Papers by Ernst Schulte-Geers
American Mathematical Monthly, 2007
Designs, Codes and Cryptography, May 25, 2012
Bundesamt für Sicherheit in der Informationstechnik (BSI) ernst.schulte-geers(at)bsi.bund.de
Electronic Journal of Combinatorics, Mar 6, 2015
We prove asymptotic normality of the distributions defined by q-supernomials, which implies asymp... more We prove asymptotic normality of the distributions defined by q-supernomials, which implies asymptotic normality of the distributions given by the central string functions and the basic specialization of fusion modules of the current algebra of sl 2 . The limit is taken over linearly scaled fusion powers of a fixed collection of irreducible representations. This includes as special instances all Demazure modules of the affine Kac-Moody algebra associated to sl 2 . Along with an available complementary result on the asymptotic normality of the basic specialization of graded tensors of the type A standard representation, our result is a central limit theorem for a serious class of graded tensors. It therefore serves as an indication towards universal behavior: The central string functions and the basic specialization of fusion and, in particular, Demazure modules behave asymptotically normal, as the number of fusions scale linearly in an asymptotic parameter, N say.
Operations Research Letters, Nov 1, 2011
The demand pooling anomaly of inventory theory of type F amounts to a kind of restricted order re... more The demand pooling anomaly of inventory theory of type F amounts to a kind of restricted order relation between the individual demands (assumed to be independent) and their average. In this paper, we present some sufficient conditions for the type F anomaly not to occur for two i.i.d. demands; furthermore we provide an asymptotic result showing whether this anomaly occurs for large n for a class of distributions containing all distributions with finite mean.
Journal of Applied Probability, Jun 1, 2016
We study in this paper a generalized coupon collector problem, which consists in analyzing the ti... more We study in this paper a generalized coupon collector problem, which consists in analyzing the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we prove that the almost uniform distribution, for which all the non-null coupons have the same drawing probability, is the distribution which stochastically minimizes the time needed to collect a fixed number of distinct coupons. Moreover, we show that in a given closed subset of probability distributions, the distribution with all its entries, but one, equal to the smallest possible value is the one, which stochastically maximizes the time needed to collect a fixed number of distinct coupons.
arXiv (Cornell University), Feb 22, 2014
We study the (random) waiting time for the appearance of the first (multi-)collision in a drawing... more We study the (random) waiting time for the appearance of the first (multi-)collision in a drawing process in detail. The results have direct implications for the assessment of generic (multi-)collision search in cryptographic hash functions.
Journal of Applied Probability, Mar 1, 2017
We show analogs of the classical arcsine theorem for the occupation time of a random walk in (-∞,... more We show analogs of the classical arcsine theorem for the occupation time of a random walk in (-∞, 0) in the case of a small positive drift. To study the asymptotic behavior of the total time spent in (-∞, 0) we consider parametrized classes of random walks, where the convergence of the parameter to zero implies the convergence of the drift to zero. We begin with shift families, generated by a centered random walk by adding to each step a shift constant a > 0 and then letting a tend to zero. Then we study families of associated distributions. In all cases we arrive at the same limiting distribution, which is the distribution of the time spent below zero of a standard Brownian motion with drift 1. For shift families this is explained by a functional limit theorem. Using fluctuation-theoretic formulas we derive the generating function of the occupation time in closed form, which provides an alternative approach. In the course also give a new form of the first arcsine law for the Brownian motion with drift.
arXiv (Cornell University), Jan 8, 2014
We show that the supremum of the successive percentages of red balls in Pólya's urn model is almo... more We show that the supremum of the successive percentages of red balls in Pólya's urn model is almost surely rational, give the set of values that are taken with positive probability and derive several exact distributional results for the all-time maximal percentage.
Operations research proceedings, 1989
arXiv (Cornell University), Feb 22, 2014
We study the (random) waiting time for the appearance of the first (multi-)collision in a drawing... more We study the (random) waiting time for the appearance of the first (multi-)collision in a drawing process in detail. The results have direct implications for the assessment of generic (multi-)collision search in cryptographic hash functions.
arXiv (Cornell University), May 28, 2022
Concentration bounds are given for throwing balls into bins independently according to a distribu... more Concentration bounds are given for throwing balls into bins independently according to a distribution p. The probability of a k-loaded bin after m balls is shown to be controlled on both sides by ρ m,k := m p k k . This gives concentration inequalities for the maximum load as well as for the waiting time until a k-loaded bin.
International Journal of Game Theory, 2021
The worst-case Lipschitz constant of an n-player k-action δ-perturbed game, λ(n, k, δ), is given ... more The worst-case Lipschitz constant of an n-player k-action δ-perturbed game, λ(n, k, δ), is given an explicit probabilistic description. In the case of k ≥ 3, λ(n, k, δ) is identified with the passage probability of a certain symmetric random walk on Z. In the case of k = 2 and n even, λ(n, 2, δ) is identified with the probability that two two i.i.d. Binomial random variables are equal. The remaining case, k = 2 and n odd, is bounded through the adjacent (even) values of n. Our characterisation implies a sharp closed form asymptotic estimate of λ(n, k, δ) as δn/k → ∞.
Journal of Applied Probability, 2017
We show analogs of the classical arcsine theorem for the occupation time of a random walk in (−∞,... more We show analogs of the classical arcsine theorem for the occupation time of a random walk in (−∞,0) in the case of a small positive drift. To study the asymptotic behavior of the total time spent in (−∞,0) we consider parametrized classes of random walks, where the convergence of the parameter to 0 implies the convergence of the drift to 0. We begin with shift families, generated by a centered random walk by adding to each step a shift constant a>0 and then letting a tend to 0. Then we study families of associated distributions. In all cases we arrive at the same limiting distribution, which is the distribution of the time spent below 0 of a standard Brownian motion with drift 1. For shift families this is explained by a functional limit theorem. Using fluctuation-theoretic formulae we derive the generating function of the occupation time in closed form, which provides an alternative approach. We also present a new form of the first arcsine law for the Brownian motion with drift.
Journal of Applied Probability, 2016
In this paper we study a generalized coupon collector problem, which consists of analyzing the ti... more In this paper we study a generalized coupon collector problem, which consists of analyzing the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we prove that the almost uniform distribution, for which all the nonnull coupons have the same drawing probability, is the distribution which stochastically minimizes the time needed to collect a fixed number of distinct coupons. Moreover, we show that in a given closed subset of probability distributions, the distribution with all its entries, but one, equal to the smallest possible value is the one which stochastically maximizes the time needed to collect a fixed number of distinct coupons.
Journal of Applied Probability, 2015
We show that the supremum of the successive percentages of red balls in Pólya's urn model is ... more We show that the supremum of the successive percentages of red balls in Pólya's urn model is almost surely rational, give the set of values that are taken with positive probability, and derive several exact distributional results for the all-time maximal percentage.
Stochastic Processes and their Applications, 1988
Operations Research Letters, 2011
The demand pooling anomaly of inventory theory of type F amounts to a kind of restricted order re... more The demand pooling anomaly of inventory theory of type F amounts to a kind of restricted order relation between the individual demands (assumed to be independent) and their average. In this paper, we present some sufficient conditions for the type F anomaly not to occur for two i.i.d. demands; furthermore we provide an asymptotic result showing whether this anomaly occurs for large n for a class of distributions containing all distributions with finite mean.
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Papers by Ernst Schulte-Geers