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A finite $n$-state Markov chain with transition matrix $P$ can be said to be ergodic, if there exists a power $a \in \mathbb{N}$ such that $P^a > 0$ entry-wise (the matrix is regular).

Are there any obvious bounds on the value of $a$ (depending on $n$), or can I construct examples for which $a$ is arbitrarily large ?

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From irreducibility, you can reach state $j$ from state $i$ in $<n$ steps.

From aperiodicity, we have two coprime periods $p_1,p_2\leq n$ in our class. For any state $i$, we find paths to and from these cycles (each $<n$), so we have three coprime-length cycles (but not-necessarily pairwise coprime) starting from $i$:

  • go to cycle 1 (but not follow it) then go to cycle 2 (again not follow it) and back.
  • go to cycle 1, follow it once, then go to cycle 2 but not following it and back.
  • go to cycle 1 but not following it then go to cycle 2, follow it once and back.

These have lengths $<3n$, $<4n$, $<4n$ respectively. So we can represent all integers $\geq c(n)$ with positive integer multiples of these three numbers, so $p_{ii}^{(m)}>0$ for all $m\geq c(n)$. We can bound $c(n)\leq2(2n-1)^2-1$ (Lewin bound on the Frobenius number).

So $a<c(n)+n<8n^2$.

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  • $\begingroup$ Thank you. Do you know if there is any reference to this fact in the literature ? $\endgroup$
    – geo.wolfer
    Jun 17 '19 at 8:12
  • $\begingroup$ I don't know if it fact is referenced anywhere. It is far easier to show irreducibility and aperiodicity separately than to compute a very high power of a matrix and check it has positive entries. $\endgroup$ Jun 17 '19 at 8:23

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