Definition:Autocovariance Matrix

Definition
Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$.

Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$:
 * $\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$

The autocovariance matrix associated with $S$ for $\sequence {s_n}$ is:


 * $\boldsymbol \Gamma_n = \begin {pmatrix}

\gamma_0 & \gamma_1 & \gamma_2 & \cdots & \gamma_{n - 1} \\ \gamma_1 & \gamma_0 & \gamma_1 & \cdots & \gamma_{n - 2} \\ \gamma_2 & \gamma_1 & \gamma_0 & \cdots & \gamma_{n - 3} \\ \vdots  & \vdots   & \vdots   & \ddots & \vdots \\ \gamma_{n - 1} & \gamma_{n - 2} & \gamma_{n - 3} & \cdots & \gamma_0 \end {pmatrix}$

where $\gamma_k$ is the autocovariance of $S$ at lag $k$.

That is, such that:
 * $\sqbrk {\Gamma_n}_{i j} = \gamma_{\size {i - j} }$

Also see

 * Definition:Autocorrelation Matrix