I'm studying theorem 6.8.8 of Durrett - convergence of transition probabilities for Harris chains and I have a (I think) pretty hard question which would help me more than words can describe if one of you nice people could help me solve.
It is the part that $W_m=S_m - T_m$ is a random walk (a sum of iid random variables) and has mean zero and furthermore that it means that $M=\inf\{n\geq 1 | S_m = T_m\}< \infty$ a.s. which I can't understand.
I can only show that $s_m=S_m-S_{m-1}$ and $t_m=T_m-T_{m-1}$ are iid for $m\geq2 $. And I know the Chung-Fuch theorem saying that if a Random Walk is one-dimensional and $S_n / n \to 0$ in probability then it is recurrent, true here by Kolmogorovs law of large numbers. This gives the result if $P(W_m=0)>0$, but that I can't realize why should be the case.
Edit3:
I've done some further realizations myself. So the result is true if for $R_m:=S_m-T_m-(S_1-T_1)$ has $R_n=S_1-T_1$ for some $n$. $R_n$ will be a recurrent random walk from the above and I found this theorem P1 saying that for all $x\in \mathbb{R}$ there exists a $N=N(x)$ such that the probability of transitioning from 0 to x in more than n steps is positive. Since recurrent means 0 i.o. we can use Theorem 6.3.3 of Durrett (page 284, 296 in scribd see below) to find that it actually hits any $x\in \mathbb{R}$ with probability 1 and therefore also $S_1-T_1$ from which the result follows. This requires though that I can show $R_m$ is strongly aperiodic, but I guess that follows from the aperiodicity of the original chain.
What do you guys think - does it stick?