Is there any intuition behind the spectral density $f(\lambda)$ of a time series, where

$$ f(\lambda)= \frac{1}{2\pi}\sum_{h=-\infty}^{+\infty}{e^{-ih\lambda}\gamma(h)}, -\infty < \lambda < \infty\\ $$, where $\gamma(h)$ is the autocovariance function.

Thanks in advance!


1 Answer 1


The Fourier transform decomposes a process into its constituent frequencies. Since we are decomposing the autocovariance, the spectral density tell us how much variance is contributed to a process by (at?) each frequency $\lambda$.

One way to see this is that if the variance of our times series is $\sigma^{2}$ then $$\sigma^{2} = \int_{-\infty}^{\infty} f(\lambda) \; d\lambda$$

(This is the same as inverse Fourier transforming the spectral density at $h = 0$, i.e., the lag-0 autocovariance.)


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