1,359,265 research outputs found
A multivariate generalization of the von Neumann-Wold decomposition
No full textLet H be a complex infinite-dimensional separable Hilbert space. If T is an isometry acting on H, then the von Neumann-Wold decomposition theorem asserts that T can be expressed as a direct sum of the unilateral shift (of some multiplicity) and a unitary operator. We establish a multivariate generalization of the von Neumann-Wold decomposition and explore some of the implications of that generalization. In particular we derive a universal representation theorem for members of a special class of spherical isometrics and verify that any member of that class is hyperreflexive
The Wold Decomposition
No full textIn Chapter 7 in Bierens (2004) the Wold decomposition was motivated by claiming that every zero-mean covariance stationary processXt can be written as Xt = P∞ j=1 βjXt−j + Ut, where E[Ut.Xt−j] = 0 for all j ≥ 1, and P∞ j=1 βjXt−j is the projection of Xt on its past. However, in general this claim is incorrect. In this note I will give a more general (and hopefully correct) proof of the Wold decomposition. 1 Projections on spaces spanned by a sequence The fundamental projection theorem states that: Theorem 1. Given a sub-Hilbert space S of a Hilbert space H and an element y ∈ H, there exists a unique element by ∈ S such that ||y − by| | = infz∈S ||y − z||. Moreover the residual u = y − by is orthogonal to any z ∈ S: hu, zi = 0. Proof: See for example Bierens (2004, Th. 7.A.3, p. 202). This result is the basis for the famous Wold (1938) decomposition for covariance stationary time series, which in its turn is the basis for time series analysis. ∗Thanks to Peter Boswijk (University of Amsterdam) for pointing out an error in a previous version of this note. Moreover, the queries of the students in my graduate time series courses have led to substantial improvements of the proof of the Wold decomposition. 1 The proof of the Wold decomposition in Anderson (1994) is more trans-parent than the original proof by Wold (1938). However, rather than follow-ing Anderson’s proof, I will in this note derive first a general Wold decompo-sition for a regular sequence1 in a general Hilbert space, and then specialize this result to the Wold decomposition for covariance stationary time series. First, we need to define sub-Hilbert spaces spanned by a sequence in a Hilbert space, as follows. Let {xk}∞k=1 be a sequence of elements of a Hilbert space H, and let Mm = span({xj}mj=1) be the space spanned by x1,..., xm, i.e.,Mm consists of all linear combination
The prediction theory of stationary random fields. III. Fourfold Wold decompositions
No full textAbstractIn this paper, we investigate various fourfold Wold-type decompositions of stationary random fields under different hypotheses of commutation properties. Spectral characterizations of the three multiplicities of the innovation subspaces are obtained. The equivalence relations between the weak commutation property, fourfold Wold-type decomposition, and quarter-plane moving average representation are proved. A complete spectral characterization of the weak commutation property is also given
The delimitation of Giffenity for the Wold-Juréen (1953) utility function using relative prices: A note
No full textIn the study of Giffen behavior or "Giffenity", there remains a paradox. On one hand, the Wold-Juréen (Demand analysis: A study in Econometrics, 1953) utility function has been touted as the progenitor of a multi-decade search for those two-good, particular utility functions, which exhibit Giffenity. On the other hand, there is no evidence that the Wold- Juréen (1953) utility function has ever been fully evaluated for Giffenity, with perhaps one minor exception, Weber (The case of a Giffen good: Comment, 1997). But there, Weber (1997) showed that the Giffenity of Good 1 depends upon the relative magnitude of income vis-à-vis the price of Good 2. Weber's precondition is so vague that it lacks broad appeal. This paper offers a new and a clear cut precondition for Giffen behavior under the Wold-Juréen (1953) utility function. That is, the author shows that if the price of Good 1 is greater than or equal to the price of Good 2, then Good 1 is a Giffen good
Multivariate Wold decompositions: a Hilbert A-module approach
No full textOrthogonal decompositions are essential tools for the study of weakly stationary time series. Some examples are given by the classical Wold decomposition of Wold (A study in the analysis of stationary time series, Almqvist & Wiksells Boktryckeri, Uppsala, 1938) and the extended Wold decomposition of Ortu et al. (Quant Econ 11(1):203–230, 2020), which permits to disentangle shocks with heterogeneous degrees of persistence from a given weakly stationary process. The analysis becomes more involved when dealing with vector processes because of the presence of different simultaneous shocks. In this paper, we recast the standard treatment of multivariate time series in terms of Hilbert A-modules (where matrices replace the field of scalars) and we prove the abstract Wold theorem for self-dual pre-Hilbert A-modules with an isometric operator. This theorem allows us to easily retrieve the multivariate classical Wold decomposition and the multivariate version of the extended Wold decomposition. The theory helps in handling matrix coefficients and computing orthogonal projections on closed submodules. The orthogonality notion is key to decompose the given vector process into uncorrelated subseries, and it implies a variance decomposition
Wold-Cramér concordance theorems for interpolation of q-variate stationary processes over locally compact Abelian groups
No full textSalehi 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.Locally compact Abelian group Matrix-valued measure q-variate stationary processes Spectral measure Linear interpolation Wold decomposition Cramer decomposition Wold-Cramer concordance theorem J0-regularity
On a Conjecture Concerning a Theorem of Cramér and Wold
No full textAbstractA conjecture concerning the Cramér–Wold device is answered in the negative by giving a Fourier-free, probabilistic proof using only elementary techniques. It is also shown how a geometric idea allows one to interpret the Cramér–Wold device as a special case of a more general concept
Wold decomposition for doubly commuting isometries
No full textIn this paper, we obtain a complete description of the class of n-tuples (n ≥ 2) of doubly commuting isometries. In particular, we present a several variables analogue of the Wold decomposition for isometries on Hilbert spaces. Our main result is a generalization of M. Slocinskiʼs Wold-type decomposition of a pair of doubly commuting isometries
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