Related papers: Random decompositions of Eulerian statistics
The development of the theories of the second-order Eulerian polynomials began with the works of Buckholtz and Carlitz in their studies of an asymptotic expansion. Gessel-Stanley introduced Stirling permutations and presented combinatorial…
Compositional data arise when count observations are normalised into proportions adding up to unity. To allow use of standard statistical methods, compositional proportions can be mapped from the simplex into the Euclidean space through the…
By considering Eulerian numbers and ordered Stirling numbers of the second and third kinds over a multiset, we generalize identities of Eulerian numbers and Stirling numbers of the second and third kinds and provide $q$-analogs of these…
We investigate the probability of observing a given pattern of $n$ rises and falls in a random stationary data series. The data are modelled as a sequence of $n+1$ independent and identically distributed random numbers. This probabilistic…
We consider sequences of polynomials that satisfy differential-difference recurrences. Polynomials satisfying such recurrences frequently appear as generating polynomials of integer valued random variables that are of interest in discrete…
A variety of techniques have been proposed to train machine learning classifiers that are independent of a given feature. While this can be an essential technique for enabling background estimation, it may also be useful for reducing…
Consider a set of order statistics that arise from sorting samples from two different populations, each with their own, possibly different distribution function. The probability that these order statistics fall in disjoint, ordered…
We apply random matrix theory to study the impact of measurement uncertainty on dynamic mode decomposition. Specifically, when the measurements follow a normal probability density function, we show how the moments of that density propagate…
We provide a unified, probabilistic approach using renewal theory to derive some novel limits of sums for the normalized binomial coefficients and for the normalized Eulerian numbers. We also investigate some corresponding results for their…
We examine the aggregate behavior of one-dimensional random walks in a model known as (one-dimensional) Internal Diffusion Limited Aggregation. In this model, a sequence of $n$ particles perform random walks on the integers, beginning at…
Normal approximations for descents and inversions of permutations of the set $\{1,2,...,n\}$ are well known. A number of sequences that occur in practice, such as the human genome and other genomes, contain many repeated elements. Motivated…
We define a new family of generalized Stirling permutations that can be interpreted in terms of ordered trees and forests. We prove that the number of generalized Stirling permutations with a fixed number of ascents is given by a natural…
This paper presents the asymptotic theory for nondegenerate $U$-statistics of high frequency observations of continuous It\^{o} semimartingales. We prove uniform convergence in probability and show a functional stable central limit theorem…
The second Eulerian numbers are defined via the descent enumerator of Stirling permutations, a class of permutations introduced by Gessel and Stanley. We give a simple and conceptual proof of two identities relating the Bernoulli numbers…
When the historical data are limited, the conditional probabilities associated with the nodes of Bayesian networks are uncertain and can be empirically estimated. Second order estimation methods provide a framework for both estimating the…
Associated to each complex-valued random variable satisfying appropriate integrability conditions, we introduce a different generalization of the Stirling numbers of the second kind. Various equivalent definitions are provided. Attention,…
Diagrammatic techniques to compute perturbatively the spectral properties of Euclidean Random Matrices in the high-density regime are introduced and discussed in detail. Such techniques are developed in two alternative and very different…
We propose and analyze a specific asymptotic stochastic order for random processes based on the measure of departure discussed in the literature. As applications, we stochastically compare mixtures of order statistics and record values…
We use elementary methods to establish three key recurrence relations: one for derangement numbers, a second for harmonic numbers, and a third for degenerate harmonic numbers. Our results not only contribute to the understanding of the…
In this paper, we characterize a duality relation between Eulerian recurrences and Eulerian recurrence systems, which generalizes and unifies Hermite-Biehler decompositions of several enumerative polynomials, including flag descent…