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Hello. After flipping through a few textbooks on birth-death processes, I can't seem to find anything about genealogical distribution of survivors (conditioned on non-extinction). What I am looking for is a statement roughly of the type, "in generation n, there is probability at least P(j,k,n) that descendants have survived from at least k distinct members of generation j".

I'm interested in the critical case of the Galton-Watson process, where the number of descendents is i.i.d. with mean 1. If necessary, assume the distribution is Bin(r,1/r).

Also, my sample question about P(j,k,n) is just the easiest thing I could think to write down. But I would like to know how likely it is for there to exist a very uniformly distributed subpopulation of size roughly n in generation n.

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"ancestors have survived from at least k distinct members of generation j". What exactly do you mean by this? Should "ancestors" really be "descendants" in this phrase? –  fedja May 8 '10 at 12:47
    
Yes, I meant "descendants" there. I will edit the post now. –  Matthew Bond May 8 '10 at 20:43
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1 Answer

Take a look at Elementary New Proofs of Classical Limit Theorems for Galton-Watson Processes by Jochen Geiger, Journal of Applied Probability, Vol. 36, No. 2 (1999), pp. 301-309. It looks to me like the construction there could be helpful to you.

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