One-step prediction forP n-weakly stationary processes

The one-step prediction problem is studied in the context ofP _n-weakly stationary stochastic processes , where is an orthogonal polynomial sequence defining a polynomial hypergroup on . This kind of stochastic processes appears when estimating the mean of classical weakly stationary processes. In particular, it is investigated whether these processes are asymptoticP _n-deterministic, i.e. the prediction mean-squared error tends to zero. Sufficient conditions on the covariance function or the spectral measure are given for being asymptoticP _n-deterministic. For Jacobi polynomialsP _n(x) the problem of being asymptoticP _n-deterministic is completely solved.     

Testing nonparametric and semiparametric hypotheses in vector stationary processes

We propose a general nonparametric approach for testing hypotheses about the spectral density matrix of multivariate stationary time series based on estimating the integrated deviation from the null hypothesis. This approach covers many important examples from interrelation analysis such as tests for noncorrelation or partial noncorrelation. Based on a central limit theorem for integrated quadratic functionals of the spectral matrix, we derive asymptotic normality of a suitably standardized version of the test statistic under the null hypothesis and under fixed as well as under sequences of local alternatives. The results are extended to cover also parametric and semiparametric hypotheses about spectral density matrices, which includes as examples goodness-of-fit tests and tests for separability.     

Some remarks on filtering and prediction of stationary processes

This note contains a some remarks concerning filtering and prediction theory. One of them is a solution to an old question of H. Furstenberg which indicates an unexpected phenomenon arising from the lack of integrability. Another gives some general results on the possibility of constructing two valued universal guessing schemes for distinguishing between classes of stochastic processes.     

Characterization of vector valued,gaussian, stationary,markov processes

We extend an old result by Doob characterizing real-valued, Gaussian, stationary, Markov processes to the vector case. In this case a deterministic component appears that consists of a system of harmonic oscillators while the random part is a collection of independent oscillator processes, modulo linear changes of coordinates.     

Wold decomposition,prediction and parameterization of stationary processes with infinite variance

Summary A discrete time stochastic process {_t} is said to be a p-stationary process (1<p2)if , for all integers n1, t ₁,...t _n,h and scalars b ₁,...b _n.The class of p-stationary processes includes the class of second-order weakly stationary stochastic processes, harmonizable stable processes of order (1<2), and p ^thorder strictly stationary processes. For any nondeterministic process in this class a finite Wold decomposition (moving average representation) and a finite predictive decomposition (autoregressive representation) are given without alluding to any notion of covariance or spectrum. These decompositions produce two unique (interrelated) sequences of scalar which are used as parameters of the process {_t}. It is shown that the finite Wold and predictive decomposition are all that one needs in developing a Kolmogorov-Wiener type prediction theory for such processes.     

Large deviation for stationary processes

A quasi-local variational characterization of the entropy for stationary processes is given. This is used to establish upper and lower large deviation estimates for arbitrary stationary processes. The upper and lower rate functions are shown to coincide for all quasi-local stationary processes. The contents of the paper is the following: 1. Introduction; 2. Notations; 3. Relative entropy of conditional expectations; 4. Relative entropy of a stationary process with respect to a covariant family of conditional expectations; 5. The role of locality and quasi-locality properties; 6. Large deviation upper estimate; 7. The Lower estimate; 8. The variational principle.     

Wold-Cramér concordance theorems for interpolation of q-variate stationary processes over locally compact Abelian groups

Salehi and Scheidt [6] have derived several Wold-Cramér concordance theorems for q-variate stationary processes over discrete groups. In this paper we characterize the concordance of the Wold decomposition with respect to families arising in the interpolation problem and the Cramér decomposition for non-full-rank q-variate stationary processes over certain nondiscrete locally compact Abelian (LCA) groups. Moreover, we give an answer to a question of Salehi and Scheidt [6, p. 319] on a characterization of the Wold-Cramér concordance with respect to J₀. As corollary we then deduce a characterization of J₀-regularity.     

