class structure

Let (Xn)n1 be a stationary Markov chainMathworldPlanetmath and let i and j be states in the indexing set. We say that i leads to j or j is accessiblePlanetmathPlanetmath from i, and write ij, if it is possible for the chain to get from state i to state j:

ijP(Xn=j:X0=i)>0for somen0

If ij and ji we say i communicates with j and write ij. is an equivalence relationMathworldPlanetmath (easy to prove). The equivalence classesMathworldPlanetmath of this relationMathworldPlanetmath are the communicating classes of the chain. If there is just one class, we say the chain is an irreducible chain.

A class C is a closed class if iC and ij implies that jC “Once the chain enters a closed class, it cannot leave it”

A state i is an absorbing state if {i} is a closed class.

Title class structure
Canonical name ClassStructure
Date of creation 2013-03-22 14:18:21
Last modified on 2013-03-22 14:18:21
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 12
Author CWoo (3771)
Entry type Definition
Classification msc 60J10
Related topic MarkovChain
Defines communicating class
Defines irreducible chain
Defines closed class
Defines absorbing state