Consider a random walk $X(n)$ on a vertex-transitive graph where the random walk has positive speed, i.e., $$ \lim\limits_{n \rightarrow \infty} \frac{d(X(n), X(0))}{n}= \alpha>0$$ almost surely. Let $A_n$ be the set of vertices having a graph distance $n$ from $X(0)$. Let $T_n$ be the difference between the first time the random walk hits $A_n$ and the last time the random walk hits $A_n$.
I need to prove that all the $T_n$ are stochastically smaller than a variable $T$, which is almost surely finite.
Is this condition true for any vertex-transitive graph, or are additional conditions necessary in order it to hold? For example, if the graph is a regular tree, this is easy to prove, as such a problem is equivalent to the same problem for biased random walk on $\mathbb{Z}$.