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 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.

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: 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 share prices In biology for describing transcription factor binding and unbinding.

See also

Markov chain List of stochastic processes topics Random telegraph signal

References

Text derived from https://en.wikipedia.org/w/index.php?oldid=1332976442 under CC BY-SA 4.0; modifications have been made.