Telegraph process


In probability theory, the telegraph process is a memoryless continuous-time stochastic process that shows two distinct values. It models burst noise. If the two possible values that a random variable can take are c_1 and c_2, 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, 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 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:

Application

This random process finds wide application in model building: