I am working on an analytic proof of Erdos-Feller-Pollard theorem. The exercise basically tells me to prove some steps in order to prove the theorem.
First, a few definitions:
Let $\{X_n\}$ be a Markov Chain where $i$ is a recurrent aperiodic state. Define $u_n=p_{ii}^{(n)}$ and $f_n=f_{ii}^{(n)}$.
Here's what I have to prove.
1) Suppose $\limsup u_n=\lambda$ and $u_{n_\gamma}\to\lambda$. Then, whenever $f_m>0$, show that $u_{n_\gamma-m}\to\lambda$ as $\gamma\to\infty$.
2) Let $M$ be the set of all finite sums of $m$ such that $f_m>0$. Then for any $k\in M$, show that $u_{n_\gamma-k}\to\lambda$ as $\gamma\to\infty$.
I have proven the first one. However, I am stuck on the second one. I understand that somehow I have to use the fact that $u_{n_\gamma}=\sum_{m:f_m>0}f_mu_{n_\gamma-m}$.
However, just because $f_{m_1}>0,f_{m_2}>0$, it doesn't mean that $f_{m_1+m_2}>0$.
But, $i$ is aperiodic! So, $\gcd(M)=1$ implying $\exists n_0$ such that $\forall n>n_0$, $n\in M$. But even if I take some such $n$, all I know is it is a sum of $m$'s such that $f_m>0$. I do not know $f_n>0$.
How does one therefore prove 2)?
In part 1, you have shown that for any subsequence $n_\gamma$ with $u_{n_\gamma}\to\lambda$ and any $m$ with $f_m>0$, you have $u_{n_\gamma-m}\to\lambda$ as $\gamma\to\infty$. If $f_{m_1}, f_{m_2}>0$, then applying this result twice, shows that
$$u_{n_\gamma-(m_1+m_2)}=u_{(n_\gamma-m_1)-m_2}\to\lambda\mbox{ as }\gamma\to\infty.$$
Note: The aperiodicity is not needed up to this point in the proof.