# For transient symmetric random walk on $\mathbb{Z}^d,$ the collision local time $V$ is finite almost surely.

I am trying to learn large deviations. I found a video lecture by Prof. Frank den Hollander on YouTube. In the video, he defines two (symmetric) random walks $$(S_k)_{k\ge 0}$$ and $$(S_k')_{k\ge 0}$$ on $$\mathbb{Z}^d, d\ge 1.$$ Let $$V=\sum\limits_{k=0}^{\infty}\mathrm{1}_{\{S_k=S_k'\}}$$ be the collision local time, that is, the number of time the two walks meet. He then comments that $$P(V<\infty)=1$$ if the walks are transient, while if the walks are recurrent then $$V=\infty$$ almost surely.

I am not able to see why this is the case?

• I got it. Anyway thanks. I do not know how to delete the question, I will delete it if I could figure out how to (otherwise I will answer it soon myself). – WhoKnowsWho Mar 26 at 4:25