Central limit theorems for additive functionals of patricia tries

Abstract General additive functionals of patricia tries are studied asymptotically in a probabilistic model with independent, identically distributed letters from a finite alphabet. Asymptotic normality is shown after normalization together with asymptotic expansions of the moments. There are two regimes depending on the algebraic structure of the letter probabilities, with and without oscillations in the expansion of moments. As applications firstly the proportion of fringe trees of patricia tries with k keys is studied, which is oscillating around $(1-\rho(k))/(2H)k(k-1)$ , where H denotes the source entropy and $\rho(k)$ is exponentially decreasing. The oscillations are identified explicitly. Secondly, the independence number of patricia tries and of tries is considered. The general results for additive functions also apply, where a leading constant is numerically approximated. The results extend work of Janson on tries by relating additive functionals on patricia tries to additive functionals on tries.