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We undertake a detailed study of the L 2 discrepancy of 2-dimensional Korobov lattices and their irrational analogues, either with or without symmetrization. We give a full characterization of such lattices with optimal L 2 discrepancy in terms of the continued fraction partial quotients, and compute the precise asymptotics whenever the continued fraction expansion is explicitly known, such as for quadratic irrationals or Euler's number e. In the metric theory, we find the asymptotics of the L 2 discrepancy for almost every irrational, and the limit distribution for randomly chosen rational and irrational lattices. [ABSTRACT FROM AUTHOR]

Optimal transport (OT) is a versatile framework for comparing probability measures, with many applications to statistics, machine learning and applied mathematics. However, OT distances suffer from computational and statistical scalability issues to high dimensions, which motivated the study of regularized OT methods like slicing, smoothing and entropic penalty. This work establishes a unified framework for deriving limit distributions of empirical regularized OT distances, semiparametric efficiency of the plug-in empirical estimator and bootstrap consistency. We apply the unified framework to provide a comprehensive statistical treatment of (i) average- and max-sliced |$p$| -Wasserstein distances, for which several gaps in existing literature are closed; (ii) smooth distances with compactly supported kernels, the analysis of which is motivated by computational considerations; and (iii) entropic OT, for which our method generalizes existing limit distribution results and establishes, for the first time, efficiency and bootstrap consistency. While our focus is on these three regularized OT distances as applications, the flexibility of the proposed framework renders it applicable to broad classes of functionals beyond these examples. [ABSTRACT FROM AUTHOR]

Let { ξ i , i ∈ Z } be a negatively superadditive dependent (NSD) sequence of positive random variables. In this paper, under suitable conditions, we obtain the limit distribution for products of sums of partial sums of linear processes generated by { ξ i , i ∈ Z }. [ABSTRACT FROM AUTHOR]

We consider one of the possible generalizations of sums of independent random variable to the case of operations on a finite set, namely quasigroup "sums" that use quasigroup operations on a given finite set instead of the addition operation. For quasigroup "sums" that contain n independent identically distributed random variables we prove that the rate of convergence of distributions to uniform distribution is exponential in n. [ABSTRACT FROM AUTHOR]

This paper considers a new mean-variance model with strong mixing errors and describes a combination test for the mean shift and variance change. Under some stationarity and symmetry conditions, the important limiting distribution for a combination test is obtained, which can derive the limiting distributions for the mean change test and variance change test. As an application, an algorithm for a three-step method to detect the change-points is given. For example, the first step is to test whether there is at least a change-point. The second and third steps are to detect the mean change-point and the variance change-point, respectively. To illustrate our results, some simulations and real-world data analysis are discussed. The analysis shows that our tests not only have high powers, but can also determine the mean change-point or variance change-point. Compared to the existing methods of cpt.meanvar and mosum from the R package, the new method has the advantages of recognition capability and accuracy. [ABSTRACT FROM AUTHOR]

The assessment of college and university counselors and student management team has been one of the important issues that colleges and universities are concerned about. This paper applies the group decision-making model to the assessment of college counselors’ student work, and establishes the group decision-making model of college counselors’ assessment work according to the actual work of counselors. Attitude consensus threshold is set, and ACT is combined with individual feedback parameters to realize the balance between individual minimum correction and group consistency. And the method of mutual evaluation and scoring by five appraisal subjects is used to construct Markov matrix, and then the limit distribution of the matrix is found through linear weighting. Finally, the appraisal of 105 counselors was empirically analyzed, and the group decision-making model was used to give practical significance to the five appraisal items, namely, sense of efficacy, dedication, leadership, innovativeness, and executive power, one by one. The results show that for the 205 college counselors tested in the appraisal, the mean value of the student management work ability index is 30.55, and the maximum value is 90.95, and the results of this appraisal pass the test and have practical reference significance.

