I've been driving myself absolutely crazy trying to do autocorrelation analysis on some data that I've got. I've read from multiple sources that the ACF of a random telegraph signal / random telegraph noise is given by an exponentially decaying function where the decay constant is the average rate of switching between states (cf. Example 7.3.2 here for example). Since I had trouble getting anything that even resembled exponential decay with real data, I decided to generate a fake signal that telegraphs between the value 0 and 1 with lifetimes centered binomially around two characteristic values for the 0 and 1 state. The autocorrelation function for this signal is still not at all exponential. In each case, the ACF ends up strongly oscillating.
I'm working in Mathematica, using CorrelationFunction to get the autocorrelation function. Based on my reading, this is a standard definition for autocorrelation.
So my question is can someone show me an example with a numerically obtained ACF from a random telegraph signal that decays exponentially? Am I missing something stupid here? Am I using an incorrect definition for the autocorrelation function?