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# f. [32, p. 98]) the notion of a regular stationary numerical

DOI link for f. [32, p. 98]) the notion of a regular stationary numerical

f. [32, p. 98]) the notion of a regular stationary numerical book

# f. [32, p. 98]) the notion of a regular stationary numerical

DOI link for f. [32, p. 98]) the notion of a regular stationary numerical

f. [32, p. 98]) the notion of a regular stationary numerical book

## ABSTRACT

Now we elaborate the preceding to stationary vector numerical sequences, or jointly stationary numerical sequences.

First, if { z ( k ) , k = 0 , 1,...,if K-1} is a collection of numerical sequences, we define

(2.10)

,W 3

1 2M + 1

1 2M + 1

(2.12)

(2.11)

B. The Wold Isomorphism

In order to introduce the Wold isomorphism let us now assume that we have a regular stationary numerical sequence x. Then for any scalar , the mean of x + Skx exists as does

||x + Skx||2 < . (2.14) This is essential but is not clearly brought out in [33]. Thus we may introduce the linear space M (x) defined as the (finite) linear span

M (x) = sp {Skx | k integer},

and on this linear space (x1, x2) of (2.5) is seen to have the properties of symmetry, additivity and homogeneity; that is, (a) (x1, x2) = (x2, x1), (b) ((x1 + x2), x3) = (x1, x2) + (x2, x3) and (c) ( x1, x2) = (x1, x2) for ev ery complex and all x1, x2, x3 in M 1(x). In order for (x1, x2) to be positive definite in the sense that || x ||2= 0 x = 0 we interpret 0 as the (equiv alence) class of sequences that are equivalent to the zero sequence (the com plex sequence with xj = 0 for every j ) . This expresses the sense of uniqueness implied by the norm induced by the scalar product (2.5). For example any two sequences x1, x2 for which z = x1 – x2 2 are clearly equivalent be cause z satisfies ||x1 – x2||2 = 0. Thus any finite sequence is equivalent to 0 and any two sequences differing in a finite number of positions are equivalent.