The role of Alexander Yakovlevich Khintchine in mathematics and especially in probability theory is undeniable. His approaches to teaching mathematics, especially in the secondary school, are still a perfect model for mathematics teachers. On the other hand, his role in the theory of infinitely divisible distributions,, distribution of the sum of independent random variables, stable distributions and the domain of attraction of the Gauss law is fundamental and influential. The purpose of this article is to introduce educational approaches and key research achievements in the 30s in mathematics.1. Introduction A. Ya. Khintchine was born on July 19, 1894 in the village Kondrovo of the Kaluga region, about one and a half hundred km southwest of Moscow. From 1911 to 1916 he was a student of the Physical-Mathematical faculty of the Moscow State University (MSU). All his scientific life was in deeply connected with this University. In the period of study at the University and in the first years of his research career Khintchine was under a strong influence of the ideas and personality of N.N. Luzin. It is known that A.Ya. Khintchine presented his first result at a meeting of the student mathematical club in November 1914.Khintchine was a member of the Soviet delegation at the International Congress of Mathematicians held in Bologna (Italy) from 3 to 10 September 1928. The Russian delegation was represented by 27 scientists including some prominent researchers like S. Bernstein (Karkhov), A.Ya. Khintchine (Moscow), V. Romanovsky (Tashkent) and E. Slutsky (Moscow). We note, however, that Khintchine did not present any communication so that he did not publish a paper in the Proceedings of the Congress (which appeared in 1929-1932). In connection with the above said motivation for our paper, it is especially important to describe the works by Khintchine because of the following reasons: • Several important results by Khintchine are forgotten and later rediscovered. • A number of results were published in inaccessible places and not in English. • The concrete and clear style of Khintchine’s work can help the readers to understand bettersome recent results.2. Main ResaltsAlexander Yakovlevich Khintchine had a constant and deep interest to the problems of teaching as in universities as in the secondary schools. His pedagogical ideas he has presented in his textbooks, monographs and special articles. In 1938-1940 he headed the physical-mathematical section of the Methodical-educational Soviet at the Ministry of Education of the Russian Federation. When the Academy of Pedagogical Sciences of the Russian Federation was founded, he became an academician ofthis Academy. He was very active as a member of the editorial board of the multi-volume “Encyclopedy of Elementary Mathematics”, some volumes of which appeared in the late 1950s. The first papers by A. Ya. Khintchine were appeared in 1924. In order to understand the role of these articles it is necessary to describe the state of Probability Theory in those years. One can recall a critical review by R. von Mises who summed up of the situation in the following words: To-day, probability theoryis not a mathematical science. The tremendous development of Probability Theory, which thus took place in the twenty years from 1920s to 1940s was, no doubt, a joint effect of the efforts of a number of mathematicians and statisticians. However, it does not seem unlikely that future historians will ascribe its development, as far as the mathematical side of the subject is concerned, above all to the creative powers of four scientists (in alphabetic order): B. de Finetti, A.Ya. Khintchine, A. N. Kolmogorov, and P. Levy. In fact, it may be said that the real turning point came with the publications of the following works: • P. Levy, Calcul des Probabilites, Gauthier-Villars, Paris, (1925) pp. viii+350. • B. de Finetti, Funzione caratteristica di un fenomeno aleatorio, Memorie della R. Accademia Nazionale dei Lincei, 4 no.5 (1930) 86-133. • A. Ya. Khintchine, Asymptotische Gesetze der Wahrscheinlichkeitsrechnung, Julius Springer, Berlin, 1933. • A. N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, Julius Springer, Berlin, 1933. • P. Levy, Sur les integrales dont leselements sont des variables aleatoires independentes, Annali della R. Scuola Normale di Pisa, 3 (1934) 337-366 and 4 (1935) 217-218. (1) $B_{0}\in T_{n}(D)$. (2) $(B_{1})_{ij}\in D$ for all $1\leq i\leq j\leq n-1$.Let $n>1$ and $f(X)=B_{k}X^{k}+\cdots+B_{1}X+B_{0}\in{\operatorname{Int}}(T_{n}(D))$. Then the following statments hold. 3. Summary of ProofsThere was no satisfactory definition of mathematical probability, and the conceptual foundations of the subject were completely obscure. Moreover, with few exceptions, mainly belonging to the French and Russian schools, writers on probability did not seem aware of the standards of rigor which in other mathematical fields, were regarded as obvious. Already in the middle of 1920s were appeared another estimate of Probability Theory as a branch of mathematical science. The Probability Theory has an integral method deeply connected with the methods of modern theory of functions, and thus the most of the recent ideas appeared in the Mathematical Analysis have a fruitful application in the Probability Theory. This optimistic opinion by A. Ya. Khintchine has got an evident justification in the next few decades. At the end of the 1930s, the picture has been radically changed. Mathematical probability theory was firmly established on an axiomatic foundation. It became a purely mathematical discipline, with problems and methods of its own, conforming the current standards of mathematical rigorism, and entering into fruitful relations with other branches of mathematics. At the same time, the fields of applications of mathematical probability were steadily and rapidly growing in number and importance.It is true that nowadays there are still some pure mathematicians who tend to look down on the applied science of probability. But this attitude is expected to disappear within a generation. The tremendous development of Probability Theory, which thus took place in the twenty years from 1920s to 1940s was, no doubt, a joint effect of the efforts of a number of mathematicians and statisticians.

