtain a central limit theorem for reward processes defined on a finite state space semi-Markov process, when reward functions assumed to have general forms and are not of constant rates. Martingale theory is the main tool which have been used for establishing the convergence of scaled and shifted reward process to a zero mean Brownian motion. The striking point in this article is considering general forms for the reward functions which are realistic in applications. The conditions needed for these results are existence of variances for sojourn times in each state and second order integrability of reward functions with respect to sojourn times distributions.     

Isoperimetric problems and roses of neighborhood for stationary flat processes

The paper yields necessary conditions for the directional distributions of stationary k–flat processes in ?^d that maximize their intersection density of order 2, that is, the mean (2k – d)–volume of their self–intersections in an observation window of unit d–volume. The conditions are given in terms of the rose of intersections (i.e., the intensity of intersections of the flat process with test flats). The notion of the rose of neighborhood is introduced which is an analogue of the rose of intersections for lower dimensional flat processes. Some properties of the rose of neighborhood are studied and an asymptotically unbiased estimator is given.     

Classification et mélanges de processusClustering and mixtures of processes

We propose a clustering method based on the estimation of mixtures of probability distributions, the new point being that the statistical units are described by probability distributions. The components of the mixtures are Dirichlet processes, normalized weighted Gamma processes, and Kraft processes. Mixtures obtained by applying some algorithms to the finite dimensional distributions of the components converge to the desired mixture as the dimension increases, since the components are mutually singular due to a theorem of Kakutani. The desired clusters are then the support of these components. To cite this article: R. Emilion, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 189–193.     

Diffusion Approximations for Queues with Markovian Bases

Consider a base family of state-dependent queues whose queue-length process can be formulated by a continuous-time Markov process. In this paper, we develop a piecewise-constant diffusion model for an enlarged family of queues, each of whose members has arrival and service distributions generalized from those of the associated queue in the base. The enlarged family covers many standard queueing systems with finite waiting spaces, finite sources and so on. We provide a unifying explicit expression for the steady-state distribution, which is consistent with the exact result when the arrival and service distributions are those of the base. The model is an extension as well as a refinement of the M/M/s-consistent diffusion model for the GI/G/s queue developed by Kimura [13] where the base was a birth-and-death process. As a typical base, we still focus on birth-and-death processes, but we also consider a class of continuous-time Markov processes with lower-triangular infinitesimal generators.     

Harmonic and Probabilistic Approaches to Zeros of Riemann's Zeta Function

Probability concepts and results are closely related to the study of zeros of the classical Riemann zeta function and its affinity to Gaussian and Gamma distributions. This is elaborated in obtaining the functional and integral equations for the zeta and in the determination first of the nonzero sets and then sets containing almost all (i.e., for the CLT probability measure) nontrivial zeros of the zeta function ζ(·). Also probability distributions determined by the zeta, based on the behavior of their finite dimensional distributions of the ζ(σ +it), σ > 0; as t varies and particularly the results of Denjoy, slightly sharpened, and also one of Salem are included. Several related opinions and comments are discussed.     

Point processes characterized by their one dimensional distributions

It is well-known that the distribution of a point process defined on a carrier space is uniquely characterised by its finite dimensional joint distributions of counts on disjoint subsets of . In this note, we investigate the common structure of point processes whose distributions are specified by their one dimensional distributions. We also show that, if is such a point process, then a sequence of point processes { _n } converges in distribution to if and only if { _n (B)} converges in distribution to (B) for a suitably rich class of sets B. Supported by ARC Discovery project number DP0209179 Mathmatics Subject Classification (2000):Primary 60G55; Secondary 60E05, 60B10 AcknowledgementI would like to thank a referee for his valuable suggestions on the presentation of this paper.     

On the approximation of continuous time threshold ARMA processes

Threshold autoregressive (AR) and autoregressive moving average (ARMA) processes with continuous time parameter have been discussed in several recent papers by Brockwellet al. (1991,Statist. Sinica,1, 401–410), Tong and Yeung (1991,Statist. Sinica,1, 411–430), Brockwell and Hyndman (1992,International Journal Forecasting,8, 157–173) and Brockwell (1994,J. Statist. Plann. Inference,39, 291–304). A threshold ARMA process with boundary width 2>0 is easy to define in terms of the unique strong solution of a stochastic differential equation whose coefficients are piecewise linear and Lipschitz. The positive boundary-width is a convenient mathematical device to smooth out the coefficient changes at the boundary and hence to ensure the existence and uniqueness of the strong solution of the stochastic differential equation from which the process is derived. In this paper we give a direct definition of a threshold ARMA processes with =0 in the important case when only the autoregressive coefficients change with the level of the process. (This of course includes all threshold AR processes with constant scale parameter.) The idea is to express the distributions of the process in terms of the weak solution of a certain stochastic differential equation. It is shown that the joint distributions of this solution with =0 are the weak limits as 0 of the distributions of the solution with >0. The sense in which the approximating sequence of processes used by Brockwell and Hyndman (1992,International Journal Forecasting,8, 157–173) converges to this weak solution is also investigated. Some numerical examples illustrate the value of the latter approximation in comparison with the more direct representation of the process obtained from the Cameron-Martin-Girsanov formula. It is used in particular to fit continuous-time threshold models to the sunspot and Canadian lynx series.Research partially supported by National Science Foundation Research Grants DMS 9105745 and 9243648.     

On processes of ornstein-uhlenbeck type in hilbert space

A process fo Ornstein-Uhlenbeck type is a mild solution of the stochastic differential system in Hilbert space dXt=AX _t dt+dZ _t, where A generates a semigroup of operators and Z _tis a process with homogeneous independent increments. The explicit integral formula for the process of O-U type is given. The main purpose is to study stationary distributions for such processes. Sufficient and necessary conditions for existence and characterization are given. The difference between finite and infinite dimensional cases is illustrated by examples     

Estimation in Stationary Markov Renewal Processes,with Application to Earthquake Forecasting in Turkey

Consider a process in which different events occur, with random inter-occurrence times. In Markov renewal processes as well as in semi-Markov processes, the sequence of events is a Markov chain and the waiting distributions depend only on the types of the last and the next event. Suppose that the state-space is finite and that the process started far in the past, achieving stationary. Weibull distributions are proposed for the waiting times and their parameters are estimated jointly with the transition probabilities through maximum likelihood, when one or several realizations of the process are observed over finite windows. The model is illustrated with data of earthquakes of three types of severity that occurred in Turkey during the 20th century.AMS 2000 Subject Classification: 60K20     

Hyperfinite Lévy Processes

A hyperfinite Lévy process is an infinitesimal random walk (in the sense of nonstandard analysis) which with probability one is finite for all finite times. We develop the basic theory for hyperfinite Lévy processes and find a characterization in terms of transition probabilities. The standard part of a hyperfinite Lévy process is a (standard) Lévy process, and we show that given a generating triplet (γ, C, μ) for standard Lévy processes, we can construct hyperfinite Lévy processes whose standard parts correspond to this triplet. Hence all Lévy laws can be obtained from hyperfinite Lévy processes. The paper ends with a brief look at Malliavin calculus for hyperfinite Lévy processes including a version of the Clark-Haussmann-Ocone formula.     

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.     

Functional limit theorems for a new class of non-stationary shot noise processes

We study a class of non-stationary shot noise processes which have a general arrival process of noises with non-stationary arrival rate and a general shot shape function. Given the arrival times, the shot noises are conditionally independent and each shot noise has a general (multivariate) cumulative distribution function (c.d.f.) depending on its arrival time. We prove a functional weak law of large numbers and a functional central limit theorem for this new class of non-stationary shot noise processes in an asymptotic regime with a high intensity of shot noises, under some mild regularity conditions on the shot shape function and the conditional (multivariate) c.d.f. We discuss the applications to a simple multiplicative model (which includes a class of non-stationary compound processes and applies to insurance risk theory and physics) and the queueing and work-input processes in an associated non-stationary infinite-server queueing system. To prove the weak convergence, we show new maximal inequalities and a new criterion of existence of a stochastic process in the space $\mathbb{D}$ given its consistent finite dimensional distributions, which involve a finite set function with the superadditive property.     

