Central limit theorems for additive functionals and fringe trees in tries
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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, c/(k(k-1)) for k>2, where c=1/(1+H) with H the entropy of the digits. 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 k→∞.
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