# Telegraph process

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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:

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

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

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

$W={\begin{pmatrix}-\lambda _{1}&\lambda _{2}\\\lambda _{1}&-\lambda _{2}\end{pmatrix}}$ is the transition rate matrix. The formal solution is constructed from the initial condition $\mathbf {P} (0)$ (that defines that at $t=t_{0}$ , the state is $x$ ) by

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

It can be shown that

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

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

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

Mean:

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

$\operatorname {var} \{X\}_{s}={\frac {(c_{1}-c_{2})^{2}\lambda _{1}\lambda _{2}}{(\lambda _{1}+\lambda _{2})^{2}}}.$ One can also calculate a correlation function:

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

This random process finds wide application in model building: