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Central limit theorems for additive functionals and fringe trees in tries

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 $k\ge2$, 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 $k\to\infty$.