Telegraph process

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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:

and

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[edit]

The master equation is compactly written in a matrix form by introducing a vector ,

where

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]

where is the identity matrix and is the average transition rate. As , the solution approaches a stationary distribution given by

Properties[edit]

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

Mean:

Variance:

One can also calculate a correlation function:

Application[edit]

This random process finds wide application in model building:

See also[edit]

References[edit]

  1. ^ a b Bondarenko, YV (2000). "Probabilistic Model for Description of Evolution of Financial Indices". Cybernetics and Systems Analysis. 36 (5): 738–742. doi:10.1023/A:1009437108439.
  2. ^ Margolin, G; Barkai, E (2006). "Nonergodicity of a Time Series Obeying Lévy Statistics". Journal of Statistical Physics. 122 (1): 137–167. arXiv:cond-mat/0504454. Bibcode:2006JSP...122..137M. doi:10.1007/s10955-005-8076-9.
  3. ^ Balakrishnan, V. (2020). Mathematical Physics: Applications and Problems. Springer International Publishing. pp. 474