On local limit theorems for q -multinomial distributions

Studies of fast protons produced in the reactions of 16O Nuclei incident on emulsion nuclei at 3.7A GeV

In experiment, the multiplicity distributions of inelastic processes are truncated due to finite energy, insufficient statistics, or special choice of events. It is shown that the moments of such truncated multiplicity distributions possess some typical features. In particular, the oscillations of cumulant moments at high ranks and their negative values at the second rank can be considered as ones most indicative of the specifics of these distributions. They allow one to distinguish between distributions of different type.

Previous article Next article On a Lower Bound for the Convergence Rate in a Local Limit TheoremV. K. MatskyavichyusV. K. Matskyavichyushttps://doi.org/10.1137/1130100PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] William Feller, An introduction to probability theory and its applications. Vol. II. , Second edition, John Wiley & Sons Inc., New York, 1971xxiv+669 42:5292 0219.60003 Google Scholar[2] A. P. Prudnikov, , Yu. A. Brychkov and , O. I. Marichev, Integrals and Series: Elementary Functions, Nauka, Moscow, 1981, (In Russian.) 0511.00044 Google Scholar[3] M. Abramowitz and , I. A. Stegun, Handbook of mathematical functions, with formulas, graphs and mathematical tables, Edited by Milton Abramowitz and Irene A. Stegun. Fifth printing, with corrections. National Bureau of Standards Applied Mathematics Series, Vol. 55, National Bureau of Standards, Washington, D.C., (for sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20402), 1966xiv+1046 34:8607 CrossrefGoogle Scholar[4] V. K. Matskyavichyus, A lower bound for the convergence rate in the central limit theorem, Theory Prob. Appl., 28 (1983), 596–601 0543.60027 LinkGoogle Scholar[5] V. K. Matskyavichyus, Letters to the editor, Theory Prob. Appl., 29 (1984), 196–197 LinkGoogle Scholar[6] V. V. Petrov, Sums of independent random variables, Springer-Verlag, New York, 1975x+346 52:9335 0322.60042 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Approximate local limit theorems with effective rate and application to random walks in random sceneryBernoulli, Vol. 23, No. 4B Cross Ref Local limit theorems in some random models from number theory1 September 2016 | Stochastic Analysis and Applications, Vol. 34, No. 6 Cross Ref Volume 30, Issue 4| 1986Theory of Probability & Its Applications History Submitted:05 August 1985Published online:28 July 2006 InformationCopyright © 1986 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1130100Article page range:pp. 810-814ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Neuronal inter-spike intervals (ISIs) have previously been described as Poisson, Gamma, inverse Gaussian or other unimodal distributions. We analyzed ISIs of rhythmic and arrhythmic neuronal spike trains in cerebellum recorded from freely behaving rats, and found that their distributions can be described as the summation or integration of multiple Gaussian distributions. The ISIs of rhythmic cerebellar Purkinje cells have a main Gaussian peak at a basic firing interval and exponentially reduced peaks at multiples of this firing period. ISIs of arrhythmic Purkinje cells can be modeled as the integration of multiple Gaussian distributions centered at continuous intervals with exponentially reduced peak amplitudes. The sources of variability are directly related to the relative timing of action potentials between neighboring cells since we show that irregularities of discharge in one cell are associated with the previous history of its discharge in time relative to another cell. Through relative phase analyses, we demonstrate that the shape and the mathematical form of the ISI distributions in cerebellum are direct result of dynamic interactions in the nearby neuronal network, in addition to intrinsic firing properties. The analysis in this paper provides a unified description of cerebellar inter-spike interval distributions which deviate from the usual Poisson assumptions. Our results suggest the existence of an intrinsic rhythmicity in cells exhibiting arrhythmic spike trains in cerebellum, and may identify an important source of variability in neuronal firing patterns that is relevant to the mechanism of neural computation in cerebellum.

The multiplicity distribution, multiplicity moment, scaled variance, entropy and reduced entropy of target evaporated fragments emitted in forward and backward hemispheres in 12 A GeV 4He, 3.7 A GeV 16O, 60 A GeV 16O, 1.7 A GeV 84Kr and 10.7 A GeV 197Au -induced emulsion heavy target (AgBr) interactions are investigated. It is found that the multiplicity distribution of target evaporated fragments emitted in both forward and backward hemispheres can be fitted by a Gaussian distribution. The multiplicity moments of target evaporated particles emitted in the forward and backward hemispheres increase with the order of the moment q, and the second-order multiplicity moment is energy independent over the entire energy range for all the interactions in the forward and backward hemisphere. The scaled variance, a direct measure of multiplicity fluctuations, is close to one for all the interactions, which indicate a correlation among the produced particles. The entropy of target evaporated fragments emitted in both forward and backward hemispheres are the same within experimental errors.

Schreiber and Yukich [Ann. Probab. 36 (2008) 363–396] establish an asymptotic representation for random convex polytope geometry in the unit ball $\mathbb{B}^{d}$, $d\geq 2$, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of generalized paraboloid growth processes. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via these growth processes we establish local functional limit theorems for the properly scaled radius-vector and support functions of convex polytopes generated by high-density Poisson samples. We show that direct methods lead to explicit asymptotic expressions for the fidis of the properly scaled radius-vector and support functions. Generalized paraboloid growth processes, coupled with general techniques of stabilization theory, yield Brownian sheet limits for the defect volume and mean width functionals. Finally we provide explicit variance asymptotics and central limit theorems for the $k$-face and intrinsic volume functionals.

Local limit theorems are derived for sums of finite range $\mathbb{Z}$-valued potential functions of an iid random field. The resulting approximations turn out to be mixtures of standard normal densities for lattice distributions supported by residue classes of integers. The mixing weights are equal to the probability that the sum of potential functions lies in such a residue class and are nonasymptotic and computable. For finite range potential functions of a stationary Gibbsian random field with bounded and finite range interactions, conditions are given under which the global central limit theorem implies the classical local limit theorem.

We study Edgeworth expansions in limit theorems for self-normalized sums. Non-uniform bounds for expansions in the central limit theorem are established while imposing only minimal moment conditions. Within this result, we address the case of non-integer moments leading to a reduced remainder. Furthermore, we provide non-uniform bounds for expansions in local limit theorems. The enhanced tail accuracy of our non-uniform bounds allows for deriving an Edgeworth-type expansion in the entropic central limit theorem as well as a central limit theorem in total variation distance for self-normalized sums.

We obtain a central limit theorem, local limit theorems and renewal theorems for stationary processes generated by skew product maps$T(\unicode[STIX]{x1D714},x)=(\unicode[STIX]{x1D703}\unicode[STIX]{x1D714},T_{\unicode[STIX]{x1D714}}x)$together with a$T$-invariant measure whose base map$\unicode[STIX]{x1D703}$satisfies certain topological and mixing conditions and the maps$T_{\unicode[STIX]{x1D714}}$on the fibers are certain non-singular distance-expanding maps. Our results hold true when$\unicode[STIX]{x1D703}$is either a sufficiently fast mixing Markov shift with positive transition densities or a (non-uniform) Young tower with at least one periodic point and polynomial tails. The proofs are based on the random complex Ruelle–Perron–Frobenius theorem from Hafouta and Kifer [Nonconventional Limit Theorems and Random Dynamics. World Scientific, Singapore, 2018] applied with appropriate random transfer operators generated by$T_{\unicode[STIX]{x1D714}}$, together with certain regularity assumptions (as functions of $\unicode[STIX]{x1D714}$) of these operators. Limit theorems for deterministic processes whose distributions on the fibers are generated by Markov chains with transition operators satisfying a random version of the Doeblin condition are also obtained. The main innovation in this paper is that the results hold true even though the spectral theory used in Aimino, Nicol and Vaienti [Annealed and quenched limit theorems for random expanding dynamical systems.Probab. Theory Related Fields162(2015), 233–274] does not seem to be applicable, and the dual of the Koopman operator of$T$(with respect to the invariant measure) does not seem to have a spectral gap.

