Semi-Selfsimilar Processes

A notion of semi-selfsimilarity of R ^d -valued stochastic processes is introduced as a natural extension of the selfsimilarity. Several topics on semi-selfsimilar processes are studied: the existence of the exponent for semi-selfsimilar processes; characterization of semi-selfsimilar processes as scaling limits; relationship between semi-selfsimilar processes with independent increments and semi-selfdecomposable distributions, and examples; construction of semi-selfsimilar processes with stationary increments; and extension of the Lamperti transformation. Semi-stable processes where all joint distributions are multivariate semi-stable are also discussed in connection with semi-selfsimilar processes. A wide-sense semi-selfsimilarity is defined and shown to be reducible to semi-selfsimilarity.     

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     

Stochastic order for alpha-permanental point processes

We establish relations of stochastic comparison among point processes elements of the set of alpha-permanental point processes. This set contains in particular, the determinantal point processes, the Poisson point processes and the permanental point processes. We show that these three classes of point processes can be ordered according to the increasing stochastic order. Elementary particles provide illustrations of some of the obtained relations of stochastic comparison.     

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.     

Constructions for planar point processes using concentric circles

Some point processes are obtained by generalising the well-known construction for a two-dimensional Poisson process which locates an event on each of a sequence of concentric circles in a particular way. The constructions considered here have, in general, a random number of events on each circle. Under certain sufficient conditions, the constructed processes are asymptotically Poisson, far from the origin. The obvious regularity in the structure of these processes can be removed at least superficially, by displacing the events independently off the concentric circles.     

Approximation of Hunt Processes by Multivariate Poisson Processes

We prove that arbitrary Hunt processes on a general state space can be approximated by multivariate Poisson processes starting from each point of the state space. The key point is that no additional regularity assumption on the state space and on the underlying transition semigroup is used.     

On Non-simple Marked Point Processes

Simple point processes are often characterized by their associated compensators or conditional intensities. Non-simple point processes are not uniquely determined by their conditional intensity and compensator, so instead one may identify with the point process its associated simple point process and corresponding conditional intensity, on an expanded mark space. Some relations between the conditional intensity on the expanded mark space and the ordinary conditional intensity are investigated here, and some classes of separable non-simple processes are presented. Transformations into simple point processes, involving thinning and rescaling, are presented.     

A class of renewal Interrupted Poisson Processes and applications to queueing systems

Switched Poisson Processes and Interrupted Poisson Processes are often employed to characterize traffic streams in distributed computer and communications systems, especially in investigations of overflow processes in telecommunication networks. With these processes, input streams having inter-segment correlations and high variance as well as state-dependent traffic can properly be modelled. In this paper we first derive an approximation method to describe the Generalized Switched Poisson processes in conjunction with a renewal assumption. As a special case of this class of processes, the class of Interrupted Poisson processes is also included in the investigation. As a result, a generalization of the well-known class of Interrupted Poisson processes is obtained. It is shown that the renewal property is also given for this general class of Interrupted Poisson processes having generally distributed off-phase. To illustrate the accuracy of the presented renewal approximation of Generalized Switched Poisson processes and to show the major properties of the General Interrupted Poisson processes, applications to some basic queueing systems are discussed by means of numerical results.This work was done while the author was with Institute of Communications Switching and Data Technics, University of Stuttgart, Seidenstrasse 36, D-7000 Stuttgart 1, FRG.     

On convergence of random walks generated by compound Cox processes to Lévy processes

A functional limit theorem is proved establishing weak convergence of random walks generated by compound doubly stochastic Poisson processes to Lévy processes in the Skorokhod space. As corollaries, theorems are proved on convergence of random walks with jumps having finite variances to Lévy processes with mixed normal distributions, and in particular, to stable Lévy processes.     

Geometric ergodicity of affine processes on cones

For affine processes on finite-dimensional cones, we give criteria for geometric ergodicity — that is exponentially fast convergence to a unique stationary distribution. Ergodic results include both the existence of exponential moments of the limiting distribution, where we exploit the crucial affine property, and finite moments, where we invoke the polynomial property of affine semigroups. Furthermore, we elaborate sufficient conditions for aperiodicity and irreducibility. Our results are applicable to Wishart processes with jumps on the positive semidefinite matrices, continuous-time branching processes with immigration in high dimensions, and classical term-structure models for credit and interest rate risk.     

