Spectral representations of sum- and max-stable processes (original) (raw)

Ergodic properties of stationary stable processes

Stochastic Processes and their Applications, 1987

We derive spectral necessary and sufficient conditions for stationary symmetric stable processes to be metrically transitive and mixing. We then consider some important classes of stationary stable processes: Sub-Gaussian stationary processes and stationary stable processes with a harmonic spectral representation are never metrically transitive, the latter in sharp contrast with the Gaussian case. Stable processes with a harmonic spectral representation satisfy a strong law of large numbers even though they are not generally stationary. For doubly stationary stable processes. sulTicient conditions are derived for metric transitivity and mixing, and necessary and sut?icient conditions for ;I strong law of large numbers. AhfS IYYO Strhjec, Cltrssi/icorirm: Primary 6OEO7. 60GlO. 47DI0, 28c)IO stable processes * ergodic theory * stationary processes l spectral representations Research supported by the Air Force Ollke of Scientific Research Contract No. AFOSR F49620 82 c 0009.

On the spectral representation of symmetric stable processes

Journal of Multivariate Analysis, 1982

The so-called spectral representation theorem for stable processes linearly imbeds each symmetric stable process of index p into Lp (0 < p < 2). We use the theory of Lp isometries for 0 < p < 2 to study the uniqueness of this representation for the non-Gaussian stable processes. We also determine the form of this representation for stationary processes and for substable processes. Complex stable processes are defined, and a complex version of the spectral representation theorem is proved. As a corollary to the complex theory we exhibit an imbedding of complex L' into real or complex Lp for 0 < p < q < 2.

On the mixing structure of stationary increment and self-similar SαS processes

1998

Mixed moving average processes appear in the ergodic decomposition of stationary symmetric α-stable (SαS) processes. They correspond to the dissipative part of "deterministic" flows generating SαS processes (Rosinski, 1995). Along these lines we study stationary increment and self-similar SαS processes. Since the classes of stationary increment and self-similar processes can be embedded into the class of stationary processes by the Masani and Lamperti transformations, respectively, we characterize these classes of SαS processes in terms of nonsingular flows and the related cocycles. We illustrate this approach considering various examples of self-similar mixed moving average SαS processes introduced in (Surgailis, Rosinski, Mandrekar and Cambanis, 1992).

On ergodicity of some Markov processes

The Annals of Probability, 2010

We formulate a criterion for the existence, uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, the weak * ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting with any initial distribution, is established. The principal assumptions are the lower bound of the ergodic averages of the transition probability function and the eproperty of the semigroup. The general result is applied to solutions of some stochastic evolution equations in Hilbert spaces. As an example we consider an evolution equation whose solution describes the Lagrangian observations of the velocity field in the passive tracer model. The weak * mean ergodicity of the respective invariant measure is used to derive the law of large numbers for the trajectory of a tracer.

A Multiplier Related to Symmetric Stable Processes

Hacettepe Journal of Mathematics and Statistics, 2016

In two recent papers [5] and [6], we generalized some classical results of Harmonic Analysis using probabilistic approach by means of a ddimensional rotationally symmetric stable process. These results allow one to discuss some boundedness conditions with weaker hypotheses. In this paper, we study a multiplier theorem using these more general results. We consider a product process consisting of a d-dimensional symmetric stable process and a 1-dimensional Brownian motion, and use properties of jump processes to obtain bounds on jump terms and the L p (R d)-norm of a new operator.

An ergodic theorem for stochastic max-plus linear systems

IFAC Proceedings Volumes, 2004

In this paper the asymptotic growth rate of stochastic max-plus linear systems is studied. For non-random irreducible max-plus linear systems the existence of this growth rate is well known. For stochastic systems conditions for the existence of the growth rate will be developed. The conditions are mild and naturaL and weaker than the conditions used in literature until now. The interpretation of the conditions is that for the asymptotic growth rate to exist the stochastic max-plus linear system should be dominated from below by a non-random irreducible max-plus linear system. The latter condition need not to be true ahvays. but should at least hold with some positive probability. The investigations in this paper are motivated by means of some examples.

Continuity of symmetric stable processes

Journal of Multivariate Analysis, 1989

The path continuity of a symmetric p-stable process is examined in terms of any stochastic integral representation for the process. When 0 < p < 1, we give necessary and suflicient conditions for path continuity in terms of any (every) representation. When 1 &p<2, we extend the known sutliciency condition in terms of metric entropy and offer a conjecture for the stable version of the Dudley-Fernique theorem. Finally, necessary and sufficient conditions for path continuity are given in terms of continuity at a point for 0 < p < 2.