Spectral Representation of Real Valued Vector Time Series

Abstract

$$X_{t}=\left ( X_{t,1},\ldots ,X_{t,m}\right ) ^{T}\in \mathbb {R}^{m}\text{ (} t\in \mathbb {Z}\text{)}$$

is called weakly stationary or second order stationary, if

$$\displaystyle\forall t\in \mathbb {Z}:E(X_{t})=\mu \in \mathbb {R}^{m},$$

and

$$\displaystyle\exists \gamma :\mathbb {Z}\rightarrow M\left ( m,m,\mathbb {R}\right ),\ k\rightarrow \gamma \left ( k \right ) =\left [\gamma _{jl}\left ( k \right )\right ]_{j,l=1,\ldots ,m} \text{ s.t.:}$$

$$\displaystyle\forall t,k\in \mathbb {Z}:cov\left ( X_{t+k,j},X_{t,l}\right ) =\gamma _{jl}\left ( k\right ) \text{ (}j,l=1,2,\ldots ,m\text{).}$$

The functions

$$\displaystyle\gamma \left ( k\right ) =\left [ \gamma _{jl}\left ( k\right ) \right ] _{j,l=1,\ldots ,m}\text{ (}k\in \mathbb {Z}\text{)}$$

and

$$\displaystyle\rho \left ( k\right ) =\left [ \rho _{jl}\left ( k\right ) \right ] _{j,l=1,\ldots ,m}=\left [ \frac {\gamma _{jl}\left ( k\right ) }{\sqrt {\gamma _{jj}\left ( 0\right ) \gamma _{ll}\left ( 0\right ) }}\right ] _{j,l=1,\ldots ,m} \text{ (}k\in \mathbb {Z}\text{)}$$

are called autocovariance and autocorrelation function of X _t respectively. Also, for j ≠ l, γ _jl and ρ _jl are called cross-autocovariance and cross-autocorrelation function respectively.