In the previous post we have seen that introducing memory into the memory-less Poisson process makes things very complicated. But we haven't yet still seen 1/f noise. So now let us consider power-law distributed inter-event times!

Over the last few posts we have been talking about the power spectral density of different random processes. We have used the fact that it is a modulus of the signal's Fourier transform squared. But what is the meaning of the power spectral density? Watch the video by "Iain Explains" below for some intuition.

In the previous post we have seen that the Poisson process generates white noise, which is not unexpected consequence of exponential distribution being a limit of geometric distribution. So, if we use non-exponential inter-event time distribution, we introduce memory into the process. As the distribution of inter-event times is no longer exponential, we now have not a Poisson process, but a point process.

Earlier we have started talking about the Poisson processes. In the few posts before the summer holidays we have driven our discussion towards waiting time paradox, which is an interesting phenomenon we encounter in our day-to-day lives. Here, on Physics of Risk we have an interest in colors of noise exhibited by variety of stochastic processes. Thus in the next few posts let us examine the power spectral density of the Poisson process.

Prof. Allen Downey has written quite a few books I have enjoyed reading ("Think Java", "Think Stats", "Think Bayes" and "Think Complexity" to name a few). In this talk at SciPy 2023 conference he gives an excellent introduction into the long-tailed distributions. I invite you to watch it!