periodicity of a Markov chain

Let {Xn} be a stationary ( Markov chainMathworldPlanetmath with state spacePlanetmathPlanetmath I. Let Pijn be the n-step transition probability that the process goes from state i at time 0 to state j at time n:


Given any state iI, define the set


It is not hard to see that if n,mN(i), then n+mN(i). The period of i, denoted by d(i), is defined as

d(i):={0if N(i)=,gcd(N(i))otherwise,

where gcd(N(i)) is the greatest common divisorMathworldPlanetmath of all positive integers in N(i).

A state iI is said to be aperiodic if d(i)=1. A Markov chain is called aperiodic if every state is aperiodic.

Property. If states i,jI communicate (, then d(i)=d(j).


We will employ a common inequality involving the n-step transition probabilities:


for any i,j,kI and non-negative integers m,n.

Suppose first that d(i)=0. Since ij, Pijn>0 and Pjim>0 for some n,m0. This implies that Piim+n>0, which forces m+n=0 or m=n=0, and hence j=i.

Next, assume d(i)>0, this means that N(i). Since ij, there are r,s0 such that Pjir>0 and Pijs>0, and so Pjjr+s>0, showing r+sN(j). If we pick any nN, we also have Pjjr+n+sPjirPiinPijs>0, or r+s+nN(j). But this means d(j) divides both r+s and r+s+n, and so d(j) divides their differencePlanetmathPlanetmath, which is n. Since n is arbitrarily picked, d(j)d(i). Similarly, d(i)d(j). Hence d(i)=d(j). ∎

Title periodicity of a Markov chain
Canonical name PeriodicityOfAMarkovChain
Date of creation 2013-03-22 16:24:28
Last modified on 2013-03-22 16:24:28
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 60J10
Defines period of a state
Defines aperiodic state
Defines aperiodic Markov chain