Abstract
We give general theorems on asymptotic normality for additive functionals of random tries generated by a sequence of independent strings. These theorems are applied to show asymptotic normality of the distribution of random fringe trees in a random trie. Formulas for asymptotic mean and variance are given. In particular, the proportion of fringe trees of size k (defined as number of keys) is asymptotically, ignoring oscillations, for , where with H the entropy of the letters. Another application gives asymptotic normality of the number of k-protected nodes in a random trie. For symmetric tries, it is shown that the asymptotic proportion of k-protected nodes (ignoring oscillations) decreases geometrically as .
Funding Statement
Supported by the Knut and Alice Wallenberg Foundation.
Acknowledgments
I thank Pawel Hitczenko for help with references on Rosenthal’s inequality.
Citation
Svante Janson. "Central limit theorems for additive functionals and fringe trees in tries." Electron. J. Probab. 27 1 - 63, 2022. https://doi.org/10.1214/22-EJP776
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