Central limit theorems for additive functionals and fringe trees in tries

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, $c∕(k(k-1))$ for $k\ge 2$, where $c=1∕(1+H)$ 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 $k\to \mathrm{\infty }$.