# Spectral Representation of Real Valued Vector Time Series

• Jan Beran
Chapter

## 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.

## References

1. Brockwell, P. J., & Davis, R. A. (1991). Time series: Theory and methods. New York: Springer.Google Scholar
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3. Hannan, E. J. (1970). Multiple time series. New York: Wiley.Google Scholar