Alternative models for stationary stochastic processes

We consider some parametric spectral estimators that can be used in a wide range of situations. Assuming the existence of fourth moments, we establish rates of convergence of the estimators, and a central limit theorem.     

Discriminant analysis for locally stationary processes

In this paper, we discuss discriminant analysis for locally stationary processes, which constitute a class of non-stationary processes. Consider the case where a locally stationary process {X_t,T} belongs to one of two categories described by two hypotheses π₁ and π₂. Here T is the length of the observed stretch. These hypotheses specify that {X_t,T} has time-varying spectral densities f(u,λ) and g(u,λ) under π₁ and π₂, respectively. Although Gaussianity of {X_t,T} is not assumed, we use a classification criterion D( f:g), which is an approximation of the Gaussian likelihood ratio for {X_t,T} between π₁ and π₂. Then it is shown that D( f:g) is consistent, i.e., the misclassification probabilities based on D( f:g) converge to zero as T→∞. Next, in the case when g(u,λ) is contiguous to f(u,λ), we evaluate the misclassification probabilities, and discuss non-Gaussian robustness of D( f:g). Because the spectra depend on time, the features of non-Gaussian robustness are different from those for stationary processes. It is also interesting to investigate the behavior of D( f:g) with respect to infinitesimal perturbations of the spectra. Introducing an influence function of D( f:g), we illuminate its infinitesimal behavior. Some numerical studies are given.     

Persistence probabilities for stationary increment processes

We study the persistence probability for processes with stationary increments. Our results apply to a number of examples: sums of stationary correlated random variables whose scaling limit is fractional Brownian motion; random walks in random sceneries; random processes in Brownian scenery; and the Matheron–de Marsily model in ${\mathbb{Z}}^2$ with random orientations of the horizontal layers. Using a new approach, strongly related to the study of the range, we obtain an upper bound of the optimal order in general and improved lower bounds (compared to previous literature) for many specific processes.     

Shift ergodicity for stationary Markov processes

In this paper shift ergodicity and related topics are studied for certain stationary processes. We first present a simple proof of the conclusion that every stationary Markov process is a generalized convex combination of stationary ergodic Markov processes. A direct consequence is that a stationary distribution of a Markov process is extremal if and only if the corresponding stationary Markov process is time ergodic and every stationary distribution is a generalized convex combination of such extremal ones. We then consider space ergodicity for spin flip particle systems. We prove space shift ergodicity and mixing for certain extremal invariant measures for a class of spin systems, in which most of the typical models, such as the Voter Models and the Contact Models, are included. As a consequence of these results we see that for such systems, under each of those extremal invariant measures, the space and time means of an observable coincide, an important phenomenon in statistical physics. Our results provide partial answers to certain interesting problems in spin systems.     

Symmetric iterative interpolation processes

Using a baseb and an even number of knots, we define a symmetric iterative interpolation process. The main properties of this process come from an associated functionF. The basic functional equation forF is thatF(t/b)=σ_n F(n/b)F(t-n). We prove thatF is a continuous positive definite function. We find almost precisely in which Lipschitz classes derivatives ofF belong. If a functiony is defined only on integers, this process extendsy continuously to the real axis asy(t=∑ _n y(n)F(t?n). Error bounds for this iterative interpolation are given.     

Asymptotic behavior of the prediction error for vector processes of less than full rank

We investigate the rate of change of the prediction error of a stationary (in wide sense) process of less than full rank as a function of the properties of the spectral density.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 79, pp. 17–37, 1978.     

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.     

Non-uniqueness of stationary measures for self-stabilizing processes

We investigate the existence of invariant measures for self-stabilizing diffusions. These stochastic processes represent roughly the behavior of some Brownian particle moving in a double-well landscape and attracted by its own law. This specific self-interaction leads to nonlinear stochastic differential equations and permits pointing out singular phenomena like non-uniqueness of associated stationary measures. The existence of several invariant measures is essentially based on the non-convex environment and requires generalized Laplace’s method approximations.