QUANTUM GAUSSIAN PROCESSES

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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.     

Stable sub-Gaussian models constructed by Poisson processes

In this paper, we first prove that one-parameter standard α-stable sub-Gaussian processes can be approximated by processes constructed by integrals based on the Poisson process with random intensity. Then we extend this result to the two-parameter processes. At last, we consider the approximation of the subordinated fractional Brownian motion.     

Ordered thinnings of point processes and random measures

This is a study of thinnings of point processes and random measures on the real line that satisfy a weak law of large numbers. The thinning procedures have dependencies based on the order of the points or masses being thinned such that the thinned process is a composition of two random measures. It is shown that the thinned process (normalized by a certain function) converges in distribution if and only if the thinning process does. This result is used to characterize the convergence of thinned processes to infinitely divisible processes, such as a compound Poisson process, when the thinning is independent and nonhomogeneous, stationary, Markovian, or regenerative. Thinning by a sequence of independent identically distributed operations is also discussed. The results here contain Renyi's classical thinning theorem and many of its extensions.     

Directed Markov Point Processes as Limits of Partially Ordered Markov Models

In this paper, we consider spatial point processes and investigate members of a subclass of the Markov point processes, termed the directed Markov point processes (DMPPs), whose joint distribution can be written in closed form and, as a consequence, its parameters can be estimated directly. Furthermore, we show how the DMPPs can be simulated rapidly using a one-pass algorithm. A subclass of Markov random fields on a finite lattice, called partially ordered Markov models (POMMs), has analogous structure to that of DMPPs. In this paper, we show that DMPPs are the limits of auto-Poisson and auto-logistic POMMs. These and other results reveal a close link between inference and simulation for DMPPs and POMMs.     

Stochastic evolution equations and related measure processes

Basic results on stochastic differential equations in Hilbert and Banach space, linear stochastic evolution equations and some classes of nonlinear stochastic evolution equations are reviewed. The emphasis is on equations relevant to the study of spacetime stochastic processes. In particular the class of measure processes, the continuous analogs of spacetime population processes, is studied in detail.     

A queueing system with random server capacity and multiple control

The authors study queueing, input and output processes in a queueing system with bulk service and state dependent service delay. The input flow of customers, modulated by a semi-Markov process, is served by a single server that takes batches of a certain fixed size if available or waits until the queue accumulates enough customers for service. In the latter case, the batch taken for service is of random size dependent on the state of the system, while service duration depends both on the state of the system and on the batch size taken. The authors establish a necessary and sufficient condition for equilibrium of the system and obtain the following results: Explicit formulas for steady state distribution of the queueing process, intensity of the input and output processes, and mean values of idle and busy periods. They employ theory of semi-regenerative processes and illustrate the results by a number of examples. In one of them an optimization problem is discussed.     

Approximate Simulation of Hawkes Processes

Hawkes processes are important in point process theory and its applications, and simulation of such processes are often needed for various statistical purposes. This article concerns a simulation algorithm for unmarked and marked Hawkes processes, exploiting that the process can be constructed as a Poisson cluster process. The algorithm suffers from edge effects but is much faster than the perfect simulation algorithm introduced in our previous work Møller and Rasmussen (2004). We derive various useful measures for the error committed when using the algorithm, and we discuss various empirical results for the algorithm compared with perfect simulations. Extensions of the algorithm and the results to more general types of marked point processes are also discussed.     

On the exponential process associated with a CARMA-type process

We study the correlation decay and the expected maximal increments of the exponential processes determined by continuous-time autoregressive moving average (CARMA)-type processes of order (p, q). We consider two background driving processes, namely fractional Brownian motions and Lévy processes with exponential moments. The results presented in this paper are significant extensions of those very recent works on the Ornstein–Uhlenbeck-type case (p = 1, q = 0), and we develop more refined techniques to meet the general (p, q). In the concluding section, we discuss the perspective role of exponential CARMA-type processes in stochastic modelling of the burst phenomena in telecommunications and the leverage effect in financial econometrics.     

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