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 and , then the process can be described by the following master equations: and where is the transition rate for going from state to state and is the transition rate for going from going from state to state . The process is also known under the names Kac process (after mathematician Mark Kac), and dichotomous random process. Solution The master equation is compactly written in a matrix form by introducing a vector , where is the transition rate matrix. The formal solution is constructed from the initial condition (that defines that at , the state is ) by . It can be shown that where is the identity matrix and is the average transition rate. As , the solution approaches a stationary distribution given by Properties Knowledge of an initial state decays exponentially. Therefore, for a time , the process will reach the following stationary values, denoted by subscript s: Mean: Variance: One can also calculate a correlation function: