How to retrieve information of autocovariance from autocovariance generating function?

Question

My lecturer introduced the concept autocovariance generating function (ACGF) of a stationary process in class.

$G(L) = \sum_{k = -\infty}^{+\infty} \gamma_kL^k$ [1]

$\gamma_k$ is autocovariance at k-th lag.

She also mentioned that we can retrieve information of $\gamma_k$ from this by differentiating this polynomial k times then evaluate the derivative at $L = 0$. Finally, divide the result by $k!$.

However, I think it is mathematically wrong as it is not possible to evaluate the derivative at $L = 0$ at any term $L^k$ where k is negative. It is possible that she mentioned another kind of ACGF which is $G(L) = \sum_{k = 0}^{+\infty} \gamma_kL^k$. Am I correct?

Furthermore, I want to know how I can retrieve information about autocovariance from the first ACGF. I can do it by deriving the spectral density function from ACGF. However, I want to know if there is another way to do it.

Answer 1

Turning comment to answer.

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If $ \langle\gamma_j\rangle_{j =-\infty}^\infty$ is absolutely summable, then the autocovariances can be summarized using a scalar-valued function, the acv generating function $$g_Y(z) := \sum_{j =-\infty}^\infty \gamma_jz^j.$$

If the linear process is $Y_t = \psi(B) \varepsilon_t,$ then acv generating function can be shown to be

\begin{align} g_Y(z) &= \sum_{j =-\infty}^\infty \gamma_jB^j\\ &= \sigma^2\psi(B) \psi(B^{-1}) ;\end{align} $\gamma_j$ can be found out by observing the coefficient of $B^j$ (also $B^{-j}$) .

Since the process is absolutely summable, one can also resort to population spectrum $$ s_Y(\omega) = \frac{1}{2\pi} \sum_{j =-\infty}^\infty \gamma_j\exp(-i\omega j) $$ to retrieve the acv s.

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Reference:

Time Series Analysis, James D. Hamilton, Princeton University Press, 1994.