Telegraph process

In probability theory, the telegraph process is a memoryless continuous-time stochastic process that shows two distinct values. It models burst noise (also called popcorn noise or random telegraph signal). If the two possible values that a random variable can take are ${\displaystyle c_{1}}$ and ${\displaystyle c_{2}}$, then the process can be described by the following master equations:

${\displaystyle \partial _{t}P(c_{1},t|x,t_{0})=-\lambda _{1}P(c_{1},t|x,t_{0})+\lambda _{2}P(c_{2},t|x,t_{0})}$

and

${\displaystyle \partial _{t}P(c_{2},t|x,t_{0})=\lambda _{1}P(c_{1},t|x,t_{0})-\lambda _{2}P(c_{2},t|x,t_{0}).}$

where ${\displaystyle \lambda _{1}}$ is the transition rate for going from state ${\displaystyle c_{1}}$ to state ${\displaystyle c_{2}}$ and ${\displaystyle \lambda _{2}}$ is the transition rate for going from going from state ${\displaystyle c_{2}}$ to state ${\displaystyle c_{1}}$. The process is also known under the names Kac process (after mathematician Mark Kac),^[1] and dichotomous random process.^[2]