# 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 values that a random variable can take are ${\displaystyle c_{1}}$ and ${\displaystyle c_{2}}$, then the process can be described by the following master equations:

${\displaystyle \partial _{t}P(c_{1},t|x,t_{0})=-\lambda _{1}P(c_{1},t|x,t_{0})+\lambda _{2}P(c_{2},t|x,t_{0})}$

and

${\displaystyle \partial _{t}P(c_{2},t|x,t_{0})=\lambda _{1}P(c_{1},t|x,t_{0})-\lambda _{2}P(c_{2},t|x,t_{0}).}$

where ${\displaystyle \lambda _{1}}$ is the transition rate for going from state ${\displaystyle c_{1}}$ to state ${\displaystyle c_{2}}$ and ${\displaystyle \lambda _{2}}$ is the transition rate for going from going from state ${\displaystyle c_{2}}$ to state ${\displaystyle c_{1}}$. The process is also known under the names Kac process (after mathematician Mark Kac),[1] and dichotomous random process.[2]

## Solution

The master equation is compactly written in a matrix form by introducing a vector ${\displaystyle \mathbf {P} =[P(c_{1},t|x,t_{0}),P(c_{2},t|x,t_{0})]}$,

${\displaystyle {\frac {d\mathbf {P} }{dt}}=W\mathbf {P} }$

where

${\displaystyle W={\begin{pmatrix}-\lambda _{1}&\lambda _{2}\\\lambda _{1}&-\lambda _{2}\end{pmatrix}}}$

is the transition rate matrix. The formal solution is constructed from the initial condition ${\displaystyle \mathbf {P} (0)}$ (that defines that at ${\displaystyle t=t_{0}}$, the state is ${\displaystyle x}$) by

${\displaystyle \mathbf {P} (t)=e^{Wt}\mathbf {P} (0)}$.

It can be shown that[3]

${\displaystyle e^{Wt}=I+W{\frac {(1-e^{-2\lambda t})}{2\lambda }}}$

where ${\displaystyle I}$ is the identity matrix and ${\displaystyle \lambda =(\lambda _{1}+\lambda _{2})/2}$ is the average transition rate. As ${\displaystyle t\rightarrow \infty }$, the solution approaches a stationary distribution ${\displaystyle \mathbf {P} (t\rightarrow \infty )=\mathbf {P} _{s}}$ given by

${\displaystyle \mathbf {P} _{s}={\frac {1}{2\lambda }}{\begin{pmatrix}\lambda _{2}\\\lambda _{1}\end{pmatrix}}}$

## Properties

Knowledge of an initial state decays exponentially. Therefore, for a time ${\displaystyle t\gg (2\lambda )^{-1}}$, the process will reach the following stationary values, denoted by subscript s:

Mean:

${\displaystyle \langle X\rangle _{s}={\frac {c_{1}\lambda _{2}+c_{2}\lambda _{1}}{\lambda _{1}+\lambda _{2}}}.}$

Variance:

${\displaystyle \operatorname {var} \{X\}_{s}={\frac {(c_{1}-c_{2})^{2}\lambda _{1}\lambda _{2}}{(\lambda _{1}+\lambda _{2})^{2}}}.}$

One can also calculate a correlation function:

${\displaystyle \langle X(t),X(u)\rangle _{s}=e^{-2\lambda |t-u|}\operatorname {var} \{X\}_{s}.}$

## Application

This random process finds wide application in model building: