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?

  • $\begingroup$ 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). $\endgroup$ – WhoKnowsWho Mar 26 at 4:25

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