Dear mathoverflow.

This is a question to a proof in a graduate text. I have asked two professors at my university without help, so I hope it suffices in difficulty for this forum otherwise I appologize.

I can't add an image but the theorem is at scribd page 336.

My problem is with the $S_m-T_m$ is a random walk part. Since the waiting times are iid after the first I know he means $S_m-T_m$ without the first step will be a random walk. What he want to use it for is to conclude it is at some point zero so $S_m=T_m$, but if we disregard the first term would we not have to have to consider the event $S_m-T_m = S_0-T_0$ instead? I know the random walk will be recurrent from what he calls Chung-Fuchs theorem.

I really hope you are willing to help, since I really need to understand this step and I don't know where else to go.