Related papers: Relating bubble sort to birthday problem
We consider the stochastic ranking process with the jump times of the particles determined by Poisson random measures. We prove that the joint empirical distribution of scaled position and intensity measure converges almost surely in the…
We compute the probability distribution of the invariant separation between nucleation centers of colliding true vacuum bubbles arising from the decay of a false de Sitter space vacuum. We find that even in the limit of a very small…
Convergence of order $O(1/\sqrt{n})$ is obtained for the distance in total variation between the Poisson distribution and the distribution of the number of fixed size cycles in generalized random graphs with random vertex weights. The…
In this short note we consider the finite-dimensional distributions of sets of states generated by dispersing billiards with a random initial condition. We establish a functional correlation bound on the distance between the…
Dispersion is a fundamental concept in statistics, yet standard approaches - especially via stochastic orders - face limitations in the discrete setting. In particular, the classical dispersive order, well-established for continuous…
The bubble size distribution is a summary statistics that can be computed from the observed 21-cm signal from the Epoch of Reionization. As it depends only on the ionization field and is not limited to gaussian information, it is an…
It is well known that a binomial $(n,p)$ can be approximated by a Poisson distribution with parameter $np$. The typical approach in undergraduate probability texts is to show a convergence result for the distribution of the binomial as $n$…
The inertial collapse of two interacting and non-translating spherical bubbles of equal size is considered. The exact analytic solution to the nonlinear ordinary differential equation that governs the bubble radii during collapse is first…
The weak limit of the normalized number of comparisons needed by the Quicksort algorithm to sort n randomly permuted items is known to be determined implicitly by a distributional fixed-point equation. We give an algorithm for perfect…
Knowledge of bubble and drop size distributions in two-phase flows is important for characterizing a wide range of phenomena, including combustor ignition, sonar communication, and cloud formation. The physical mechanisms driving the…
Consider an urn model where at each step one of $q$ colors is sampled according to some probability distribution and a ball of that color is placed in an urn. The distribution of assigning balls to urns may depend on the color of the ball.…
In this article, we discuss the ubiquity of Big Data and machine learning in society and propose that it evinces the need of further investigation of their fundamental limitations. We extend the ``too much information tends to behave like…
We overview results on the topic of Poisson approximation that are missed in existing surveys. The topic of Poisson approximation to the distribution of a sum of integer-valued random variables is presented as well. We do not restrict…
We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a…
Consider the family of power divergence statistics based on $n$ trials, each leading to one of $r$ possible outcomes. This includes the log-likelihood ratio and Pearson's statistic as important special cases. It is known that in certain…
In the framework of the Gibbs statistical theory, the question of the size of the particles forming the statistical system is investigated. This task is relevant for a wide variety of applications. The distribution for particle sizes and…
We address the problem of clustering words (or constructing a thesaurus) based on co-occurrence data, and using the acquired word classes to improve the accuracy of syntactic disambiguation. We view this problem as that of estimating a…
The Gumbel trick is a method to sample from a discrete probability distribution, or to estimate its normalizing partition function. The method relies on repeatedly applying a random perturbation to the distribution in a particular way, each…
Using techniques from Poisson approximation, we prove explicit error bounds on the number of permutations that avoid any pattern. Most generally, we bound the total variation distance between the joint distribution of pattern occurrences…
We consider the classical problem of sorting an input array containing $n$ elements, where each element is described with a $k$-bit comparison-key and a $w$-bit payload. A long-standing open problem is whether there exist $(k + w) \cdot o(n…