This paper discusses the problem of the mean changes in time series with heavy-tailed AR(p) noise. Firstly, it proposes a modified ratio-type test statistic, and the results show that under the null hypothesis of no mean change, the asymptotic distribution of the modified statistic is a functional of Lévy processes and the consistency under the alternative hypothesis is obtained. However, a heavy-tailed index exists in the asymptotic distribution and is difficult to estimate. This paper uses bootstrap sampling, jackknife sampling, and subsampling to approximate the distribution under the null hypothesis, and obtain more accurate critical values and empirical power. In addition, some results from a small simulation study and a practical example give an idea of the finite sample behavior of the proposed statistic. [ABSTRACT FROM AUTHOR]

Focusing on a continuous-time quantum walk on Z = 0 , ± 1 , ± 2 , ... , we analyze a probability distribution with which the quantum walker is observed at a position. The walker launches off at a localized state, and its system is operated by a spatially periodic Hamiltonian. As a result, we see an asymmetric probability distribution. To catch a long-time behavior, we also try to find a long-time limit theorem and realize that the limit distribution holds a symmetric density function. [ABSTRACT FROM AUTHOR]

We consider a critical Galton – Watson branching process starting with N particles; the number of offsprings is supposed to have the distribution pk=(k + 1)−τ−(k + 2)−τ, k=0, 1, 2, ... Limit distributions of the maximal tree size are obtained for the corresponding Galton – Watson forest with N trees and n non-root vertices as N, n → ∞, n/Nτ ⩾ C > 0. [ABSTRACT FROM AUTHOR]

Poor drug adherence is considered one of major barriers to achieving the clinical and public health benefits of many pharmacotherapies. In the current paper, we aim to investigate the impact of dose omission on the plasma concentrations of two-compartment pharmacokinetic models with two typical routes of drug administration, namely the intravenous bolus and extravascular first-order absorption. First, we reformulate the classical two-compartment pharmacokinetic models with a new stochastic feature, where a binomial random model of dose intake is integrated. Then, we formalize the explicit expressions of expectation and variance for trough concentrations and limit concentrations, with the latter proved of the existence and uniqueness for steady-state distribution. Moreover, we mathematically demonstrate the strict stationarity and ergodicity of trough concentrations as a Markov chain. In addition, we numerically simulate the impact of drug non-adherence to different extents on the variability and regularity of drug concentration and compare the drug pharmacokinetic preference between one and two compartment pharmacokinetic models. The results of sensitivity analysis also suggest the drug non-adherence as one of the most sensitive model parameters to the expectation of limit concentration. Our modelling and analytical approach can be integrated into the chronic disease models to estimate or quantitatively predict the therapy efficacy with drug pharmacokinetics presumably affected by random dose omissions. [ABSTRACT FROM AUTHOR]

