I'd like to model a two state process with a random variable $X$ which can take values $a$ and $b$.
The simplest case of such a model is the dichotomous markov noise, also called the telegraph process. This is incredibly well studied and documented, and it gives exponential residence periods in each of the two states.
However, I'm interested in the case where one of the states has a heavy-tailed residence time distribution, while the other remains exponential.
Do any standard models in stochastic processes come to mind which have this property? Any advice is appreciated!
