What is a multi-parameter renewal process?

The concept of the renewal property is extended to processes indexed by a multidimensional time parameter. The definition given includes not only partial sum processes, but also Poisson processes and many other point processes whose jump points are not totally ordered. A new version of the waiting time paradox is proven for multidimensional Poisson processes, and is shown to imply the renewal property. Finally, martingale properties of renewal processes are studied.     

Measuring deviations from stationarity

Various parameters for measuring the deviation from stationarity for processes belonging to two classes of nonstationarity processes are proposed. Several new results for the two types of processes are obtained. Points of contact are established with the class of oscillatory processes and with the Hamiltonian equation of motion in quantum mechanics. The relation to processes of normal type and to innovations stable processes is also discussed.     

Sampling and reconstruction for shift-invariant stochastic processes

In this paper, combining stochastic processes with shift-invariant spaces, we introduce shift-invariant stochastic processes. It is a general case of the classical band-limited stochastic processes and a kind of non-band-limited stochastic processes. Two sampling theorems are obtained for the shift-invariant stochastic processes. The results for band-limited stochastic processes and shift-invariant spaces are generalized by our new results.     

Catalytic Discrete State Branching Models and Related Limit Theorems

Catalytic discrete state branching processes with immigration are defined as strong solutions of stochastic integral equations. We provide main limit theorems of those processes using different scalings. The class of limit processes of the theorems includes essentially all continuous state catalytic branching processes and spectrally positive regular affine processes.      

On permanental processes

Permanental processes can be viewed as a generalization of squared centered Gaussian processes. We analyze the connections of these processes with the local time process of general Markov processes. The obtained results are related to the notion of infinite divisibility.     

Functional Central Limit Theorem for the Super-Brownian Motion with Super-Brownian Immigration

The functional central limit theorems are proved for super-Brownian motion with immigration and their occupation time processes. For the lower dimensions, the limiting processes are Gaussian processes; For the critical dimension, the limiting processes consist of two ingredient processes of different types. Interestingly, for the higher dimensions, the limiting process for the occupation time process is of a new type.     

THE STATIONARY DIFFUSION PROCESSES GENERATED BY SECOND ORDER DIFFERENTIAL OPERATIONS AND THEIR PROBABILITY CURRENTS

In this paper,we are concerned with the stationary Markov processes generated by second order differential operators under the local boundary conditions, It is proved that all those processes have constnt probability currents, known as circulations of the processes, and hence the processes are called single circulation processes. The invariant measures and the circulation values of those processes are calculated in all cases of boundary classification. It is shown that thr circulation value is an elementary characteristic of irreversible stationary Markov processes and that all the reversible Markov processes in the same problem are just the special ones of the single circulation processes whose circulation values are equal to zero and whose ergodic limits in the sense of weak convergence are not trivial.     

Group Self-Similar Stable Processes in R d

Group self-similar processes are introduced. The spectral representation of such processes in the class of multidimensional strictly stable processes is determined. The uniqueness of such representations in terms of the corresponding group actions and cocycles is established.     

Gaussian and sparse processes are limits of generalized Poisson processes

The theory of sparse stochastic processes offers a broad class of statistical models to study signals, far beyond the more classical class of Gaussian processes. In this framework, signals are represented as realizations of random processes that are solution of linear stochastic differential equations driven by Lévy white noises. Among these processes, generalized Poisson processes based on compound-Poisson noises admit an interpretation as random L-splines with random knots and weights. We demonstrate that every generalized Lévy process—from Gaussian to sparse—can be understood as the limit in law of a sequence of generalized Poisson processes. This enables a new conceptual understanding of sparse processes and suggests simple algorithms for the numerical generation of such objects.     

Markov cubature rules for polynomial processes

We study discretizations of polynomial processes using finite state Markov processes satisfying suitable moment matching conditions. The states of these Markov processes together with their transition probabilities can be interpreted as Markov cubature rules. The polynomial property allows us to study such rules using algebraic techniques. Markov cubature rules aid the tractability of path-dependent tasks such as American option pricing in models where the underlying factors are polynomial processes.     

Coupling for Markovian switching jump-diffusions

This work is concerned with coupling for a class of Markovian switching jump-diffusion processes. The processes under consideration can be regarded as a number of jump-diffusion processes modulated by a Markovian switching device. For this class of processes, we construct a successful coupling and an order-preserving coupling.     

On Quadratic g-Evaluations/Expectations and Related Analysis

This work is concerned with coupling and exponential convergence rate for a class of Markovian switching jump-diffusion processes. The processes under consideration can be thought of as a number of jump-diffusion processes modulated by a Markovian switching device. For this class of processes, we construct some order-preserving couplings. Furthermore, by virtue of the coupling results, we also provide an estimate of exponential convergence rate for the Markovian switching jump-diffusion processes without Gaussian noise.     

