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),[1] and dichotomous random process.[2]
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[3]
where is the identity matrix and is the average transition rate. As , the solution approaches a stationary distribution given by
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:
This random process finds wide application in model building:
Balakrishnan, V. (2020). Mathematical Physics: Applications and Problems. Springer International Publishing. pp. 474