Poisson Limits Research Articles

A random graph growth model

AbstractA growing random graph is constructed by successively sampling without replacement an element from the pool of virtual vertices and edges. At start of the process, the pool contains virtual vertices and no edges. Each time a vertex is sampled and occupied, the edges linking the vertex to previously occupied vertices are added to the pool of virtual elements. We focus on the edge‐counting at times when the graph has occupied vertices. Two different Poisson limits are identified for and . For the bulk of the process, when , the scaled number of edges is shown to fluctuate about a deterministic curve, with fluctuations being of the order of and approximable by a Gaussian bridge.

Hitting Times and Positions in Rare Events

We establish abstract limit theorems which provide sufficient conditions for a sequence $(A_{l})$ of rare events in an ergodic probability preserving dynamical system to exhibit Poisson asymptotics, and for the consecutive positions inside the $A_{l}$ to be asymptotically iid (spatiotemporal Poisson limits). The limit theorems only use information on what happens to $A_{l}$ before some time $\tau_{l}$ which is of order $o(1/\mu(A_{l}))$. In particular, no assumptions on the asymptotic behavior of the system akin to classical mixing conditions are used. We also discuss some general questions about the asymptotic behaviour of spatial and spatiotemporal processes, and illustrate our results in a setup of simple prototypical systems.

Universality of Poisson Limits for Moduli of Roots of Kac Polynomials

AbstractWe give a new proof of a recent resolution [18] by Michelen and Sahasrabudhe of a conjecture of Shepp and Vanderbei [19] that the moduli of roots of Gaussian Kac polynomials of degree $n$, centered at $1$ and rescaled by $n^2$, should form a Poisson point process. We use this new approach to verify a conjecture from [18] that the Poisson statistics are in fact universal.

Improving sensitivity to low-mass dark matter in LUX using a novel electrode background mitigation technique

This paper presents a novel technique for mitigating electrode backgrounds that limit the sensitivity of searches for low-mass dark matter (DM) using xenon time projection chambers. In the LUX detector, signatures of low-mass DM interactions would be very low energy ($\sim$keV) scatters in the active target that ionize only a few xenon atoms and seldom produce detectable scintillation signals. In this regime, extra precaution is required to reject a complex set of low-energy electron backgrounds that have long been observed in this class of detector. Noticing backgrounds from the wire grid electrodes near the top and bottom of the active target are particularly pernicious, we develop a machine learning technique based on ionization pulse shape to identify and reject these events. We demonstrate the technique can improve Poisson limits on low-mass DM interactions by a factor of $2$-$7$ with improvement depending heavily on the size of ionization signals. We use the technique on events in an effective $5$ tonne$\cdot$day exposure from LUX's 2013 science operation to place strong limits on low-mass DM particles with masses in the range $m_{\chi}\in0.15$-$10$ GeV. This machine learning technique is expected to be useful for near-future experiments, such as LZ and XENONnT, which hope to perform low-mass DM searches with the stringent background control necessary to make a discovery.

Two-neutron transfer reactions and the quantum chaos measure of nuclear spectra

A new statistical interpretation of the nuclear collective states is suggested and applied recently in rare earths and actinide nuclei by the two-neutron transfer reactions in terms of the nearest neighbor-spacing distributions (NNSDs). Experimental NNSDs were obtained by using the complete and pure sequences of the collective states through an unfolding procedure. The two-neutron transfer reactions allow to obtain such a sequence of the collective states that meets the requirements for a statistical analysis. Their theoretical analysis is based on the linear approximation of a repulsion level density within the Wigner-Dyson theory. This approximation is successful to evaluate separately the Wigner chaos and Poisson order contributions. We found an intermediate behavior of NNSDs between the Wigner and Poisson limits. NNSDs turn out to be shifted from a chaos to order with increasing the length of spectra and the angular momentum of collective states. Perspectives for the statistical analysis of the symmetry breaking of states with the fixed projection of angular momenta $K$ are discussed.

The first-principle study of mechanical, optical and thermoelectric properties of SnZrO3 and SnHfO3 for renewable energy applications

The electronic, optical and thermoelectric properties of SnZrO3 and SnHfO3 for renewable energy applications have been studied within the framework of density functional and Boltzmann transport theories. The structural, thermodynamic and mechanical stability in the cubic phase has been confirmed by tolerance factor, formation energy and Born mechanical stability criteria. The ductile strength has been checked by Poisson and Pugh's ratios critical limits. The optical properties have been elucidated by dispersion, absorption, reflection, optical conductivity and loss factor per unit length. The thermoelectric performance has been explored by the Wiedemann-Franz law, Seebeck coefficient and power factor by classical theory based BoltzTrap code.

Spectral Fluctuations in A=32 Nuclei Using the Framework of the Nuclear Shell Model

Chaotic properties of nuclear energy spectra in A=32 nuclei are investigated via the framework of the nuclear shell model. The energies (the main object of this investigation) are calculated through accomplishing shell model calculations employing the OXBASH computer code with the realistic effective interaction of W in the isospin formalism. The A=32 nuclei are supposed to have an inert 16O core with 16 nucleons move in the 1d5/2, 2s1/2 and 1d3/2 orbitals. For full hamiltonian calculations, the spectral fluctuations (i.e., the nearest neighbor level spacing distributions P(S) and the Δ3 statistics) are well characterized by the Gaussian orthogonal ensemble of random matrices. Besides, they show no dependency on the spin J and isospin T. For unperturbed hamiltonian calculations, we find a regular behavior for the distribution of P(S) and an intermediate behavior between the GOE and the Poisson limits for the Δ3 statistics.

Portmanteau inequalities on the Poisson space: mixed regimes and multidimensional clustering

Using Malliavin operators together with an interpolation technique inspired by Arratia, Goldstein and Gordon (1989), we prove a new inequality on the Poisson space, allowing one to measure the distance between the laws of a general random vector, and of a target random element composed of Gaussian and Poisson random variables. Several consequences are deduced from this result, in particular: (1) new abstract criteria for multidimensional stable convergence on the Poisson space, (2) a class of mixed limit theorems, involving both Poisson and Gaussian limits, (3) criteria for the asymptotic independence of U-statistics following Gaussian and Poisson asymptotic regimes. Our results generalize and unify several previous findings in the field. We provide an application to joint sub-graph counting in random geometric graphs.

A non-parametric estimation approach in the investigation of spectral statistics

We have used kernel density estimation (KDE) technique to analyze the spectral statistics of nuclear systems with emphasis on the nearest neighbor spacing distribution. The deviations to regular and chaotic dynamics are described by closer distances to Poisson and Wigner limits, respectively which have calculated via Kullback–Leibler divergence measure. The level statistics of nuclei provide empirical evidences for three dynamical symmetry limits of interacting boson model, considering oblate and prolate nuclei. The predictions of KDE technique suggest a considerable reduction in the uncertainties of chaocity degrees and also more regular dynamics in comparison with other estimation methods for considered systems.

Some notes on poisson limits for empirical point processes

AbstractThe authors define the scaled empirical point process. They obtain the weak limit of these point processes through a novel use of a dimension‐free method based on the convergence of compensators of multiparameter martingales. The method extends previous results in several directions. They obtain limits at points where the density may be zero, but has regular variation. The joint limit of the empirical process evaluated at distinct points is given by independent Poisson processes. They provide applications both to nearest‐neighbour density estimation in high dimensions, and to the asymptotic behaviour of multivariate extremes such as those arising from bivariate normal copulas. The Canadian Journal of Statistics 37: 347–360; 2009 © 2009 Statistical Society of Canada

A Note on Embedding Certain Bernoulli Sequences in Marked Poisson Processes

A sequence of independent Bernoulli random variables with success probabilities a / (a + b + k − 1), k = 1, 2, 3, …, is embedded in a marked Poisson process with intensity 1. Using this, conditional Poisson limits follow for counts of failure strings.

A Note on Embedding Certain Bernoulli Sequences in Marked Poisson Processes

A sequence of independent Bernoulli random variables with success probabilities a / (a + b + k − 1), k = 1, 2, 3, …, is embedded in a marked Poisson process with intensity 1. Using this, conditional Poisson limits follow for counts of failure strings.

Poisson limits of sums of point processes and a particle-survivor model

We present sufficient conditions for sums of dependent point processes to converge in distribution to a Poisson process. This extends the classical result of Grigelionis [Theory Probab. Appl. 8 (1963) 172--182] for sums of uniformly null point processes that have Poisson limits. Included is an application in which a particle-survivor point process converges to a Poisson process. This result sheds light on the ``surprising'' Poisson limit of the species competition process of Durrett and Limic [Stochastic Process. Appl. 102 (2002) 301--309].

Compound Poisson limits for household epidemics

We consider epidemics in populations that are partitioned into small groups known as households. Whilst infectious, a typical infective makes global and local contact with individuals chosen independently and uniformly from the whole population or their own household, as appropriate. Previously, the classical Poisson approximation for the number of survivors of a severe epidemic has been extended to the household model. However, in the current work we exploit a Sellke-type construction of the epidemic process, which enables the derivation of sufficient conditions for the existence of a compound Poisson limit theorem for the survivors of the epidemic. The results are specialised to the Reed-Frost and general stochastic epidemic models.

