telegraph process

{{short description|Memoryless continuous-time stochastic process that shows two distinct values}}

{{about|the probabilistic model|the electrical transmission line|Telegrapher's equations}}

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),{{cite journal | doi = 10.1023/A:1009437108439 | last1 = Bondarenko | first1 = YV | year = 2000 | title = Probabilistic Model for Description of Evolution of Financial Indices | journal = Cybernetics and Systems Analysis | volume = 36 | issue = 5| pages = 738–742 | s2cid = 115293176 }} and dichotomous random process.{{cite journal | last1 = Margolin | first1 = G | last2 = Barkai | first2 = E | year = 2006 | title = Nonergodicity of a Time Series Obeying Lévy Statistics | journal = Journal of Statistical Physics | volume = 122 | issue = 1| pages = 137–167 | doi =10.1007/s10955-005-8076-9 |bibcode=2006JSP...122..137M|arxiv = cond-mat/0504454 | s2cid = 53625405 }}

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 thatBalakrishnan, V. (2020). Mathematical Physics: Applications and Problems. Springer International Publishing. pp. 474

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

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:

In physics, spin systems and fluorescence intermittency show dichotomous properties. But especially in single molecule experiments probability distributions featuring algebraic tails are used instead of the exponential distribution implied in all formulas above.
In finance for describing stock prices
In biology for describing transcription factor binding and unbinding.

References

Category:Stochastic differential equations

telegraph process

Solution

Properties

Application

See also

References