Let $X_n$ be a simple random walk on the discrete torus $\Bbb{T}_N^d$,
$K \subset \subset \Bbb{Z}^d $ and $K^*=\phi(K)$, with $\phi:\Bbb{Z}^d \to \Bbb{T}_N^d$
Let also $H_A= inf\{n \ge0: X_n(\omega)\in A\}$ and $T_A= inf\{n \ge 0: X_n(\omega)\notin A\}$
How do I use Doob's submartingale inequality to go from the first limit to the second one?
for any $x\in K$
$lim_{N \to \infty}P_x[H_{\phi^{-1}(K^*)\setminus K}\le n]=0$
for any $x\in K$ and for any $\epsilon>0$
$lim_{N \to \infty}P_x[T_{B(x,n^{(1+\epsilon)/2}) }\le n]=0$