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 c_1 and c_2, then the process can be described by the following master equations: and where \lambda_1 is the transition rate for going from state c_1 to state c_2 and \lambda_2 is the transition rate for going from going from state c_2 to state c_1. 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 t=t_0, the state is x) by It can be shown that where I 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: