Law of the iterated logarithm for the periodogram

We consider the almost sure asymptotic behavior of the periodogram of stationary and ergodic sequences. Under mild conditions we establish that the limsup of the periodogram properly normalized identifies almost surely the spectral density function associated with the stationary process. Results for a specified frequency are also given. Our results also lead to the law of the iterated logarithm for the real and imaginary parts of the discrete Fourier transform. The proofs rely on martingale approximations combined with results from harmonic analysis and techniques from ergodic theory. Several applications to linear processes and their functionals, iterated random functions, mixing structures and Markov chains are also presented.     

On completeness of the spectral domain of harmonizable processes

Summary The spectral domain of harmonizable processes is studied and a criterion for its completeness is obtained. It is shown that even for periodically correlated processes the spectral domain need not be complete.This work was funded in part by Office of Naval Research grant N00014-89-J-1824 and in part by US Army Research Office grant DAAL 03-91-G-0238. Part of these results has been presented at the May 1993 AMS meeting in DeKalb; ref. nr. 882-28-69     

Existence and uniqueness of stationary Lévy-driven CARMA processes

Necessary and sufficient conditions for the existence of a strictly stationary solution of the equations defining a general Lévy-driven continuous-parameter ARMA process with index set R$\mathbb{R}$ are determined. Under these conditions the solution is shown to be unique and an explicit expression is given for the process as an integral with respect to the background driving Lévy process. The results generalize results obtained earlier for second-order processes and for processes defined by the Ornstein–Uhlenbeck equation.     

Power variation from second order differences for pure jump semimartingales

We introduce power variation constructed from powers of the second-order differences of a discretely observed pure-jump semimartingale processes. We derive the asymptotic behavior of the statistic in the setting of high-frequency observations of the underlying process with a fixed time span. Unlike the standard power variation (formed from the first-order differences of the process), the limit of our proposed statistic is determined solely by the jump component of the process regardless of the activity of the latter. We further show that an associated Central Limit Theorem holds for a wider range of activity of the jump process than for the standard power variation. We apply these results for estimation of the jump activity as well as the integrated stochastic scale.     

A sampling theorem for multivariate stationary processes

The notion of sampling for second-order q-variate processes is defined. It is shown that if the components of a q-variate process (not necessarily stationary) admits a sampling theorem with some sample spacing, then the process itself admits a sampling theorem with the same sample spacing. A sampling theorem for q-variate stationary processes, under a periodicity condition on the range of the spectral measure of the process, is proved in the spirit of Lloy's work. This sampling theorem is used to show that if a q-variate stationary process admits a sampling theorem, then each of its components will admit a sampling theorem too.     

Point processes in the plane

In this survey paper, two-parameter point processes are studied in connection with martingale theory and with respect to the partial-order induced by the Cartesian coordinates of the plane. Point processes are characterized by jump stopping times and by their two-parameter compensators. Properties of the doubly stochastic Poisson process, such as predictability, are discussed. A definition for the Palm measure of a two-parameter stationary point process is proposed.     

On almost sure limit inferior for B-valued stochastic processes and applications

Summary A criterion on almost sure limit inferior for the increments of B-valued stochastic processes is presented. Applications to processes of independent increments and to Gaussian processes with stationary increments are given. In particular, an exact limit inferior bound is established for increments of infinite series of independent Ornstein-Uhlenbeck processes.Work supported by an NSERC Canada grant at Carleton UniversityWork supported by the Fok Yingtung Education Foundation of China     

On the ergodicity and mixing of max-stable processes

Max-stable processes arise in the limit of component-wise maxima of independent processes, under appropriate centering and normalization. In this paper, we establish necessary and sufficient conditions for the ergodicity and mixing of stationary max-stable processes. We do so in terms of their spectral representations by using extremal integrals.     

Volterra-type Ornstein–Uhlenbeck processes in space and time

We propose a novel class of temporo-spatial Ornstein–Uhlenbeck processes as solutions to Lévy-driven Volterra equations with additive noise and multiplicative drift. After formulating conditions for the existence and uniqueness of solutions, we derive an explicit solution formula and discuss distributional properties such as stationarity, second-order structure and short versus long memory. Furthermore, we analyze in detail the path properties of the solution process. In particular, we introduce different notions of càdlàg paths in space and time and establish conditions for the existence of versions with these regularity properties. The theoretical results are accompanied by illustrative examples.     

Correlation cascades,ergodic properties and long memory of infinitely divisible processes

In this paper, we investigate the properties of the recently introduced measure of dependence called correlation cascade. We show that the correlation cascade is a promising tool for studying the dependence structure of infinitely divisible processes. We describe the ergodic properties (ergodicity, weak mixing, mixing) of stationary infinitely divisible processes in the language of the correlation cascade and establish its relationship with the codifference. Using the correlation cascade, we investigate the dependence structure of four fractional α$\alpha$-stable stationary processes. We detect the property of long memory and verify the ergodic properties of the discussed processes.     

Discrete-time stochastic processes on Riesz spaces

It has been recognised that order is closely linked with probability theory, with lattice theoretic approaches being used to study Markov processes but, to our knowledge, the complete theory of (sub, super) martingales and their stopping times has not been formulated on Riesz spaces. We generalize the concepts of stochastic processes, (sub, super) martingales and stopping times to Riesz spaces. In this paper we consider discrete time processes with bounded stopping times.     

On the extremes of a class of non-linear processes with heavy tailed innovations

The aim of this paper is to look at the limiting form of certain empirical point processes induced by a particular class of non-linear processes generated by heavy tailed innovations. Such asymptotic results will be highly useful in obtaining the weak limiting behavior of various functionals of the underlying process including the asymptotic distribution of upper and lower order statistics. In particular, we investigate the maximum limiting distribution and its corresponding extremal index. The results are applied to the study of the extremal properties of bilinear processes.     

Fractional-order regularization and wavelet approximation to the inverse estimation problem for random fields

The least-squares linear inverse estimation problem for random fields is studied in a fractional generalized framework. First, the second-order regularity properties of the random fields involved in this problem are analysed in terms of the fractional Sobolev norms. Second, the incorporation of prior information in the form of a fractional stochastic model, with covariance operator bicontinuous with respect to a certain fractional Sobolev norm, leads to a regularization of this problem. Third, a multiresolution approximation to the class of linear inverse problems considered is obtained from a wavelet-based orthogonal expansion of the input and output random models. The least-squares linear estimate of the input random field is then computed using these orthogonal wavelet decompositions. The results are applied to solving two important cases of linear inverse problems defined in terms of fractional integral operators.     

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.     

Stopping for two-dimensional stochastic processes

The aim of this paper is to introduce some techniques that can be used in the study of stochastic processes which have as parameter set the positive quadrant of the plane R²₊. We define stopping lines and derive an interesting property of measurability for them. The notion of predictability is developed, and we show the connection between predictable processes, fields associated with stopping lines, and predictable stopping lines. We also give a theorem of section for predictable sets. Extension to processes indexed by any partially ordered set with some regularity assumptions can be carried out quite easily with the same techniques.     

Quasi-invariance properties of a class of subordinators

We study absolute-continuity relationships for a class of stochastic processes, including the gamma and the Dirichlet processes. We prove that the laws of a general class of non-linear transformations of such processes are locally equivalent to the law of the original process and we compute explicitly the associated Radon–Nikodym densities. This work unifies and generalizes to random non-linear transformations several previous quasi-invariance results for gamma and Dirichlet processes.     

Characterization of the finite variation property for a class of stationary increment infinitely divisible processes

We characterize the finite variation property for stationary increment mixed moving averages driven by infinitely divisible random measures. Such processes include fractional and moving average processes driven by Lévy processes, and also their mixtures. We establish two types of zero–one laws for the finite variation property. We also consider some examples to illustrate our results.     

Maximal Inequalities for Fractional Lévy and Related Processes

In this article we study processes that are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred Lévy process, which covers the popular class of fractional Lévy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding “convoluted martingale” is p-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale.     

Path and semimartingale properties of chaos processes

The present paper characterizes various properties of chaos processes which in particular include processes where all time variables admit a Wiener chaos expansion of a fixed finite order. The main focus is on the semimartingale property, p$p$-variation and continuity. The general results obtained are finally used to characterize when a moving average is a semimartingale.