Describing the color and textural information of a person image is one of the most crucial aspects of person re-identification. In this paper, we present a novel descriptor based on a hierarchical distribution of pixel features. A hierarchical covariance descriptor has been successfully applied for image classification. However, the mean information of pixel features, which is absent in covariance, tends to be major discriminative information of person images. To solve this problem, we describe a local region in an image via hierarchical Gaussian distribution in which both means and covariances are included in their parameters. More specifically, we model the region as a set of multiple Gaussian distributions in which each Gaussian represents the appearance of a local patch. The characteristics of the set of Gaussians are again described by another Gaussian distribution. In both steps, unlike the hierarchical covariance descriptor, the proposed descriptor can model both the mean and the covariance information of pixel features properly. The results of experiments conducted on five databases indicate that the proposed descriptor exhibits remarkably high performance which outperforms the state-of-the-art descriptors for person re-identification.

Word embedding can represent a word as a point vector or a Gaussian distribution in high-dimensional spaces. Gaussian distribution is innately more expressive than point vector owing to the ability to additionally capture semantic uncertainties of words, and thus can express asymmetric relations among words more naturally (e.g., animal entails cat but not the reverse. However, previous Gaussian embedders neglect inner-word conceptual knowledge and lack tailored Gaussian contextualizer, leading to inferior performance on both intrinsic (context-agnostic) and extrinsic (context-sensitive) tasks. In this paper, we first propose a novel Gaussian embedder which explicitly accounts for inner-word conceptual units (sememes) to represent word semantics more precisely; during learning, we propose Gaussian Distribution Attention over Gaussian representations to adaptively aggregate multiple sememe distributions into a word distribution, which guarantees the Gaussian linear combination property. Additionally, we propose a Gaussian contextualizer to utilize outer-word contexts in a sentence, producing contextualized Gaussian representations for context-sensitive tasks. Extensive experiments on intrinsic and extrinsic tasks demonstrate the effectiveness of the proposed approach, achieving state-of-the-art performance with near 5.00% relative improvement.

The present paper deals with the interactions of 16 O and 28 Si nuclei with emulsion heavy target nuclei (silver (Ag) and bromine (Br)) at 4.5A GeV/c. The results have been analyzed to identify interesting characteristics of the total disintegration (TD) events produced in these interactions. Events having a number of heavily ionizing particles Nh 28 have been selected in this study. The integral multiplicity distributions for the disintegrated particles from the target nuclei have been used to separate the inelastic interactions with the free hydrogen (H), the light (CNO), and the heavy (AgBr) nuclei. The average multiplicities of the different secondary particles emitted in these interactions have been compared with the corresponding ones of different projectiles nearly at the same momentum. The average multiplicity of the grey particles is found to increase with the mass of the projectile, while the average multiplicity of the black particles is found to decrease with increasing mass of the projectile; this result agrees with the prediction of the fireball model. Also, the dependence of the shower, grey, and black particles produced in the TD events on the mass of the projectile has been studied. Moreover, the dependence of the probability of the TD for the projectile mass as well as the target fragments has also been discussed. The results show that the multiplicity distributions of the shower, grey, and black particles present a Gaussian distribution. Finally, the linear dependence of the grey and compound multiplicities on the black, heavy, and shower particles has also been observed.

We consider recurrent diffusive random walks on a strip. We present constructive conditions on Green functions of finite sub-domains which imply a Central Limit Theorem with polynomial error bound, a Local Limit Theorem, and mixing of environment viewed by the particle process. Our conditions can be verified for a wide class of environments including independent environments, quasiperiodic environments, and environments which are asymptotically constant at infinity. The conditions presented deal with a fixed environment, in particular, no stationarity conditions are imposed.

Central and local limit theorems for the coefficients of polynomials of binomial type

We study the asymptotic behavior of a multidimensional random walk in a general cone. We find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone. The main step in the proof consists in constructing a positive harmonic function for our random walk under minimal moment restrictions on the increments. For the proof of tail asymptotics and integral limit theorems, we use a strong approximation of random walks by Brownian motion. For the proof of local limit theorems, we suggest a rather simple approach, which combines integral theorems for random walks in cones with classical local theorems for unrestricted random walks. We also discuss some possible applications of our results to ordered random walks and lattice path enumeration.