Let {$a_n$} be a wide-sense stationary process and {$X_n$} be its spectral representation (discrete Fourier transformation? ).

Let $b_n$ be the covariance function of $X_n$.

According to the Bochner Theorem, there exists a measure $\rho$ defined on the borel sigma algebra on $R$ s.t. $$ b(n) = \int_{[0, 1)}e^{2\pi i \lambda n}d\rho(\lambda)$$

How should I understand measure $\rho$?


1 Answer 1


It might be easier to instead look at $b(n)/b(0)$, which would then give $\frac{1}{b(0)}d\rho(\lambda)$ as your measure. This is now a CDF when you define $F(\lambda):=\frac{1}{b(0)}\rho([0,\lambda])$, and $b(n)/b(0)$ is the associated characteristic function. So then $\frac{1}{b(0)}d\rho$ becomes the spectral density. Explicitely $b(n)/b(0) = E[e^{inX}]$, where $X$ is, up to a constant multiple, the random variable whose distribution is the spectral density.

  • $\begingroup$ Thanks for the comment. Whats $\rho(\lambda)$ mean again? I thought it is only legit to write $d\rho(\lambda)$ $\endgroup$ Commented Aug 10, 2018 at 21:01
  • $\begingroup$ Whoops. Dropped a $d$. Fixed $\endgroup$
    – Alex R.
    Commented Aug 10, 2018 at 23:09

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