Compound Poisson limits for household epidemics

We consider epidemics in populations that are partitioned into small groups known as households. Whilst infectious, a typical infective makes global and local contact with individuals chosen independently and uniformly from the whole population or their own household, as appropriate. Previously, the classical Poisson approximation for the number of survivors of a severe epidemic has been extended to the household model. However, in the current work we exploit a Sellke-type construction of the epidemic process, which enables the derivation of sufficient conditions for the existence of a compound Poisson limit theorem for the survivors of the epidemic. The results are specialised to the Reed-Frost and general stochastic epidemic models.

Convergence in Distribution for Subset Counts Between Random Sets

Erdős posed the problem of how many random subsets need to be chosen from a set of $n$ elements, each element appearing in each subset with probability $p=1/2$, in order that at least one subset is contained in another. Rényi answered this question, but could not determine the limiting probability distribution for the number of subset counts because the higher moments diverge to infinity. The model considered by Rényi with $p$ arbitrary is denoted by ${\cal P}(m,n,p)$, where $m$ is the number of random subsets chosen. We give a necessary and sufficient condition on $p(n)$ and $m(n)$ for subset counts to be asymptotically Poisson and find rates of convergence using Stein's method. We discuss how Poisson limits can be shown for other statistics of ${\cal P}(m,n,p)$.

Det-Det correlations for quantum maps: Dual pair and saddle-point analyses

An attempt is made to clarify the ballistic nonlinear sigma model formalism recently proposed for quantum chaotic systems, by looking at the spectral determinant Z(s)=Det(1−sU) for quantized maps U∈U(N), and studying the correlator ωU(s)=∫dθ|Z(eiθs)|2. By identifying U(N) as one member of a dual pair acting in the spinor representation of Spin(4N), the expansion of ωU(s) in powers of s2 is shown to be a decomposition into irreducible characters of U(N). In close analogy with the ballistic nonlinear sigma model, a coherent-state integral representation of ωU(s) is developed. For generic U this integral has (N2N) saddle points and the leading-order saddle-point approximation turns out to reproduce ωU(s) exactly, up to a constant factor. This miracle is explained by interpreting ωU(s) as a character of U(2N), and arguing that the leading-order saddle-point result corresponds to the Weyl character formula. Unfortunately, the Weyl decomposition behaves nonsmoothly in the semiclassical limit N→∞, and to make further progress some additional averaging needs to be introduced. Several schemes are investigated, including averaging over basis states and an “isotropic” average. The saddle-point approximation applied in conjunction with these schemes is demonstrated to give incorrect results in general, one notable exception being a semiclassical averaging scheme, for which all loop corrections vanish identically. As a side product of the dual pair decomposition with isotropic averaging, the crossover between the Poisson and CUE limits is obtained.

Poisson limits for U-statistics

We study Poisson limits for U-statistics with non-negative kernels. The limit theory is derived from the Poisson convergence of suitable point processes of U-statistics structure. We apply these results to derive infinite variance stable limits for U-statistics with a regularly varying kernel and to determine the index of regular variation of the left tail of the kernel. The latter is known as correlation dimension. We use the point process convergence to study the asymptotic behavior of some standard estimators of this dimension.

Overdispersion in nuclear statistics

The modern statistical distribution theory is applied to the development of the overdispersion theory in ionizing-radiation statistics for the first time. The physical nuclear system is treated as a sequence of binomial processes, each depending on a characteristic probability, such as probability of decay, detection, etc. The probabilities fluctuate in the course of a measurement, and the physical reasons for that are discussed. If the average values of the probabilities change from measurement to measurement, which originates from the random Lexis binomial sampling scheme, then the resulting distribution is overdispersed. The generating functions and probability distribution functions are derived, followed by a moment analysis. The Poisson and Gaussian limits are also given. The distribution functions belong to a family of generalized hypergeometric factorial moment distributions by Kemp and Kemp, and can serve as likelihood functions for the statistical estimations. An application to radioactive decay with detection is described and working formulae are given, including a procedure for testing the counting data for overdispersion. More complex experiments in nuclear physics (such as solar neutrino) can be handled by this model, as well as distinguishing between the source and background.

On the distribution of tail array sums for strongly mixing stationary sequences

This paper concerns the asymptotic distributions of tail array sums of the form $\Sigma \Psi_n (X_i - u_n)$ where ${X_i}$ is a strongly mixing stationary sequence, $\Psi_n$ are real functions which are constant for negative arguments, $\Psi_n (x) = \Psi_n (X_+)$ and ${u_n}$ are levels with $u_n \to \infty$. Compound Poisson limits for rapid convergence of $u_n \to \infty (nP{X_1 > u_n} \to \tau u_n) \to \infty)$. The work is motivated by (1) the modeling of damage due to very high and moderately high extremes and (2) the provision of probabilistic theory for application to problems of tail inference for stationary sequences.