Central limit theorems for additive functionals and fringe trees in tries

03/05/2020

∙

by   Svante Janson, et al.

∙

0

∙

share

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→∞.