Summary
Binary trees are grown by adding one node at a time, an available node at height i being added with probability proportional to c -i, c>1. We establish both a “strong law of large numbers” and a “central limit theorem” for the vector X(t)=(X i(t)), where X i(t) is the proportion of nodes of height i that are available at time t. We show, in fact, that there is a deterministic process x i(t) such that
and such that if c2\(\tfrac{1}{2}\),
and Z n(t)=(Z ni (t)), then Z n(t) converges weakly to a Gaussian diffusion Z(t). The results are applied to establish asymptotic normality in the unbiased coin-tossing case for an entropy estimation procedure due to J. Ziv, and to obtain results on the growth of the maximum height of the tree.
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The work of the first author supported in part by NSF grant # MCS84-03239
The work of the second author supported in part by NSF grants # MCS83-03253 and # DMS85-07189 and in part by a Fulbright fellowship to visit the Mathematics Institute of the Hungarian Academy of Sciences, Budapest
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Aldous, D., Shields, P. A diffusion limit for a class of randomly-growing binary trees. Probab. Th. Rel. Fields 79, 509–542 (1988). https://doi.org/10.1007/BF00318784
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DOI: https://doi.org/10.1007/BF00318784