Telegraph process

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In probability theory, the telegraph process is a memoryless continuous-time stochastic process that shows two distinct values.

If these are called a and b, the process can be described by the following master equations:

\partial_t P(a, t|x, t_0)=-\lambda P(a, t|x, t_0)+\mu P(b, t|x, t_0)

and

\partial_t P(b, t|x, t_0)=\lambda P(a, t|x, t_0)-\mu P(b, t|x, t_0).

The process is also known under the names Kac process[1] , dichotomous random process.[2]

Properties

Knowledge of an initial state decays exponentially. Therefore for a time in the remote future, the process will reach the following stationary values, denoted by subscript s:

Mean:

\langle X \rangle_s = \frac {a\mu+b\lambda}{\mu+\lambda}.

Variance:

 \operatorname{var} \{ X \}_s = \frac {(a-b)^2\mu\lambda}{(\mu+\lambda)^2}.

One can also calculate a correlation function:

\langle X(t),X(s)\rangle_s = \exp(-(\lambda+\mu)|t-s|) \operatorname{var} \{ X \}_s.

Application

This random process finds wide application in model building:

See also

References

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