The Periodic BMAP/PH/c Queue

In queueing theory, most models are based on time-homogeneous arrival processes and service time distributions. However, in communication networks arrival rates and/or the service capacity usually vary periodically in time. In order to reflect this property accurately, one needs to examine periodic rather than homogeneous queues. In the present paper, the periodic BMAP/PH/c queue is analyzed. This queue has a periodic BMAP arrival process, which is defined in this paper, and phase-type service time distributions. As a Markovian queue, it can be analysed like an (inhomogeneous) Markov jump process. The transient distribution is derived by solving the Kolmogorov forward equations. Furthermore, a stability condition in terms of arrival and service rates is proven and for the case of stability, the asymptotic distribution is given explicitly. This turns out to be a periodic family of probability distributions. It is sketched how to analyze the periodic BMAP/M _t /c queue with periodically varying service rates by the same method.     

Functional central limit theorem for super α-stable processes

A functional central limit theorem is proved for the centered occupation time process of the super α-stable processes in the finite dimensional distribution sense. For the intermediate dimensions α < d < 2α (0 < α ≤ 2), the limiting process is a Gaussian process, whose covariance is specified; for the critical dimension d= 2α and higher dimensions d < 2α, the limiting process is Brownian motion. Zhang Mei, Functional central limit theorem for the super-brownian motion with super-Brownian immigration, J. Theoret. Probab., to appear.     

Deterministic Noises that can be Statistically Distinguished from the Random Ones

Empirical measures generated by random sequences with deterministic and random noises have same asymptotic distributions provided that the noises have same asymptotic distributions (cf., Davydov and Zitikis, 2004, Proc. Am. Math. Soc. 132, 1203–1210). This phenomenon has raised an intriguing question about the possibility of distinguishing the two types of noises based only on their asymptotic distributions. In the present paper we suggest an answer to the question by considering asymptotic variances, and distributions, of the appropriately centered and normalized empirical measures and processes. In final form 6 January 2005     

On Geometric Infinite Divisibility and Stability

The purpose of this paper is to study geometric infinite divisibility and geometric stability of distributions with support in Z ₊ and R ₊. Several new characterizations are obtained. We prove in particular that compound-geometric (resp. compound-exponential) distributions form the class of geometrically infinitely divisible distributions on Z ₊ (resp. R ₊). These distributions are shown to arise as the only solutions to a stability equation. We also establish that the Mittag-Leffler distributions characterize geometric stability. Related stationary autoregressive processes of order one (AR(1)) are constructed. Importantly, we will use Poisson mixtures to deduce results for distributions on R ₊ from those for their Z ₊-counterparts.     

Study of Dependence for Some Stochastic Processes

Abstract

This article is concerned with studying the following problem: Consider a multivariate stochastic process whose law is characterized in terms of some infinitesimal characteristics, such as the infinitesimal generator in case of finite Markov chains. Under what conditions imposed on these infinitesimal characteristics of this multivariate process, the univariate components of the process agree in law with given univariate stochastic processes. Thus, in a sense, we study a stochastic processe' counterpart of the stochastic dependence problem, which in case of real valued random variables is solved in terms of Sklar's theorem.     

One-armed bandit models with continuous and delayed responses

One-armed bandit processes with continuous delayed responses are formulated as controlled stochastic processes following the Bayesian approach. It is shown that under some regularity conditions, a Gittins-like index exists which is the limit of a monotonic sequence of break-even values characterizing optimal initial selections of arms for finite horizon bandit processes. Furthermore, there is an optimal stopping solution when all observations on the unknown arm are complete. Results are illustrated with a bandit model having exponentially distributed responses, in which case the controlled stochastic process becomes a Markov decision process, the Gittins-like index is the Gittins index and the Gittins index strategy is optimal. Acknowledgement.We thank an anonymous referee for constructive and insightful comments, especially those related to the notion of the Gittins index.Both authors are funded by the Natural Sciences and Engineering Research Council (NSERC) of Canada.