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I am looking for the definition of the ‚branching property‘ of a Galton Watson process. Can someone give me an example about it?

It looks to me like an independence.

I have a branching processes book, but there is the branching property only for lines defined. I need it special for a Galton-Watson Process.

BR, Fynn

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  • $\begingroup$ en.wikipedia.org/wiki/Galton%E2%80%93Watson_process $\endgroup$
    – Liviu Nicolaescu
    Commented Oct 7, 2021 at 8:37
  • $\begingroup$ I don’t see on Wikipedia the definition of the branching property. $\endgroup$
    – Fynn13
    Commented Oct 7, 2021 at 10:17
  • $\begingroup$ There are $X_n$ individuals (bacteria) in the $n$-th generation. Each individual $j$ in the $n$-th generation yields $\xi_j^{(n))$ branches, or successors. Hence the word branching process. Draw a picture. $\endgroup$
    – Liviu Nicolaescu
    Commented Oct 7, 2021 at 13:17
  • $\begingroup$ @LiviuNicolaescu fine - but that still doesn't tell you what people customarily mean when they use the phrase "branching property"! $\endgroup$
    – James Martin
    Commented Oct 8, 2021 at 5:22

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Indeed the "branching property" is an independence property for the Galton-Watson tree (or "Bienaymé tree"). People typically use the phrase to refer to some version of the following fact. Condition on the event that the root vertex has $k$ children. Call those children $v_1, v_2, \dots, v_k$. Then the $k$ subtrees rooted at $v_1, v_2, \dots, v_k$ are independent, and each one has the distribution of the original tree.

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  • $\begingroup$ Do you maybe know where i can find the formalised definition of your comment? $\endgroup$
    – Fynn13
    Commented Oct 8, 2021 at 16:52

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