Volterra Processes Research Articles

Sampling Properties of U-statistics for a Class of Stationary Nonlinear Processes

We consider the sampling properties of U-statistics based on a sample of realization from a class of stationary nonlinear processes which include, in particular, linear, bilinear and finite order volterra processes. It is shown that if the size n of the realization tends to infinity then certain normalized versions of the U-statistics tend to be distributed normally with zero means and finite variances.

Large sample properties of spectral estimators for a class of stationary nonlinear processes

Abstract. We consider the standard spectral estimators based on a sample from a class of strictly stationary nonlinear processes which include, in particular, the bilinear and Volterra processes. It is shown that these estimators, under certain mild regularity conditions are both consistent and asymptotically normal.

WEAK DEPENDENCE: MODELS AND APPLICATIONS TO ECONOMETRICS

In this paper we discuss weak dependence and mixing properties of some popular models. We also develop some of their econometric applications. Autoregressive models, autoregressive conditional heteroskedasticity (ARCH) models, and bilinear models are widely used in econometrics. More generally, stationary Markov modeling is often used. Bernoulli shifts also generate many useful stationary sequences, such as autoregressive moving average (ARMA) or ARCH(∞) processes. For Volterra processes, mixing properties obtain given additional regularity assumptions on the distribution of the innovations.We recall associated probability limit theorems and investigate the nonparametric estimation of those sequences.We first thank the editor for the huge amount of additional editorial work provided for this review paper. The efficiency of the numerous referees was especially useful. The error pointed out in Hall and Horowitz (1996) was the origin of the present paper, and we thank the referees for asking for a more detailed treatment of a correct proof for this paper in Section 2.3. Also we thank Marc Henry and Rafal Wojakowski for a very careful rereading of the paper. An anonymous referee has been particularly helpful in the process of revision of the paper. The authors thank him for his numerous suggestions of improvement, including important results on negatively associated sequences and a thorough update in standard English.

Stochastic integration with respect to Volterra processes

We construct the basis of a stochastic calculus for so-called Volterra processes, i.e., processes which are defined as the stochastic integral of a time-dependent kernel with respect to a standard Brownian motion. For these processes which are natural generalization of fractional Brownian motion, we construct a stochastic integral and show some of its main properties: regularity with respect to time and kernel, transformation under an absolutely continuous change of probability, possible approximation schemes and Itô formula.

Equivalence of Volterra processes

In this paper we study necessary and sufficient conditions for the equivalence of Volterra Gaussian processes. Though this topic has already been studied in the literature, we provide new proofs, precisions and new theorems. We also give some examples of equivalent Volterra processes all related to the extensively studied fractional Brownian motion. Finally, we give an extension to general Gaussian processes of a recent regularization theorem by P. Cheridito.

Population Dynamics and Spatial Scale: Effects of System Size on Population Persistence

This paper explores the limitations of ecological models based on differential equations. We test three different model implementations of the processes represented by the Lotka–Volterra predator–prey equations. We use these models to investigate the system dynamics suppressed by the assumptions contained in differential-equation-based models; specifically, we examine the effects of space and demographic stochasticity. The differential equation implementation of the Lotka–Volterra processes assumes that there are enough individuals in the system that the effects of demographic stochasticity can be ignored and that the interactions between individuals always occur as though the system were continually well mixed or spatially random. The model predicts a neutrally stable equilibrium and infinite persistence, regardless of the size of the system. We use a stochastic birth–death (SBD) model to explore the possible effects of demographic stochasticity on the basic Lotka–Volterra model. The results show that, over a wide range of system sizes, the time to extinction is finite and increases linearly with the total size of the system. We introduce a new type of spatially explicit individual-based model called the Heuristic Asynchronous Discrete Event Simulation or HADES. HADES adds an explicit component of space to the SBD version of our process. We compare the system dynamics of the SBD and HADES models over the same system sizes, using the same demographic parameters, thereby partitioning the effects of demographic stochasticity and space. As system sizes are increased, the dynamics of the HADES model with respect to the SBD model, as measured by time to extinction, move from equivalent to less stable to much more stable. We analyze the effects of space on the system dynamics using animation, statistical point process analysis, and predicted system dynamics (spatial effects on the predation rate). We then show that both the destabilizing and stabilizing effects of space with respect to the SBD dynamics can be accounted for by specific nonrandom spatial patterns. We contrast our results with three commonly invoked mechanisms that affect the stability of predator–prey systems. We comment on the strengths and limitations of differential-equation-based models.

POPULATION DYNAMICS AND SPATIAL SCALE: EFFECTS OF SYSTEM SIZE ON POPULATION PERSISTENCE

This paper explores the limitations of ecological models based on differential equations. We test three different model implementations of the processes represented by the Lotka–Volterra predator–prey equations. We use these models to investigate the system dynamics suppressed by the assumptions contained in differential-equation-based models; specifically, we examine the effects of space and demographic stochasticity. The differential equation implementation of the Lotka–Volterra processes assumes that there are enough individuals in the system that the effects of demographic stochasticity can be ignored and that the interactions between individuals always occur as though the system were continually well mixed or spatially random. The model predicts a neutrally stable equilibrium and infinite persistence, regardless of the size of the system. We use a stochastic birth–death (SBD) model to explore the possible effects of demographic stochasticity on the basic Lotka–Volterra model. The results show that, over a wide range of system sizes, the time to extinction is finite and increases linearly with the total size of the system. We introduce a new type of spatially explicit individual-based model called the Heuristic Asynchronous Discrete Event Simulation or HADES. HADES adds an explicit component of space to the SBD version of our process. We compare the system dynamics of the SBD and HADES models over the same system sizes, using the same demographic parameters, thereby partitioning the effects of demographic stochasticity and space. As system sizes are increased, the dynamics of the HADES model with respect to the SBD model, as measured by time to extinction, move from equivalent to less stable to much more stable. We analyze the effects of space on the system dynamics using animation, statistical point process analysis, and predicted system dynamics (spatial effects on the predation rate). We then show that both the destabilizing and stabilizing effects of space with respect to the SBD dynamics can be accounted for by specific nonrandom spatial patterns. We contrast our results with three commonly invoked mechanisms that affect the stability of predator–prey systems. We comment on the strengths and limitations of differential-equation-based models.

Fractional Brownian Motions, Fractional Noises and Applications

Fractional Brownian Motions, Fractional Noises and Applications