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Discrete kernel smoothing is now gaining importance in nonparametric statistics. In this paper, we investigate some asymptotic properties of the normalised discrete associated-kernel estimator of a probability mass function and make comparisons. We show, under some regularity and non-restrictive assumptions on the associated-kernel, that the normalising random variable converges in mean square to 1. We then derive the consistency and the asymptotic normality of the proposed estimator. Various families of discrete kernels already exhibited satisfy the conditions, including the refined CoM-Poisson which is underdispersed and of second-order. Finally, the first-order binomial kernel is discussed and, surprisingly, its normalised estimator has a suitable asymptotic behaviour through simulations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a first-order autoregressive process with a general autoregressive coefficient. Asymptotic behaviors of an M-estimator of the autoregressive coefficient are established for the nearly stationary and mildly explosive cases, respectively. The rate of convergence of the robust estimators for the two cases are provided. The results extend ones for the least squares and least absolute deviation estimators to the robust estimator under the weaker initial conditions in the literature. Some simulations are carried out to assess the performance of our procedure. [ABSTRACT FROM AUTHOR]

Let $ G $ be a Lie group, and let $ \Gamma $ be a lattice in $ G $. We introduce the notion of locally unipotent invariant measures on $ G/\Gamma $. We then prove that under some conditions, the limit measure supported on the image of polynomial trajectories on $ G/\Gamma $ is locally unipotent invariant, thus give a partial answer to an equidistribution problem for higher dimensional polynomial trajectories on homogeneous spaces, which was raised by Shah in [15].The proof relies on Ratner's measure classification theorem, a linearization technique for polynomial trajectories near singular sets, and a twisting technique of Shah. [ABSTRACT FROM AUTHOR]

We consider a critical GaltonnWatson branching process starting with N particles and such that the number of offsprings of each particle is distributed as pk = (k +1))) (k +2))), k = 0, 1, 2, . . . . For the corresponding GaltonnWatson forest with N trees and n nonroot vertices, we find the limit distributions for the number of trees of a given size as N, n, n/NN C > 0. [ABSTRACT FROM AUTHOR]

This paper considers a new mean-variance model with strong mixing errors and describes a combination test for the mean shift and variance change. Under some stationarity and symmetry conditions, the important limiting distribution for a combination test is obtained, which can derive the limiting distributions for the mean change test and variance change test. As an application, an algorithm for a three-step method to detect the change-points is given. For example, the first step is to test whether there is at least a change-point. The second and third steps are to detect the mean change-point and the variance change-point, respectively. To illustrate our results, some simulations and real-world data analysis are discussed. The analysis shows that our tests not only have high powers, but can also determine the mean change-point or variance change-point. Compared to the existing methods of cpt.meanvar and mosum from the R package, the new method has the advantages of recognition capability and accuracy.

We study the limiting behaviour of the maximum of a bivariate (finite or infinite) moving average model, based on discrete random variables. We assume that the bivariate distribution of the iid innovations belong to the Anderson's class (Anderson, 1970). The innovations have an impact on the random variables of the INMA model by binomial thinning. We show that the limiting distribution of the bivariate maximum is also of Anderson's class, and that the components of the bivariate maximum are asymptotically independent. [ABSTRACT FROM AUTHOR]

Let T n be the set of all mappings T : { 1 , 2 , ... , n } → { 1 , 2 , ... , n }. The corresponding graph of T is a union of disjoint connected unicyclic components. We assume that each T ∈ T n is chosen uniformly at random (i.e., with probability n − n). The cycle of T contained within its largest component is called the deepest one. For any T ∈ T n , let ν n = ν n (T) denote the length of this cycle. In this paper, we establish the convergence in distribution of ν n / n and find the limits of its expectation and variance as n → ∞. For n large enough, we also show that nearly 55% of all cyclic vertices of a random mapping T ∈ T n lie in its deepest cycle and that a vertex from the longest cycle of T does not belong to its largest component with approximate probability 0.075. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of the mean changes in time series with heavy-tailed AR(p) noise. Firstly, it proposes a modified ratio-type test statistic, and the results show that under the null hypothesis of no mean change, the asymptotic distribution of the modified statistic is a functional of Lévy processes and the consistency under the alternative hypothesis is obtained. However, a heavy-tailed index exists in the asymptotic distribution and is difficult to estimate. This paper uses bootstrap sampling, jackknife sampling, and subsampling to approximate the distribution under the null hypothesis, and obtain more accurate critical values and empirical power. In addition, some results from a small simulation study and a practical example give an idea of the finite sample behavior of the proposed statistic.

In this paper, we investigate the mean change-point models based on associated sequences. Under some weak conditions, we obtain a limit distribution of CUSUM statistic which can be used to judge the mean change-mount δ n is satisfied or dissatisfied n 1 / 2 δ n = o (1). We also study the consistency of sample covariances and change-point location statistics. Based on Normality and Lognormality data, some simulations such as empirical sizes, empirical powers and convergence are presented to test our results. As an important application, we use CUSUM statistics to do the mean change-point analysis for a financial series. [ABSTRACT FROM AUTHOR]

In this paper, on the basis of constructing a new shift operator S and choosing the Grover coin G as coin operator C , we get the standard evolution operator U on cycles. Using U , we not only got the analytical expression of wavefunction ψ s , j , but also obtained the conclusion that the limit distribution π (ν) of N , which is not uniform distribution, regardless of N is odd or even. [ABSTRACT FROM AUTHOR]

In this paper, we consider an expanding sparse dynamic network model where the time evolution is governed by a Markovian structure. Transition of the network from time t to t + 1 involves three components where a new node joins the existing network, some of the existing edges drop out, and new edges are formed with the incoming node. We consider long term behavior of the network density and establish its limit. We also study asymptotic distributions of the maximum likelihood estimators of key model parameters. We report results from a simulation study to investigate finite sample properties of the estimators. [ABSTRACT FROM AUTHOR]

A continuous-state branching process with immigration having branching mechanism $\Psi$ and immigration mechanism $\Phi$ , a CBI $(\Psi,\Phi)$ process for short, may have either of two different asymptotic regimes, depending on whether $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u or $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$. When $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u , the CBI process has either a limit distribution or a growth rate dictated by the branching dynamics. When $\scriptstyle\int_{0}\tfrac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$ , immigration overwhelms branching dynamics. Asymptotics in the latter case are studied via a nonlinear time-dependent renormalization in law. Three regimes of weak convergence are exhibited. Processes with critical branching mechanisms subject to a regular variation assumption are studied. This article proves and extends results stated by M. Pinsky in 'Limit theorems for continuous state branching processes with immigration' (Bull. Amer. Math. Soc.78, 1972). [ABSTRACT FROM AUTHOR]

In the paper, the limit distribution of an integral square deviation between two kernel type of Nadaraya-Watson estimators of the Bernoulli regression function for the group data is established. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the CUSUM statistic of change point under the negatively associated (NA) sequences. By establishing the consistency estimators for mean and covariance functions respectively, the limit distribution of the CUSUM statistic is proved to be a standard Brownian bridge, which extends the results obtained under the case of an independent normal sample and the moving average processes. Finally, the finite sample properties of the CUSUM statistic are given to show the efficiency of the method by simulation studies and an application on a real data analysis. [ABSTRACT FROM AUTHOR]

