Ralph Neininger

On the contraction method with degenerate limit equation

A class of random recursive sequences (Yn) with slowly varying variances as arising for parameters of random trees or recursive algorithms leads after normalizations to degenerate limit equations of the form

The Wiener Index of Random Trees

X\stackrel {\mathcal{L}}{=}X

A functional limit theorem for the profile of search trees

. For nondegenerate limit equations the contraction method is a main tool to establish convergence of the scaled sequence to the “unique” solution of the limit equation. In this paper we develop an extension of the contraction method which allows us to derive limit theorems for parameters of algorithms and data structures with degenerate limit equation. In particular, we establish some new tools and a general convergence scheme, which transfers information on mean and variance into a central limit law (with normal limit). We also obtain a convergence rate result. For the proof we use selfdecomposability properties of the limit normal distribution which allow us to mimic the recursive sequence by an accompanying sequence in normal variables.

On a functional contraction method

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, asymptotic correlations, and bivariate limit laws for the Wiener index and the internal path length are given. AMS subject classifications. Primary: 60F05, 05C12; secondary: 05C05, 68Q25.

Density approximation and exact simulation of random variables that are solutions of fixed-point equations

We study the profile X-n,X-k of random search trees including binary search trees and m-ary search trees. Our main result is a functional limit theorem of the normalized profile X-n,X-k/EXn,k for ...

Distances and Finger Search in Random Binary Search Trees

Methods for proving functional limit laws are developed for sequences of stochastic processes which allow a recursive distributional decomposition either in time or space. Our approach is an extension of the so-called contraction method to the space C[0,1] of continuous functions endowed with uniform topology and the space D[0,1] of cadlag functions with the Skorokhod topology. The contraction method originated from the probabilistic analysis of algorithms and random trees where characteristics satisfy natural distributional recurrences. It is based on stochastic fixed-point equations, where probability metrics can be used to obtain contraction properties and allow the application of Banach’s fixed-point theorem. We develop the use of the Zolotarev metrics on the spaces C[0,1] and D[0,1] in this context. Applications are given, in particular, a short proof of Donsker’s functional limit theorem is derived and recurrences arising in the probabilistic analysis of algorithms are discussed.

Analysis of Algorithms by the Contraction Method: Additive and Max-recursive Sequences

An algorithm is developed for exact simulation from distributions that are defined as fixed points of maps between spaces of probability measures. The fixed points of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic analysis of algorithms. Approximating sequences for the densities of the fixed points with explicit error bounds are constructed. The sampling algorithm relies on a modified rejection method.

Rates of convergence for Quicksort

For the random binary search tree with n nodes inserted the number of ancestors of the elements with ranks k and

Pólya Urns Via the Contraction Method

\ell

On binary search tree recursions with monomials as toll functions

,