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تسجيل مستخدم جديدLeaf multiplicity in a Bienayme-Galton-Watson tree
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This note defines a notion of multiplicity for nodes in a rooted tree and presents an asymptotic calculation of the maximum multiplicity over all leaves in a Bienayme-Galton-Watson tree with critical offspring distribution $xi$, conditioned on the tree being of size $n$. In particular, we show that if $S_n$ is the maximum multiplicity in a conditional Bienayme-Galton-Watson tree, then $S_n = Omega(log n)$ asymptotically in probability and under the further assumption that ${bf E}{2^xi} < infty$, we have $S_n = O(log n)$ asymptotically in probability as well. Explicit formulas are given for the constants in both bounds. We conclude by discussing links with an alternate definition of multiplicity that arises in the root-estimation problem.
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We consider a biased random walk $X_n$ on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant $gamma= gamma(beta) in (0,1)$, depending on the bias $beta$, such that $X_n$ is of order $n^{gamma
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