Evaluating the autocorrelation function of a telegraph signal to get exponential decay? 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?
A semi-random telegraph signal is one for which $X(0) = 0$ and $X(t)$ equals $0$ or $1$
according as there have been an even number or an odd number of arrivals in a Poisson Process in the interval $(0,T]$.  A simple physical model would be the least significant
bit (LSB) of a digital counter that is counting the Poisson arrivals.  A random telegraph signal is almost the same except that $X(0)$ is equally likely to have value $0$ or $1$.
(Feel free to translate this to signals having values $\pm 1$ if that is more convenient
or more in accordance with your notation).

With this as background, you are encountering problems because your simulation is
not modeling the random telegraph signal correctly. You should instead simulate a 
Poisson process and obtain the random telegraph signal from that.