Ergodic properties of max-infinitely divisible processes

We prove that a stationary max-infinitely divisible process is mixing (ergodic) iff its dependence function converges to 0 (is Cesàro summable to 0). These criteria are applied to some classes of max-infinitely divisible processes.     

An empirical central limit theorem for dependent sequences

We prove a central limit theorem for the d$d$-dimensional distribution function of a class of stationary sequences. The conditions are expressed in terms of some coefficients which measure the dependence between a given σ$\sigma$-algebra and indicators of quadrants. These coefficients are weaker than the corresponding mixing coefficients, and can be computed in many situations. In particular, we show that they are well adapted to functions of mixing sequences, iterated random functions, and a class of dynamical systems.     

Approximation of stationary solutions to SDEs driven by multiplicative fractional noise

In a previous paper, we studied the ergodic properties of an Euler scheme of a stochastic differential equation with a Gaussian additive noise in order to approximate the stationary regime of such an equation. We now consider the case of multiplicative noise when the Gaussian process is a fractional Brownian motion with Hurst parameter H>1/2$H>1/2$ and obtain some (functional) convergence properties of some empirical measures of the Euler scheme to the stationary solutions of such SDEs.     

Convergence to Lévy stable processes under some weak dependence conditions

For a strictly stationary sequence of random vectors in R^d${\mathbb{R}}^d$ we study convergence of partial sum processes to a Lévy stable process in the Skorohod space with _J1$J_1$-topology. We identify necessary and sufficient conditions for such convergence and provide sufficient conditions when the stationary sequence is strongly mixing.     

Asymptotic behavior of the quadratic variation of the sum of two Hermite processes of consecutive orders

Hermite processes are self-similar processes with stationary increments which appear as limits of normalized sums of random variables with long range dependence. The Hermite process of order 1 is fractional Brownian motion and the Hermite process of order 2 is the Rosenblatt process. We consider here the sum of two Hermite processes of orders q≥1$q\ge 1$ and q+1$q+1$ and of different Hurst parameters. We then study its quadratic variations at different scales. This is akin to a wavelet decomposition. We study both the cases where the Hermite processes are dependent and where they are independent. In the dependent case, we show that the quadratic variation, suitably normalized, converges either to a normal or to a Rosenblatt distribution, whatever the order of the original Hermite processes.     

A central limit theorem for stationary random fields

This paper establishes a central limit theorem and an invariance principle for a wide class of stationary random fields under natural and easily verifiable conditions. More precisely, we deal with random fields of the form _Xk=g(_εk−s,s∈Z^d)$X_k=g({\epsilon }_{k-s},s\in {\mathbb{Z}}^d)$, k∈Z^d$k\in {\mathbb{Z}}^d$, where _(εi)i∈Zd${\left({\epsilon }_i\right)}_{i\in {\mathbb{Z}}^d}$ are iid random variables and g$g$ is a measurable function. Such kind of spatial processes provides a general framework for stationary ergodic random fields. Under a short-range dependence condition, we show that the central limit theorem holds without any assumption on the underlying domain on which the process is observed. A limit theorem for the sample auto-covariance function is also established.     

An invariance principle for stationary random fields under Hannan’s condition

We establish an invariance principle for a general class of stationary random fields indexed by Z^d${\mathbb{Z}}^d$, under Hannan’s condition generalized to Z^d${\mathbb{Z}}^d$. To do so we first establish a uniform integrability result for stationary orthomartingales, and second we establish a coboundary decomposition for certain stationary random fields. At last, we obtain an invariance principle by developing an orthomartingale approximation. Our invariance principle improves known results in the literature, and particularly we require only finite second moment.     

The Lamperti correspondence extended to Lévy processes and semi-stable Markov processes in locally compact groups

We establish Lamperti representations for semi-stable Markov processes in locally compact groups. We also study the particular cases of processes with values in R$\mathbb{R}$ and C$\mathbb{C}$ under the hypothesis that they do not visit 0. These Lamperti representations yield some properties of these semi-stable Markov processes.     

Tempering stable processes

A tempered stable Lévy process combines both the α$\alpha$-stable and Gaussian trends. In a short time frame it is close to an α$\alpha$-stable process while in a long time frame it approximates a Brownian motion. In this paper we consider a general and robust class of multivariate tempered stable distributions and establish their identifiable parametrization. We prove short and long time behavior of tempered stable Lévy processes and investigate their absolute continuity with respect to the underlying α$\alpha$-stable processes. We find probabilistic representations of tempered stable processes which specifically show how such processes are obtained by cutting (tempering) jumps of stable processes. These representations exhibit α$\alpha$-stable and Gaussian tendencies in tempered stable processes and thus give probabilistic intuition for their study. Such representations can also be used for simulation. We also develop the corresponding representations for Ornstein–Uhlenbeck-type processes.     

Monotonicity properties of multi-dimensional reflected diffusions in random environment and applications

We consider an N$N$-dimensional reflected process, modeling an infinite capacity fluid queues network, of which service and input rates depend on the queue levels as well as on the state of an exterior ergodic stationary process. N$N$ is the number of queues in the network. We prove a monotonicity result for such a process, from which we deduce stability results for networks of queues. Particular attention is paid to the case N=2$N=2$. Next, we give some applications of those stability results.     

Ergodicity for time-changed symmetric stable processes

In this paper we study ergodicity and related semigroup property for a class of symmetric Markov jump processes associated with time-changed symmetric α$\alpha$-stable processes. For this purpose, explicit and sharp criteria for Poincaré type inequalities (including Poincaré, super Poincaré and weak Poincaré inequalities) of the corresponding non-local Dirichlet forms are derived. Moreover, our main results, when applied to a class of one-dimensional stochastic differential equations driven by symmetric α$\alpha$-stable processes, yield sharp criteria for their various ergodic properties and corresponding functional inequalities.     

Limit theorems for occupation time fluctuations of branching systems II: Critical and large dimensions

We give functional limit theorems for the fluctuations of the rescaled occupation time process of a critical branching particle system in R^d${\mathbb{R}}^d$ with symmetric α$\alpha$-stable motion in the cases of critical and large dimensions, d=2α$d=2\alpha$ and d>2α$d>2\alpha$. In a previous paper [T. Bojdecki, L.G. Gorostiza, A. Talarczyk, Limit theorems for occupation time fluctuations of branching systems I: long-range dependence, Stochastic Process. Appl., this issue.] we treated the case of intermediate dimensions, α<d<2α$\alpha <d<2\alpha$, which leads to a long-range dependence limit process. In contrast, in the present cases the limits are generalized Wiener processes. We use the same space–time random field method of the previous paper, the main difference being that now the tightness requires a new approach and the proofs are more difficult. We also give analogous results for the system without branching in the cases d=α$d=\alpha$ and d>α$d>\alpha$.     

A class of asymptotically self-similar stable processes with stationary increments

We generalize the BM-local time fractional symmetric α$\alpha$-stable motion introduced in Cohen and Samorodnitsky (2006) by replacing the local time with a general continuous additive functional (CAF). We show that the resulting process is again symmetric α$\alpha$-stable with stationary increments. Depending on the CAF, the process is either self-similar or lies in the domain of attraction of the BM-local time fractional symmetric α$\alpha$-stable motion. We also show that the process arises as a weak limit of a discrete “random rewards scheme” similar to the one described by Cohen and Samorodnitsky.     

Tempered stable distributions and processes

We investigate the class of tempered stable distributions and their associated processes. Our analysis of tempered stable distributions includes limit distributions, parameter estimation and the study of their densities. Regarding tempered stable processes, we deal with density transformations and compute their p$p$-variation indices. Exponential stock models driven by tempered stable processes are discussed as well.     

Generalized Hermite processes,discrete chaos and limit theorems

We introduce a broad class of self-similar processes {Z(t),t≥0}$\{Z\left(t\right),t\ge 0\}$ called generalized Hermite processes. They have stationary increments, are defined on a Wiener chaos with Hurst index H∈(1/2,1)$H\in (1/2,1)$, and include Hermite processes as a special case. They are defined through a homogeneous kernel g$g$, called the “generalized Hermite kernel”, which replaces the product of power functions in the definition of Hermite processes. The generalized Hermite kernels g$g$ can also be used to generate long-range dependent stationary sequences forming a discrete chaos process {X(n)}$\left\{X\left(n\right)\right\}$. In addition, we consider a fractionally-filtered version Z^β(t)$Z^{\beta }\left(t\right)$ of Z(t)$Z\left(t\right)$, which allows H∈(0,1/2)$H\in (0,1/2)$. Corresponding non-central limit theorems are established. We also give a multivariate limit theorem which mixes central and non-central limit theorems.     

Weak convergence of the tail empirical process for dependent sequences

This paper proves weak convergence in D$D$ of the tail empirical process–the renormalized extreme tail of the empirical process–for a large class of stationary sequences. The conditions needed for convergence are (i) moment restrictions on the amount of clustering of extremes, (ii) restrictions on long range dependence (absolute regularity or strong mixing), and (iii) convergence of the covariance function. We further show how the limit process is changed if exceedances of a nonrandom level are replaced by exceedances of a high quantile of the observations. Weak convergence of the tail empirical process is one key to asymptotics for extreme value statistics and its wide range of applications, from geoscience to finance.     

Stationarity of multivariate particle systems

A particle system is a family of i.i.d. stochastic processes with values translated by Poisson points. We obtain conditions that ensure the stationarity in time of the particle system in R^d${\mathbb{R}}^d$ and in some cases provide a full characterisation of the stationarity property. In particular, a full characterisation of stationary multivariate Brown–Resnick processes is given.     

Limit Theory for the Sample Autocovariance for Heavy-Tailed Stationary Infinitely Divisible Processes Generated by Conservative Flows

This study aim to develop limit theorems on the sample autocovariances and sample autocorrelations for certain stationary infinitely divisible processes. We consider the case where the infinitely divisible process has heavy tail marginals and is generated by a conservative flow. Interestingly, the growth rate of the sample autocovariances is determined not only by heavy tailedness of the marginals but also by the memory length of the process. Although this feature was first observed by Resnick et al. (Stoch Process Appl 85:321–339, 2000) for some very specific processes, we will propose a more general framework from the viewpoint of infinite ergodic theory. Consequently, the asymptotics of the sample autocovariances can be more comprehensively discussed.     

On some transformations between positive self-similar Markov processes

A path decomposition at the infimum for positive self-similar Markov processes (pssMp) is obtained. Next, several aspects of the conditioning to hit 0 of a pssMp are studied. Associated to a given pssMp X$X$, that never hits 0, we construct a pssMp X^↓$X^{\downarrow }$ that hits 0 in a finite time. The latter can be viewed as X$X$ conditioned to hit 0 in a finite time, and we prove that this conditioning is determined by the pre-minimum part of X$X$. Finally, we provide a method for conditioning a pssMp that hits 0 by a jump to do it continuously.