Wold Decomposition Research Articles

Predictable processes and Wold's decomposition: A review

The concepts of predictability and band-limitedness are reexamined and a simple proof of Wold's decomposition is presented in the context of mean-square estimation.

Complements to models for noncommuting operators

This paper develops a dilation theory for two noncommutative operators. Many of the results and techniques in dilation theory for one operator are extended to this setting. This is done by extending the Schaffer construction for one operator to the Fock space setting. A new Wold decomposition for two isometrics with orthogonal range is obtained. This Wold decomposition and the Schaffer construction is used to extend the Sz.-Nagy-Foias lifting theorem and the characteristic function theory to the noncommutative setting. The characteristic function is operator valued and defined in a Fock space.

Prediction of stable processes: Spectral and moving average representations

For stable processes which are Fourier transforms of processes with independent increments, we obtain a Wold decomposition, we characterize their regularity and singularity, and, in the discrete-parameter case, we derive their linear predictors. In sharp contrast with the Gaussian case, regular stable processes which are Fourier transforms of processes with independent increments are not moving averages of stable motion.

Inference robustness of ARIMA models under non-normality —Special application to stock price data

Wold's decomposition theorem [Wold] states that every weakly stationary stochastic process can be written as a linear combination of orthogonal shocks. For practical reasons, however, it is desirable to employ models which use parameters parsimoniously.Box andJenkins [1970] show how parsimony can be achieved by representing the linear process in terms of a small number of autoregressive and moving average terms (ARIMA-models). The Gaussian hypothesis assumes that the shocks follow a normal distribution with fixed mean and variance. In this case the process is characterized by first and second order moments. The normality assumption seems reasonable for many kinds of series. However, it was pointed out byKendall [1953],Mandelbrot [1963, 1967],Fama [1965],Mandelbrot andTaylor [1967] that particularly for stock price data the distribution of the shocks appears leptokurtic:In this paper we investigate the sensitivity of ARIMA models to non-normality of the distribution of the shocks. We suppose that the distribution function of the shocks is a member of the symmetric exponential power family, which includes the normal as well as leptokurtic and platikurtic distributions. A Bayesian approach is adopted and the inference robustness of ARIMA models with respect to i) the estimation of parameters ii) the forecasts of future observations is discussed.

On the construction of Wold decomposition for multivariate stationary processes

A method to construct the Wold decomposition for multivariate stationary stochastic processes x k , k ∈ Z, is presented. The method is based on orthogonal decompositions for x k , k ∈ Z, obtained by forming orthogonal projections of x k , k ∈ Z, onto its component processes x k j , k ∈ Z, j = 1, …, q. The method does not give a complete solution to the Wold decomposition problem.

Orthogonal Decomposition of Isometries in a Banach Space

In this paper the Wold decomposition theorem is proved for a class of isometries in smooth reflexive Banach spaces. The class in particular contains all isometries of ${L^p}(\mu )$ spaces for arbitrary measures $\mu$.

Orthogonal decomposition of isometries in a Banach space

In this paper the Wold decomposition theorem is proved for a class of isometries in smooth reflexive Banach spaces. The class in particular contains all isometries of L p ( μ ) {L^p}(\mu ) spaces for arbitrary measures μ \mu .

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 J 0. As corollary we then deduce a characterization of J 0-regularity.

Interpolation of q-variate weakly stationary stochastic processes over a locally compact abelian group

Let G be a discrete locally compact abelian group. Let ( x g ) g∈G be a q-variate weakly stationary stochastic process indexed by elements g of G. A. N. Kolmogorov, P. Masani, H. Salehi and others have derived numerous results on the minimality and interpolation of random processes indexed by the integers. The main efforts of this work are to derive similar results for multivariate processes indexed by the group G. Although the ideas and concepts used here are similar to the ones used earlier by other authors, some of the techniques are different, since the integers are ordered and singly generated, whereas an arbitrary group need not be. Results such as Kolmogorov's Minimality Theorem, the Wold Decomposition Theorem, the Wold-Cramér Concordance Theorem and several theorems on interpolation of ( x g ) g∈G are obtained.

Orthogonal Decompositions of Multivariate Weakly Stationary Stochastic Processes

In this paper we shall study the relations between the ranks of g-variate, discrete-parameter, weakly stationary stochastic processes x, y, and z satisfying the condition1.1and derive from them a characterization for the Wold decomposition and conditions for the concordance of the Wold and the Lebesgue-Cramér decompositions.

Properties of the Wold Decomposition of Stationary Stochastic Processes

The basic results of the paper (Theorems 11–13) treat the representation of the quantities $\hat x_{t + \alpha } $ – the best predictors of the quantities $x_{t + \alpha } $ of a process, which is stationary in the wide sense, from the quantities $x_s , s \leqq t$ – in the form of a series \[ \hat x_{t + \alpha } \sim \sum\limits_{s = 0}^\infty {k_s x_{t - s} } ,\]where the coefficients ${k_s }$ satisfy the condition $\sum |k_s |^2 < \infty $. Certain properties of the sequences $\{ w_t \} \sum ,{|w_t |}^2 < \infty $, are derived first.