That's my first question here.
Currently I'm reading a paper of T. Lyons about recurrence/tranciense of Markov chains and Simple Random Walks on $\mathbb{Z}^{2+\epsilon}$ dimensional grids. It's quit interesting, you can find a link to the paper below.
I've been about to finish the first implication of the proof to the Royden's criterion, but now I've got stucked in the last step that completes the proof.
The setting is like this:
- Probability space $X$.
- Reversible Markov chain $(Y_n)_{n\in\mathbb{N}}$ with transitions $p_{ij}$ for $i,j\in X$.
- For reversibility there exist strictly positive weights $\pi_i, i\in X$ with $\pi_ip_{ij}=\pi_jp_{ji}$.
- We denote the conditioned expected value and probability on $i\in X$ as $\mathbb{E}^i$ and $\mathbb{P}^i$.
Furthermore we have constructed a function $W:X \to \mathbb{R}$ with the following properties:
(i) $W_{i_0}=0$
(ii) $W\neq 0$
(iii) $W_i=\sum_{j\in X}p_{ij}W_j$ for $i\neq i_0$
(iv) $\sum_{i,j\in X}\pi_ip_{ij}(W_i-W_j)^2<\infty$
Let $T$ be the first time $Y$ hits $i_0$.
Now the proof says that from $$\sup_k\mathbb{E}^i((W(Y_{k\land T})-W(Y_0))^2)<\infty$$ follows that $$W(Y_{k\land T})=0$$ and therefore $$W=0$$ How is this? I feel like it should be possible to show with the martingale convergence theorem or the optional sampling theorem, but it didn't work out for me.
Anyone an idea? Thanks a lot.
Consider the Martingale $M_k:=W(Y_{k \wedge T})$, where $T$ is the first time $Y$ hits $i_0$, and $Y_0=i$. From $E[(M_k-M_0)^2]<\infty$, the Martingale is bounded in $L^2$, so it is uniformly integrable. Since it converges a.s. to $0$, the expectations must converge to $0$ as well: $W(i)=M_0=E(M_k) \to 0$, so $W \equiv 0$, as $i$ is arbitrary.
Re-reading Lyons' paper, I think he had a different argument in mind: A martingale $\{M_k\}$ bounded in $L^2$ converges a.s. and in $L^2$ to a limit $M_\infty$, with $E(M_k^2) \uparrow E(M_\infty^2)$ as $k \to \infty$. Thus if $M_\infty=0$ a.s., we must have $E(M_k^2)=0$ for all $k$, whence $M_k:=W(Y_{k \wedge T})=0$ a.s. for all $k$.
For an elementary exposition of this Theorem of Lyons, see Theorem 2 of [1] (Proved in Section 4 there.)
[1] Levin, David A., and Yuval Peres. "Pólya’s theorem on random walks via Pólya’s urn." The American Mathematical Monthly 117, no. 3 (2010): 220-231. https://www.tandfonline.com/doi/abs/10.4169/000298910X480072 Also available at https://pages.uoregon.edu/dlevin/polya.pdf as a preprint