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单词 RecurrenceInAMarkovChain
释义

recurrence in a Markov chain


Let {Xn} be a stationary (http://planetmath.org/StationaryProcess) Markov chainMathworldPlanetmath and I the state spacePlanetmathPlanetmath. Given i,jI and any non-negative integer n, define a number Fijn as follows:

Fijn:={0if n=0,P(Xn=j and Xmj for 0<m<nX0=i)otherwise.

In other words, Fijn is the probability that the process first reaches state j at time n from state i at time 0.

From the definition of Fijn, we see that the probability of the process reaching state j within and including time n from state i at time 0 is given by

m=0nFijm.

As n, we have the limiting probability of the process reaching j eventually from the initial state of i at 0, which we denote by Fij:

Fij:=m=0Fijm.

Definitions. A state iI is said to be recurrent or persistent if Fii=1, and transient otherwise.

Given a recurrent state i, we can further classify it according to “how soon” the state i returns after its initial appearance. Given Fiin, we can calculate the expected number of steps or transitions required to return to state i by time n. This expectation is given by

m=0nmFiim.

When n, the above expression may or may not approach a limit. It is the expected number of transitions needed to return to state i at all from the beginning. We denote this figure by μi:

μi:=m=0mFiim.

Definitions. A recurrent state iI is said to be or strongly ergodic if μi<, otherwise it is called null or weakly ergodic. If a stronly ergodic state is in additionPlanetmathPlanetmath aperiodic (http://planetmath.org/PeriodicityOfAMarkovChain), then it is said to be an ergodic state.

Titlerecurrence in a Markov chain
Canonical nameRecurrenceInAMarkovChain
Date of creation2013-03-22 16:24:43
Last modified on2013-03-22 16:24:43
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id5
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 60J10
Synonymnull recurrent
Synonympositive recurrent
Synonymstrongly ergodic
Synonymweakly ergodic
Definesrecurrent state
Definespersistent state
Definestransient state
Definesnull state
Definespositive state
Definesergodic state

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