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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. (SOLVED in the comment section. I confused myself this part)

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.

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.

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. (SOLVED in the comment section. I confused myself this part)