Set-indexed Processes Research Articles

Simulations for Karlin random fields

We investigate the simulation methods for a large family of stable random fields that appeared in the recent literature, known as the Karlin stable set-indexed processes. We exploit a new representation and implement the procedure introduced by Asmussen and Rosinski (2001) by first decomposing the random fields into large-jump and small-jump parts, and simulating each part separately. As special cases, simulations for several manifold-indexed processes are considered, and adjustments are introduced accordingly in order to improve the computational efficiency.

Stable Processes with Stationary Increments Parameterized by Metric Spaces

A new family of stable processes indexed by metric spaces with stationary increments is introduced. They are special cases of a new family of set-indexed stable processes with Chentsov representation. At the heart of the representation, a result on the so-called measure definite kernels is of independent interest. A limit theorem for set-indexed processes is also established.

Local Hölder regularity for set-indexed processes

In this paper, we study the Holder regularity of set-indexed stochastic processes defined in the framework of Ivanoff–Merzbach. The first key result is a Kolmogorov-like Holder-continuity Theorem, whose novelty is illustrated on an example which could not have been treated with anterior tools. Increments for set-indexed processes are usually not simply written as XU − XV, hence we considered different notions of Holder-continuity. Then, the localization of these properties leads to various definitions of Holder exponents, which we compare to one another.

A fractional Brownian field indexed by [formula omitted] and a varying Hurst parameter

Using structures of Abstract Wiener Spaces, we define a fractional Brownian field indexed by a product space (0,1/2]×L2(T,m), (T,m) a separable measure space, where the first coordinate corresponds to the Hurst parameter of fractional Brownian motion. This field encompasses a large class of existing fractional Brownian processes, such as Lévy fractional Brownian motions and multiparameter fractional Brownian motions, and provides a setup for new ones. We prove that it has satisfactory incremental variance in both coordinates and derive certain continuity and Hölder regularity properties in relation with metric entropy. Also, a sharp estimate of the small ball probabilities is provided, generalizing a result on Lévy fractional Brownian motion. Then, we apply these general results to multiparameter and set-indexed processes, proving the existence of processes with prescribed local Hölder regularity on general indexing collections.

The set-indexed Lévy process: Stationarity, Markov and sample paths properties

We present a satisfactory definition of the important class of Lévy processes indexed by a general collection of sets. We use a new definition for increment stationarity of set-indexed processes to obtain different characterizations of this class. As an example, the set-indexed compound Poisson process is introduced. The set-indexed Lévy process is characterized by infinitely divisible laws and a Lévy–Khintchine representation. Moreover, the following concepts are discussed: projections on flows, Markov properties, and pointwise continuity. Finally the study of sample paths leads to a Lévy–Itô decomposition. As a corollary, the semi-martingale property is proved.

A Notion of Stopping Line for Set-Indexed Processes

In this paper, we introduce the class of $$\mathcal {A}$$ -stopping lines which generalize the planar stopping lines in Merzbach [(1980), Stochastic Process. Appl. 10, 49–63] by replacing the positive quadrant of the plane by a collection $$\mathcal {A}$$ of compact subsets of a fixed topological space. Our notion of stopping line also compliments and generalizes the stopping sets defined in Ivanoff and Merzbach [(1995), Stochastic Process. Appl. 57, 83–98].

The invariance principle for a class of dependent random fields

Sufficient conditions for the tightness of a family of distributions of partial sum set-indexed processes constructed from symmetric random fields are obtained in this paper. We require that the moments of order s, s > 2, exist. The dependence structure of the field is described by the β1-mixing coefficients decreasing with a power rate. Assuming that a field is stationary and applying a result of D. Chen (1991) on the convergence of finite-dimensional distributions of the processes we obtain the invariance principle.

The set-indexed Ito integral

We construct an Ito-type stochastic integral where the integrator is a process indexed by a semilattice of compact subsets of a fixed topological spaceT and the integrands, which are indexed by the points inT, possess a natural form of predictability. The definition of the integral involves, among other things, an Ito-type isometry defined in terms of the set-indexed quadratic variation of the integrator. The martingale property and quadratic variation for the resulting integral process are derived. In addition, employing the notion of stopping set from Ivanoff and Merzbach (1995), we construct and study a set-indexed local integral. A novel and flexible notion of predictability for set-indexed processes is defined and characterized, permitting the integration of a set-indexed integrand against a set-indexed process.

