Telegraph process explained

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

and

c2

, then the process can be described by the following master equations:

\partialtP(c1,t|x,t0)=-λ1P(c1,t|x,t0)2P(c2,t|x,t0)

and

\partialtP(c2,t|x,t0)1P(c1,t|x,t0)2P(c2,t|x,t0).

where

λ1

is the transition rate for going from state

c1

to state

c2

and

λ2

is the transition rate for going from going from state

c2

to state

c1

. 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

P=[P(c1,t|x,t0),P(c2,t|x,t0)]

,
dP
dt

=WP

where

W=\begin{pmatrix} 1&λ2\\ λ1&2 \end{pmatrix}

is the transition rate matrix. The formal solution is constructed from the initial condition

P(0)

(that defines that at

t=t0

, the state is

x

) by

P(t)=eWtP(0)

.

It can be shown that[3]

eWt=I+W

(1-e-2λ)

where

I

is the identity matrix and

λ=(λ1+λ2)/2

is the average transition rate. As

tinfty

, the solution approaches a stationary distribution

P(tinfty)=Ps

given by

Ps=

1

\begin{pmatrix} λ2\\ λ1 \end{pmatrix}

Properties

Knowledge of an initial state decays exponentially. Therefore, for a time

t\gg()-1

, the process will reach the following stationary values, denoted by subscript s:

Mean:

\langleX\rangles=

c2+c1
λ1+λ2

.

Variance:

\operatorname{var}\{X\}s=

(c
2
1-c
(λ
2
2)
1+λ

.

One can also calculate a correlation function:

\langleX(t),X(u)\rangles=e-2λ\operatorname{var}\{X\}s.

Application

This random process finds wide application in model building:

See also

References

  1. 10.1023/A:1009437108439 . Bondarenko . YV . 2000 . Probabilistic Model for Description of Evolution of Financial Indices . Cybernetics and Systems Analysis . 36 . 5. 738–742 . 115293176 .
  2. Margolin . G . Barkai . E . 2006 . Nonergodicity of a Time Series Obeying Lévy Statistics . Journal of Statistical Physics . 122 . 1. 137–167 . 10.1007/s10955-005-8076-9 . 2006JSP...122..137M. cond-mat/0504454 . 53625405 .
  3. [V. Balakrishnan (physicist)|Balakrishnan, V.]