Power-law distribution from superposition of exponential distributions

In the previous post we have asked a simple question - what if gap duration is distributed according to a power-law distribution. Our answer was an apparently pure 1/f noise. But how could the power-law distribution arise? While in some materials some quantities indeed follow power-law distribution (such as blinking times in quantum dot experiments [1]), most materials have exponentially distributed generation and/or recombination times. So then, can we construct a power-law distribution from superposition of exponential distributions?

Power-law gap times in random telegraph noise

In the last post we have assumed that the event rate of the individual processes making up the superposition needs to be distributed according the bounded Pareto distribution. And we have seen that in \( \alpha = 0 \) case, 1/f noise will be obtained. But superposition of such processes seems to be very demanding assumption, maybe we can choose another assumption? Here we will see what happens if we assume that gap times are assumed to follow bounded Pareto distribution, while the pulse times will still follow exponential distribution.

Mathemaniac: Queuing theory and Poisson process

Recently we have been looking a lot into Poisson process and other related more physical topics. But the things we have been talking about also applies to our earlier more social science related works. This video by Mathemaniac discusses how the statistical methodology behind the Poisson process can be used to understand queues. As usual, we invite you to watch this fascinating video!

Superposition of Lorentzians with fixed height pulses

In the last post we have seen that 1/f noise can be obtained from superposition of Lorentzian power spectral densities. Though we have imposed an odd requirement of scaling pulse height for different characteristic rates \( \lambda \). In this post let us explore how to achieve the same thing with fixed height pulses.

Obtaining 1/f noise from superposition of Lorentzians

Earlier we have taken a look at a power spectral density of a signal generated by a single charge carrier. We have seen that it generates a signal composed of pulses and gaps. If the pulse and gap duration is distributed according to the exponential distribution (i.e., Poisson process model applies), then the power spectral density of the signal has a characteristic Lorentzian shape. Which is nothing alike 1/f noise, which we are looking for. But maybe we can still obtain it?