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
, then the process can be described by the following master equations: and where
λ
is the transition rate for going from state
is the transition rate for going from going from 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
) by It can be shown that[3] where
is the identity matrix and
is the average transition rate.
, 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: