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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 us to establish a limit law… (More)

We characterize all limit laws of the quicksort type random variables defined recursively by X n d = X In + X * n−1−In + T n when the " toll function " T n varies and satisfies general conditions, When the " toll function " T n (cost needed to partition the original problem into smaller subproblems) is small (roughly lim sup n→∞ log E(T n)/ log n ≤ 1/2), X… (More)

- Ralph Neininger
- 2003

In the first part of this paper we give an introduction to the contraction method for the analysis of additive recursive sequences of divide and conquer type. Recently some general limit theorems have been obtained by this method based on a general transfer theorem. This allows to conclude from the recursive structure and the asymp-totics of first moment(s)… (More)

We prove convergence in distribution for the profile (the number of nodes at each level), normalized by its mean, of random recursive trees when the limit ratio ˛ of the level and the logarithm of tree size lies in OE0; e/. Convergence of all moments is shown to hold only for ˛ 2 OE0; 1 (with only convergence of finite moments when ˛ 2 .1; e/). When the… (More)

The contraction method for recursive algorithms is extended to the multivariate analysis of vectors of parameters of recursive structures and algorithms. We prove a general multivariate limit law which also leads to an approach to asymptotic covariances and correlations of the parameters. As an application the asymptotic correlations and a bivariate limit… (More)

The Wiener index is analyzed for random recursive trees and random binary search trees in the uniform probabilistic models. We obtain the expectations, asymptotics for the variances, and limit laws for this parameter. The limit distributions are characterized as the projections of bivariate measures that satisfy certain fixed-point equations. Covariances,… (More)

It is proved that the internal path length of a d{ dimensional quad tree after normalization converges in distribution. The limiting distribution is characterized as a xed point of a random aane operator. We obtain convergence of all moments and of the Laplace transforms. The moments of the limiting distribution can be evaluated from the recursion and lead… (More)

We study a random fragmentation process and its associated random tree. The process has earlier been studied by Dean and Majumdar [7], who found a phase transition: the number of fragmentations is asymptotically normal in some cases but not in others, depending on the position of roots of a certain characteristic equation. This parallels the behaviour of… (More)

We propose an approach to analyze the asymptotic behavior of Pólya urns based on the contraction method. For this a combinatorial discrete time embedding of the evolution of the composition of the urn into random rooted trees is used. A decomposition of the trees leads to a system of recursive distributional equations which capture the distributions of the… (More)

The complexity of the Quicksort algorithm is usually measured by the number of key comparisons used during its execution. When operating on a list of n data, permuted uniformly at random, the appropriately normalized complexity Yn is known to converge almost surely to a non-degenerate random limit Y. This assumes a natural embedding of all Yn on one… (More)