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 states are called a and b, the process can be described by the following master equations:

${\displaystyle \partial _{t}P(a,t|x,t_{0})=-\lambda P(a,t|x,t_{0})+\mu P(b,t|x,t_{0})}$

and

${\displaystyle \partial _{t}P(b,t|x,t_{0})=\lambda P(a,t|x,t_{0})-\mu P(b,t|x,t_{0}).}$

The process is also known under the names Kac process[1] , dichotomous random process.[2]

Properties

Knowledge of an initial state decays exponentially. Therefore, for a time in the remote future, the process will reach the following stationary values, denoted by subscript s:

Mean:

${\displaystyle \langle X\rangle _{s}={\frac {a\mu +b\lambda }{\mu +\lambda }}.}$

Variance:

${\displaystyle \operatorname {var} \{X\}_{s}={\frac {(a-b)^{2}\mu \lambda }{(\mu +\lambda )^{2}}}.}$

One can also calculate a correlation function:

${\displaystyle \langle X(t),X(s)\rangle _{s}=\exp(-(\lambda +\mu )|t-s|)\operatorname {var} \{X\}_{s}.}$

Application

This random process finds wide application in model building: