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


The process is also known under the names Kac process (after mathematician Mark Kac),[1] and dichotomous random process.[2]


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:



One can also calculate a correlation function:


This random process finds wide application in model building:

See also[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.