Inprobability theory, thetelegraph process is amemoryless continuous-timestochastic process that shows two distinct values. It modelsburst noise (also called popcorn noise or random telegraph signal). If the two possible values that arandom variable can take are${\displaystyle c_1}$ and${\displaystyle c_2}$, then the process can be described by the followingmaster equations:

${\displaystyle {\mathrm{\partial }}_{t}P(c_1,t|x,t_0)=-{\lambda }_{1}P(c_1,t|x,t_0)+{\lambda }_{2}P(c_2,t|x,t_0)}$

and

${\displaystyle {\mathrm{\partial }}_{t}P(c_2,t|x,t_0)={\lambda }_{1}P(c_1,t|x,t_0)-{\lambda }_{2}P(c_2,t|x,t_0).}$

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 namesKac process (after mathematicianMark Kac),^[1] anddichotomous random process.^[2]

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

is thetransition 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${\displaystyle t\to \mathrm{\infty }}$, the solution approaches a stationary distribution${\displaystyle \mathbf{P}(t\to \mathrm{\infty })={\mathbf{P}}_s}$ given by

${\displaystyle {\mathbf{P}}_s=\frac{1}{2\lambda }\left(\begin{array}{c}{\lambda }_2\\ {\lambda }_1\end{array}\right)}$

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

Mean:

${\displaystyle ⟨X{⟩}_s=\frac{c_1{\lambda }_2+c_2{\lambda }_1}{{\lambda }_1+{\lambda }_2}.}$

Variance:

${\displaystyle \mathrm{var}\{X{\}}_s=\frac{(c_1-c_2{)}^2{\lambda }_1{\lambda }_2}{({\lambda }_1+{\lambda }_2{)}^2}.}$

One can also calculate acorrelation function:

${\displaystyle ⟨X(t),X(u){⟩}_s=e^{-2\lambda |t-u|}\mathrm{var}\{X{\}}_s.}$

This random process finds wide application in model building:

• Inphysics,spin systems andfluorescenceintermittency show dichotomous properties. But especially insingle molecule experimentsprobability distributions featuringalgebraic tails are used instead of theexponential distribution implied in all formulas above.
• Infinance for describingstock prices^[1]
• Inbiology for describingtranscription factor binding and unbinding.

• Markov chain
• List of stochastic processes topics
• Random telegraph signal

• Stochastic differential equations

• Articles with short description
• Short description matches Wikidata