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26
Mellin transforms and asymptotics: Finite differences and Rice's integrals
, 1995
"... High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combin ..."
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Cited by 82 (8 self)
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High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combinatorics and the analysis of algorithms like digital tries, digital search trees, quadtrees, and distributed leader election.
Euler Sums and Contour Integral Representations
, 1998
"... This paper develops an approach to the evaluation of Euler sums that involve harmonic numbers, either linearly or nonlinearly. We give explicit formulæ for several classes of Euler sums in terms of Riemann zeta values. The approach is based on simple contour integral representations and residue comp ..."
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Cited by 30 (1 self)
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This paper develops an approach to the evaluation of Euler sums that involve harmonic numbers, either linearly or nonlinearly. We give explicit formulæ for several classes of Euler sums in terms of Riemann zeta values. The approach is based on simple contour integral representations and residue computations.
On a multivariate contraction method for random recursive structures with applications to Quicksort
, 2001
"... 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 appl ..."
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Cited by 30 (16 self)
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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 law for the number of key comparisons and exchanges of medianof(2t + 1) Quicksort is given. Moreover, for the Quicksort programs analyzed by Sedgewick the exact order of the standard deviation and a limit law follow, considering all the parameters counted by Sedgewick.
The Wiener index of random trees
, 2001
"... 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 ..."
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Cited by 18 (4 self)
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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 fixedpoint equations. Covariances, asymptotic correlations, and bivariate limit laws for the Wiener index and the internal path length are given.
On the internal path length of ddimensional quad trees
, 1999
"... 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 limit ..."
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Cited by 17 (9 self)
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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 lead to first order asymptotics for the moments of the internal path lengths. The analysis is based on the contraction method. In the final part of the paper we state similar results for general split tree models if the expectation of the path length has a similar expansion as in the case of quad trees. This applies in particular to the mary search trees.
Density Approximation and Exact Simulation of Random Variables that are Solutions of FixedPoint Equations
 Adv. Appl. Probab
, 2002
"... An algorithm is developed for the exact simulation from distributions that are defined as fixedpoints of maps between spaces of probability measures. The fixedpoints of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic an ..."
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Cited by 11 (6 self)
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An algorithm is developed for the exact simulation from distributions that are defined as fixedpoints of maps between spaces of probability measures. The fixedpoints 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 fixedpoints with explicit error bounds are constructed. The sampling algorithm relies on a modified rejection method. AMS subject classifications. Primary: 65C10; secondary: 65C05, 68U20, 11K45.
Normal Approximations of the Number of Records in Geometrically Distributed Random Variables
 Alg
, 1998
"... We establish the asymptotic normality of the number of upper records in a sequence of iid geometric random variables. Large deviations and local limit theorems as well as approximation theorems for the number of lower records are also derived. 1 ..."
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Cited by 8 (1 self)
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We establish the asymptotic normality of the number of upper records in a sequence of iid geometric random variables. Large deviations and local limit theorems as well as approximation theorems for the number of lower records are also derived. 1
WIDTH AND MODE OF THE PROFILE FOR SOME RANDOM TREES OF LOGARITHMIC HEIGHT
 SUBMITTED TO THE ANNALS OF APPLIED PROBABILITY
, 2005
"... We propose a new, direct, correlationfree approach based on central moments of profiles to the asymptotics of width (size of the most abundant level) in some random trees of logarithmic height. The approach is simple but gives precise estimates for expected width, central moments of the width, and ..."
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Cited by 7 (1 self)
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We propose a new, direct, correlationfree approach based on central moments of profiles to the asymptotics of width (size of the most abundant level) in some random trees of logarithmic height. The approach is simple but gives precise estimates for expected width, central moments of the width, and almost sure convergence. It is widely applicable to random trees of logarithmic height, including recursive trees, binary search trees, quadtrees, planeoriented ordered trees and other varieties of increasing trees.
Partial match queries in random quadtrees
 SIAM J. Comput
, 2003
"... We propose a simple, direct approach for computing the expected cost of random partial match queries in random quadtrees. The approach gives not only an explicit expression for the leading constant in the asymptotic approximation of the expected cost but also more terms in the asymptotic expansion i ..."
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Cited by 7 (1 self)
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We propose a simple, direct approach for computing the expected cost of random partial match queries in random quadtrees. The approach gives not only an explicit expression for the leading constant in the asymptotic approximation of the expected cost but also more terms in the asymptotic expansion if desired. Key words. Quadtrees, partial match queries, binomial transform, Mellin transform, Euler transform, Rice’s integral.