# Information, rate of generation of

A quantity characterizing the amount of information (cf. Information, amount of) emanating from an information source in unit time. The rate of generation of information of an information source with discrete time generating the communication , formed by a sequence of random variables taking values from some discrete set , is defined by the equation

(*) |

if this limit exists. Here is the entropy of the random variable . The variable , defined by (*), is also called the entropy (in a symbol) of the information source.

In certain cases one can successfully prove the existence of the limit in (*) and calculate it explicitly, e.g. in the case of stationary sources. Explicit formulas for have been obtained for stationary Markov sources and Gaussian sources. The concept of a rate of generation of information is closely related to that of redundancy of an information source.

If is a stationary ergodic information source with finite number of states, then the following property of asymptotic uniform distribution (the McMillan theorem, [1]) holds. Let , where are the values of in an information interval of length . For any , there is an such that for all ,

#### References

[1] | J. Wolfowitz, "Coding theorems of information theory" , Springer (1961) |

[2] | R. Gallagher, "Information theory and reliable communication" , Wiley (1968) |

[3] | A.A. Feinstein, "Foundations of information theory" , McGraw-Hill (1958) |

#### Comments

#### References

[a1] | I. Csiszar, J. Körner, "Information theory. Coding theorems for discrete memoryless systems" , Akad. Kiado (1981) |

[a2] | A. Rényi, "A diary on information theory" , Akad. Kiado & Wiley (1987) |

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Information, rate of generation of.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Information,_rate_of_generation_of&oldid=15080