PP09 (VII)

Last time we have defined the spectrum of a point process.

In general, spectral properties of stochastic processes are an interesting and difficult topic. Interesting because of the amount of information that can be extracted by analysis in frequency domain, and difficult mainly because it is quite difficult to measure the spectrum of a stochastic process.

Why it is difficult? To measure the spectrum, one idea could be to sample the stochastic process for a very long time, to plug the observed values into the definition of the Fourier transform, and take this as an estimation of the spectrum.

Does it work? If you do this, you're actually computing the periodogram of the stochastic process: as you can read in wikipedia, the periodogram has the problem to be a non-consistent estimator of the spectrum, and this fact can't be avoided but by collecting more observations of the process.

So, surprisingly, the key is not to observe the process for a long time, but to collect many independent observations of them.

Of course, observing for long times also is useful, e.g. to reduce spectral leakage, but this is another story...