Memoryless continuous-time stochastic process that shows two distinct values
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
and
, 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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6247a18800d9dc418bef26314a6a770114b6e3a0)
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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a841410aac1e43a66ee2ea24d57180aa1b9c3df2)
where
is the transition rate for going from state
to state
and
is the transition rate for going from going from state
to state
. 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 {\frac {d\mathbf {P} }{dt))=W\mathbf {P} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/93d00d83f64b333f96a25ec5922ca35907caf43e)
where
![{\displaystyle W={\begin{pmatrix}-\lambda _{1}&\lambda _{2}\\\lambda _{1}&-\lambda _{2}\end{pmatrix))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c81312665c4f825eb078340913649ec7b6d7597)
is the transition rate matrix. The formal solution is constructed from the initial condition
(that defines that at
, the state is
) by
.
It can be shown that[3]
![{\displaystyle e^{Wt}=I+W{\frac {(1-e^{-2\lambda t})}{2\lambda ))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1948e24554aff618317b409142996fcc531afd3b)
where
is the identity matrix and
is the average transition rate. As
, the solution approaches a stationary distribution
given by
![{\displaystyle \mathbf {P} _{s}={\frac {1}{2\lambda )){\begin{pmatrix}\lambda _{2}\\\lambda _{1}\end{pmatrix))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba49a5f5ab9438bb956611caea003e59ace63a71)
Properties
Knowledge of an initial state decays exponentially. Therefore, for a time
, 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))}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a710e1f6d2c6998e95122abcbc7cb86cc1887fc)
Variance:
![{\displaystyle \operatorname {var} \{X\}_{s}={\frac {(c_{1}-c_{2})^{2}\lambda _{1}\lambda _{2)){(\lambda _{1}+\lambda _{2})^{2))}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f96688df072d18ae7c2db7c944e0604a1f381b4b)
One can also calculate a correlation function:
![{\displaystyle \langle X(t),X(u)\rangle _{s}=e^{-2\lambda |t-u|}\operatorname {var} \{X\}_{s}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e75efe0d7c0563bd20c41b09ba1daf34b38564bb)
Application
This random process finds wide application in model building: