A General Contraction Theorem and Asymptotic Normality in Combinatorial Structures

Limit laws are proven by the contraction method for random vectors of a recursive nature as they arise as parameters of combinatorial structures such as random trees or recursive algorithms, where we use the Zolotarev metric. In comparison to previous applications of this method a general transfer theorem is derived, which allows to establish a limit law on the basis of the recursive structure and using the asymptotics of the rst and second moments of the sequence. In particular a general asymptotic normality result is obtained by this theorem, which typically cannot be handled by the more common`2-metrics. As applications we derive quite automatically many asymptotic normality results ranging from the size of tries or m-ary search trees and path lengths in digital structures to mergesort and parameters of random recursive trees, which were previously shown by diierent methods one by one. We also obtain a related local density approximation result as well as a global approximation result. For the proof of these we establish that a smoothed density distance as well as a smoothed total variation distance can be estimated from above by the Zolotarev metric which is the main tool in this paper.