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
c1
c2
\partialtP(c1,t|x,t0)=-λ1P(c1,t|x,t0)+λ2P(c2,t|x,t0)
and
\partialtP(c2,t|x,t0)=λ1P(c1,t|x,t0)-λ2P(c2,t|x,t0).
where
λ1
c1
c2
λ2
c2
c1
The master equation is compactly written in a matrix form by introducing a vector
P=[P(c1,t|x,t0),P(c2,t|x,t0)]
dP | |
dt |
=WP
where
W=\begin{pmatrix} -λ1&λ2\\ λ1&-λ2 \end{pmatrix}
is the transition rate matrix. The formal solution is constructed from the initial condition
P(0)
t=t0
x
P(t)=eWtP(0)
It can be shown that[3]
eWt=I+W
(1-e-2λ) | |
2λ |
where
I
λ=(λ1+λ2)/2
t → infty
P(t → infty)=Ps
Ps=
1 | |
2λ |
\begin{pmatrix} λ2\\ λ1 \end{pmatrix}
Knowledge of an initial state decays exponentially. Therefore, for a time
t\gg(2λ)-1
Mean:
\langleX\rangles=
c1λ2+c2λ1 | |
λ1+λ2 |
.
Variance:
\operatorname{var}\{X\}s=
| |||||||||||||
|
.
One can also calculate a correlation function:
\langleX(t),X(u)\rangles=e-2λ\operatorname{var}\{X\}s.
This random process finds wide application in model building: