This is a very basic concept, which I'm confused with. I'm struggling to understand why a chain having the transition probabilities shown below would be considered irreducible as I thought every element in the Transition Matrix had to be positive, but in this case there are elements with Zeros.

Given the Markov Chain with state space ${0,1,2,3,4,5,6}$ and transition probabilities $p(0,0) = 0.75$, $p(0,1) = 0.25$, $p(1,0) = 0.5$, $p(1,1) = 0.25$, $p(1,2) = 0.25$, $p(6,0) = 0.25$, $p(6,5) = 0.25$, $p(6,6) = 0.5$, $p(j,0) = p(j,j-1) = p(j,j) = p(j,j+1) = 0.25$

**The answer for this question is that it is irreducible and every state communicates with each other.** I am trying to understand what this statement means.

Thank You

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#### Best Answer

The path $0to1to2to3to4to5to6to0$ has positive probability hence the chain is irreducible.

(Additionally, the transition $0to0$ has positive probability hence the chain is aperiodic.)