Boundary estimation based on set-indexed empirical processes

We observe independent random variables taken at the nodes of a grid in the d-dimensional unit cube. It is assumed that there exists a partition of the unit cube into two disjoint regions. Observations with nodes in a particular region stem from a common distribution, whereas observations from different regions differ in distribution. The problem is to estimate the common topological boundary of the two regions. Our estimator is defined as maximizer of a certain set-indexed empirical process. It induces a partition from which we show that the number of misclassified data is stochastically bounded as the sample sizes increase to infinity.

Q-Markov random probability measures and their posterior distributions

In this paper, we use the Markov property introduced in Balan and Ivanoff (J. Theoret. Probab. 15 (2002) 515) for set-indexed processes and we prove that a Markov prior distribution leads to a Markov posterior distribution. In particular, by proving that a neutral to the right prior distribution leads to a neutral to the right posterior distribution, we extend a fundamental result of Doksum (Ann. Probab. 2 (1974) 183) to arbitrary sample spaces.

Set-indexed processes with independent increments

Set-indexed process with independent increments are described by ‘convolution systems’; the construction of such a process is given using Kolmogorov's extension theorem. When we specialize to the case of Lévy processes, we obtain a double characterization in terms of their characteristic functions and in terms of their generators.

A Markov Property For Set-Indexed Processes

We consider a type of Markov property for set-indexed processes which is satisfied by all processes with independent increments and which allows us to introduce a transition system theory leading to the construction of the process. A set-indexed generator is defined such that it completely characterizes the distribution of the process.

On different topologies for set-indexing collections

We underline some topological properties of set-indexing collections in order to provide new assumptions for set-indexed stochastic processes. We show that any of these topological assumptions is equivalent to another one already used in the framework of a set-indexed process. These may replace usual metric assumptions needed to derive some known stopping results. We provide an example related to a class of birth-and-growth processes.

A strong Markov property for set-indexed processes

We introduce adapted sets and optional sets and we study a type of strong Markov property for set-indexed processes that can be associated with the sharp Markov property defined by Ivanoff and Merzbach (Proceedings of the International Conference on Stochastic Models, June 1998, Carleton University, Can. Math. Soc. Conf. Proc. 26 (2000) 217).

Asymptotic Behavior of Continuous Set-Indexed Martingales

In this paper, following the Knight's approach, we solve a convergence problem for set-indexed martingales. For this purpose, we first define a tightness criterion for set-indexed continuous processes. The core of this characterization is connected with a weaker definition of continuity and hence the use of the corresponding topology, and with the fact that indices take values in a semilattice of closed subsets. Then, we give an effective tightness criterion by means of an estimate for a majorizing measure defined on the space. We finally prove under this set-indexed framework a theorem similar to the Knight's.

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.

Stochastic integration for set-indexed processes

In this paper, we are concerned with the construction of a stochastic integral, when the integrator is a set-indexed stochastic process. For this purpose, we adopt Blei’s point of view and we get this integral by means of the bimeasure he introduced. This enables us to enlarge the class of integrators. Here, the integral is first defined for point-indexed integrands and later for set-indexed ones. Then we generalize this definition to local processes.

A Maximal Inequality and a Functional Central Limit Theorem for set-indexed empirical processes

For the tail probabilities of a general set-indexed empirical process in an arbitrary sample space a maximal inequality is derived. In the case that the class of sets by which the process is indexed possesses a total ordering, the application of our inequality yields an elementary proof for a functional central limit theorem without involving such advanced techniques as symmetrization, stratification, chaining or Gaussian domination. Analogously, the inequality leads to a weak uniform law of large numbers (including convergence rate).

On the Mode-Change Problem for Random Measures

Abstract The classical change-point problem in modern terms, i.e., the mode-change problem, is stated for sufficiently general set-indexed random processes, namely for random measures. A method is shown for solving this problem both in the general form and for the intensity of compound Poisson random measures. The results obtained are novel for the change-point problem, too.

On the mode-change problem for random measures

The classical change-point problem in modern terms, i.e., the mode-change problem, is stated for sufficiently general set-indexed random processes, namely for random measures. A method is shown for solving this problem both in the general form and for the intensity of compound Poisson random measures. The results obtained are novel for the change-point problem, too.