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Asymptotic normality of fringe subtrees in conditioned Galton--Watson trees

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We consider conditioned Galton-Watson trees and show asymptotic normality of the number of fringe subtrees isomorphic to any given tree, and joint asymptotic normality for several such subtree counts. (This improves a law of large numbers shown by Aldous.) By a truncation argument, this extends to additive functionals that are defined by toll functions that are not too large. The offspring distribution defining the random tree is assumed to have expectation 1 and finite variance; no further moment condition is assumed. The methods include transformation to a problem on sums of m-dependent variables, conditioned on the value of another sum.

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