Ralph Neininger

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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 Xn d = XIn + X ∗ n−1−In + Tn when the “toll function” Tn varies and satisfies general conditions, where (Xn), (X ∗ n ), (In, Tn) are independent, Xn d = X∗ n , and In is uniformly distributed over {0, . . . , n − 1}. When the “toll function” Tn (cost needed to(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)
We investigate random distances in a random binary search tree. Two types of random distance are considered: the depth of a node randomly selected from the tree, and distance between randomly selected pairs of nodes. By a combination of classical methods and modern contraction techniques we arrive at a Gaussian limit law for normed random distances between(More)
The normalized number of key comparisons needed to sort a list of randomly permuted items by the Quicksort algorithm is known to converge in distribution. We identify the rate of convergence to be of the order Θ(ln(n)/n) in the Zolotarev metric. This implies several ln(n)/n estimates for other distances and local approximation results as for characteristic(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)
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 asymptotics of first moment(s)(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)
For the random binary search tree with n nodes inserted the number of ancestors of the elements with ranks k and `, 1 ≤ k < ` ≤ n, as well as the path distance between these elements in the tree are considered. For both quantities central limit theorems for appropriately rescaled versions are derived. For the path distance the condition ` − k → ∞ as n → ∞(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 fixed point of a random affine 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(More)