Entropy Maximization for Markov and Semi-Markov Processes

The literature about maximum of entropy for Markov processes deals mainly with discrete-time Markov chains. Very few papers dealing with continuous-time jump Markov processes exist and none dealing with semi-Markov processes. It is the aim of this paper to contribute to fill this lack. We recall the basics concerning entropy for Markov and semi-Markov processes and we study several problems to give an overview of the possible directions of use of maximum entropy in connection with these processes. Numeric illustrations are presented, in particular in application to reliability.     

Exponential Martingales and Changes of Measure for Counting Processes

We give sufficient criteria for the Doléans-Dade exponential of a stochastic integral with respect to a counting process local martingale to be a true martingale. The criteria are adapted particularly to the case of counting processes and are sufficiently weak to be useful and verifiable, as we illustrate by several examples. In particular, the criteria allow for the construction of for example nonexplosive Hawkes processes, counting processes with stochastic intensities depending on diffusion processes as well as inhomogeneous finite-state Markov processes.     

关于截断型分布族中估计的经验过程的渐近分布的进一步研究

但是,若分布 F 中含有未知参数θ,即 F=F(x;θ),那么为计算经验过程,就必须对θ进行适当的估计,把估计(?)_n 代入(2),(1)中,便得到一估计的经验过程(?)_n(t)。那么,这一估计的经验过程的渐近分布如何?Durbin 研究了这一问题。对一般的分布族 F(x;θ),θ∈(?),在一定的假设条件下(主要是所谓条件 A_2),他证明了这一估计的经验过程(?)(t)其渐近过程是一较复杂的正态过程,这个正态过程一般是依赖于 F 的,甚至依赖于未知参数θ的。但该文指出,当θ是位置、刻度参数时,渐近过程可与θ无关。尔后,Durbin,Schneider 等又具体地研究了指数分布族、Gamma 分布族,得到具体结果,     

The invariance principle for fractionally integrated processes with strong near-epoch dependent innovations

In this paper, we show the invariance principle for the partial sum processes of fractionally integrated processes, otherwise known as I(d + m) processes, where |d| < 1/2 and m is a nonnegative integer, with strong near-epoch dependent innovations. The results are applied to the test of unit root. The conditions given improve previous results in the literature concerning fractionally integrated processes.     

Asymptotics for General Nonstationary Fractionally Integrated Processes Without Prehistoric Influence

This paper derives a functional limit theorem for general nonstationary fractionally integrated processes having no influence from prehistory. Asymptotic distributions of sample autocovariances and sample autocorrelations based on these processes are also investigated. The problem arises naturally in discussing fractionally integrated processes when the processes starts at a given initial date.     

On score-functions and goodness-of-fit tests for stochastic processes

The problems of the construction of asymptotically distribution free goodness-of-fit tests for two diffusion processes are considered. The null hypothesis is composite parametric. All tests are based on the score-function processes, where the unknown parameter is replaced by the maximum likelihood estimators. We show that a special change of time transforms the limit score-function processes into the Brownian bridge. This property allows us to construct asymptotically distribution-free tests for dynamical systems with small noise and ergodic diffusion processes. The proposed tests are in some sense universal. We discuss the possibilities of the construction of asymptotically distribution free tests for inhomogeneous Poisson processes and nonlinear AR time series.     

Zur Überlagerung von Erneuerungsprozessen

Summary If a finite sequence of independent (not necessarily stationary) renewal processes is given, a superposition process can easily be defined as the union of all point sequences represented by the given processes. The properties of such superposition processes are investigated. First, a necessary and sufficient condition for a superposition process to be a renewal process is given. Essentially, this condition reads thus: the given processes must be Poisson processes. The main result given in this paper is a limit theorem for superposition processes which shows that, even with largely arbitrary renewal processes superimposed, the superposition process has local properties which approach the properties of the Poisson process as the number of given processes increases. The theorem contains some well-known special theorems of this type [e.g. Khintchine, 1960; Franken, 1963].

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Von der Fakultät für Allgemeine Wissenschaften der T. H. München angenommene Habilitationsschrift (Auszug).     

On Markov-Additive Jump Processes

In 1995, Pacheco and Prabhu introduced the class of so-called Markov-additive processes of arrivals in order to provide a general class of arrival processes for queueing theory. In this paper, the above class is generalized considerably, including time-inhomogeneous arrival rates, general phase spaces and the arrival space being a general vector space (instead of the finite-dimensional Euclidean space). Furthermore, the class of Markov-additive jump processes introduced in the present paper is embedded into the existing theory of jump processes. The best known special case is the class of BMAP arrival processes.