Aperiodicity for Markov chains ensures that powers of self-loop probabilities $[P^{k}]_{ii}$ remain strictly positive from a certain power k onward (for the transition matrix $P$)
The proof relies on generalizing Bezout's lemma to the following result:
additively closed set lemma: If $A \subseteq Z^{+}$ is a coprime set of positive integers (GCD of its elements is 1) closed under addition, then there exists a positive integer N such that $A \supseteq [N, +\infty]$.
The proof of the additively closed set lemma uses Bezout's lemma (sketch given only)
One can show that $A$ must have a finite coprime subset $B$ (details omitted).
By Bezout, there exist integers $k_1, ..., k_n$ such that $k_1b_1 + \ldots, k_M b_M = 1$. Without loss of generality, assume that $k_1, \ldots, k_L > 0$ and $k_{L+1}, \ldots, k_M < 0$.
Let $a_1 = k_1 b_1 + \ldots k_Lb_L$ and $a_2 = -(k_{L+1} b_{L+1} + \ldots k_Mb_M)$. Clearly, both belong to $B$ and $a_1 - a_2 = 1$.
If $a_2 = 0$ then the result is trivial.
Otherwise, use Eucledian division and assume that $N \geq a_{2}^{2}$ to obtain that $N = la_2 + r$ for $0 \leq r < a_2$. From this we can deduce that N must belong to by rewriting $N$ to $N = (l-r)a_2 + ra1$ (relying on $a_1 - a_2 = 1)$. It can be shown that $(l-r) \geq 0$ thus $N \in A$ from $N \geq a_2^2$ onward.
One then applies the additively closed set lemma to the set $A_i = \{n \geq 1|\, [P^n]_{ii} >0\}$ for any state $i$.
The question is the following: for reversible markov chains, one can demonstrate that probabilities of self loop paths are the same, irrespective of the direction followed on the path. For instance a self-loop path $v_1 \rightarrow \ldots \rightarrow v_{j - 1} \rightarrow v_1$ with probability $p_{12}p_{23} \ldots p_{(j-1)1}$ has the same probability as the reversed self-loop path $v_1 \rightarrow v_{j-1} \rightarrow \ldots \rightarrow v_2 \rightarrow v_1$, i.e. for reversible markov chains: $$p_{12}p_{23} \ldots p_{(j-1)1} = p_{1(j-1)} \ldots p_{32}p_{21}$$.
So, for reversible markov chains the notion of a positive direction on a self-loop and a negative direction could be used (with both directions having the same probability of being run through). This seems to indicate that instead of the generalized lemma discussed above, a more direct application of Bezout might be used to show that powers $[P^{k}]_{ii}$ remain strictly positive from a certain point on. The negative coefficients in Bezout would in this context correspond to running in the "negative" direction of a self-loop. Is this intuition valid? I.e. does the result of positive self-loop powers follow more easily for reversible markov chains?
More precisely: in the above proof, the use of $a_1 = k_1 b_1 + \ldots k_Lb_L$ and $a_2 = -(k_{L+1} b_{L+1} + \ldots k_Mb_M)$, where $a_1 - a_2 =1$ (via Bezout), may not need a further application of Eucledian division. Perhaps a more direct application of interpreting $a_1$ as following a series of positive self-loops and $a_2$ as following a series of negative self-loops can be used for reversible Markov chains?
For reversible markov chains the probabilities for reversed self-loops are the same as the probability of following the original self-loops. Hence if one is positive the other must be too. Could the "negative" self-loops used in Bezout, simply be reversed to yield a proof that the powers $[P^{k}]_{ii}$ remain strictly positive from a certain point on?
Is this type of simplified proof given somewhere? (reference?) or is there an easy way to use this intuition and get the same result, known as part of the folklore on reversible markov chains? Perhaps a result obtaining a sharper estimate than $N \geq a_{2}^{2}$?
