7.1 THE SPECTRAL DENSITY MATRIX AND ITS INTERPRETATION

In this chapter we extend the results of Chapter 5 to cover the case of the joint behavior of second-order statistics based on various components of a vector-valued stationary time series.

Let X(*t*), t=0,±1,… be an *r* vector-valued series with component series Xa (t) , t=0,±1,… for a=1,… ,r . Suppose EX (t) = cX 7.1.1 E [X (t+u) − cX ] [X (t) − cX ]τ =cov {X (t+u), X (t) } = cXX (u) for t,u=0,±1,…. 7.1.2 Indicate the individual entries of cX by ca , a=1,… ,r so ca =E Xa (t) is the mean of the series Xa (t) , t=0,±1,… . Denote the entry in row *a*, column *b* of cXX (u) by cab (u) , a,b=1,… ,r , so cab (u) is the cross-covariance function of the series Xa (t) with the series Xb (t) . Note that cXX (u)τ =cov {X (t),X (t+u) } = cXX (−u) for u=0,±1,…. 7.1.3