Width and mode of the profile for some random trees of logarithmic height

We propose a new, direct, correlation-free 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, quad trees, plane-oriented ordered trees and other varieties of increasing trees.Comment: Published at http://dx.doi.org/10.1214/105051606000000187 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Local limit theorems for finite and infinite urn models

Local limit theorems are derived for the number of occupied urns in general finite and infinite urn models under the minimum condition that the variance tends to infinity. Our results represent an optimal improvement over previous ones for normal approximation.Comment: Published in at http://dx.doi.org/10.1214/07-AOP350 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Sur la répartition des valeurs des fonctions arithmétiques. Le nombre de facteurs premiers d'un entier

RésuméThis paper is concerned with the quantityN(x,m), the number of positive integersn, 1⩽n⩽x, for whichΩ(n)=m, whereΩ:(n) denotes the total number of prime factors (counted with multiplicities) ofn. The main purpose of this article is to present three powerful analytic methods, due, respectively, to Selberg, van der Waerden, and the author, the combination of which allows one to completely characterize the asymptotic behavior of the quantityN(x,m) as the second parameter varies through all its possible values, namely 1⩽m⩽(logx)/log2. These methods constitute, in a certain sense, a compact and effective set of analytical tools and apply also to the distribution function associated withn(x,m). All these methods are quite general and applicable to many other arithmetic functions

Uniform asymptotics of some Abel sums arising in coding theory

AbstractWe derive uniform asymptotic expressions of some Abel sums appearing in some problems in coding theory and indicate the usefulness of these sums in other fields, like empirical processes, machine maintenance, analysis of algorithms, probabilistic number theory, queuing models, etc

Sharp Bounds on the Runtime of the (1+1) EA via Drift Analysis and Analytic Combinatorial Tools

The expected running time of the classical (1+1) EA on the OneMax benchmark function has recently been determined by Hwang et al. (2018) up to additive errors of $O((\log n)/n)$. The same approach proposed there also leads to a full asymptotic expansion with errors of the form $O(n^{-K}\log n)$ for any $K>0$. This precise result is obtained by matched asymptotics with rigorous error analysis (or by solving asymptotically the underlying recurrences via inductive approximation arguments), ideas radically different from well-established techniques for the running time analysis of evolutionary computation such as drift analysis. This paper revisits drift analysis for the (1+1) EA on OneMax and obtains that the expected running time $E(T)$, starting from $\lceil n/2\rceil$ one-bits, is determined by the sum of inverse drifts up to logarithmic error terms, more precisely $$\sum_{k=1}^{\lfloor n/2\rfloor}\frac{1}{\Delta(k)} - c_1\log n \le E(T) \le \sum_{k=1}^{\lfloor n/2\rfloor}\frac{1}{\Delta(k)} - c_2\log n,$$ where $\Delta(k)$ is the drift (expected increase of the number of one-bits from the state of $n-k$ ones) and $c_1,c_2 >0$ are explicitly computed constants. This improves the previous asymptotic error known for the sum of inverse drifts from $\tilde{O}(n^{2/3})$ to a logarithmic error and gives for the first time a non-asymptotic error bound. Using standard asymptotic techniques, the difference between $E(T)$ and the sum of inverse drifts is found to be $(e/2)\log n+O(1)$.Comment: 33 pages; preprint of a paper that will be published in the proceedings of FOGA 2019; v2: minor correction