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I am trying to study the Elementary new proofs of classical limit theorems for Galton Watson processes written by Jochen Geiger. I don't understand what Z_(n,i) stand for. And in the proof of Theorem 3.2, they discuss n-N_n, I really can't get this. I have been trying to find some documents to understand this problems but failed. I know their definitions, I just can't visualize what they are in the shape of the backward tree they discuss earlier in the paper.

Please anyone could help me?

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See the third paragraph on page 304: $Z_{n,i}$, $i \ge 1$, are defined to be independent random variables with the same law as $Z_n$.

$N_n$ is defined at the bottom of page 305 to be $N_n = \min\{j \ge 0 \mid \widetilde{Z}_j = \widetilde{Z}_n\}$, where $\widetilde{Z}_n$ is as defined in equation (2.4).

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