We consider the model of an N-vertex configuration graph where the number of edges is at most n and the degrees of vertices are independent and identically distributed (i.i.d.) random variables (r.v.'s). The distribution of the r.v. ?, which is defined as the degree of any given vertex, is assumed to satisfy the condition pk = P{? = k} ~ L/kg lnh k as k = ∞, where L > 0, g > 1, h &#8805: 0. Limit theorems for the number of vertices of a given degree as N, n = ∞ are proved. [ABSTRACT FROM AUTHOR]

We present the probability distribution function of the nodal domain number for the right-isosceles triangle billiard. This is the first explicit result for a nonseparable plane polygonal billiard. The distribution function has a peak at (the same as for integrable billiards) and also at . The only exact results previously known in this direction correspond to integrable billiards that are separable. [ABSTRACT FROM AUTHOR]

Abstract We study the minimum Skorohod distance estimation θε∗ $\theta _{\varepsilon}^{\ast }$ and minimum L1 $L_{1}$-norm estimation θε˜ $\widetilde {\theta _{\varepsilon}}$ of the drift parameter θ of a stochastic differential equation dXt=θXtdt+εdLtd $dX_{t}=\theta X_{t}\,dt+\varepsilon \,dL^{d}_{t}$, X0=x0 $X_{0}=x_{0}$, where {Ltd,0≤t≤T} $\{L^{d}_{t},0\leq t\leq T\}$ is a fractional Lévy process, ε∈(0,1] $\varepsilon \in (0,1]$. We obtain their consistency and limit distribution for fixed T, when ε→0 $\varepsilon \rightarrow 0$. Moreover, we also study the asymptotic laws of their limit distributions for T→∞ $T\rightarrow \infty $.

The classical coin tossing experiment is studied. Limit theorems are obtained concerning the head-runs containing certain number of tails. It is proved that the limit of the number of those runs of length n which contain at most T tails is compound Poisson. Accompanying distributions are obtained for the length of the longest head-runs containing at most T tails. To this end a two parameter family of accompanying distributions is offered. [ABSTRACT FROM AUTHOR]

We undertake a detailed study of the L 2 discrepancy of 2-dimensional Korobov lattices and their irrational analogues, either with or without symmetrization. We give a full characterization of such lattices with optimal L 2 discrepancy in terms of the continued fraction partial quotients, and compute the precise asymptotics whenever the continued fraction expansion is explicitly known, such as for quadratic irrationals or Euler's number e . In the metric theory, we find the asymptotics of the L 2 discrepancy for almost every irrational, and the limit distribution for randomly chosen rational and irrational lattices., (© The Author(s) 2024.)

This article investigates the long-time behavior of conservative affine processes on the cone of symmetric positive semidefinite $d\times d$ matrices. In particular, for conservative and subcritical affine processes we show that a finite $\log$ -moment of the state-independent jump measure is sufficient for the existence of a unique limit distribution. Moreover, we study the convergence rate of the underlying transition kernel to the limit distribution: first, in a specific metric induced by the Laplace transform, and second, in the Wasserstein distance under a first moment assumption imposed on the state-independent jump measure and an additional condition on the diffusion parameter. [ABSTRACT FROM AUTHOR]

The problem of iterated partial summations is solved for some discrete distributions defined on finite supports. The power method, usually used as a computational approach to the problem of finding matrix eigenvalues and eigenvectors, is in some cases an effective tool to prove the existence of the limit distribution, which is then expressed as a solution of a system of linear equations. Some examples are presented. [ABSTRACT FROM AUTHOR]

Group testing has been widely used in epidemiology and related fields to estimate prevalence of rare diseases. Parametric binary regression models are used in group testing to estimate the covariate adjusted prevalence. Unlike the standard binary regression model (viz. , logistic, complementary log–log, etc.), the regression model for group testing data connects the maximum of a group of independent binary responses to its covariate values. Recently, in group testing literature, it has been extensively used for estimating covariate adjusted prevalence of a disease making the tacit assumptions that (i) the regression model is identifiable, and (ii) the standard asymptotic theory is valid for the maximum likelihood estimators of the regression parameters. Verifying these assumptions is found to be non-trivial theoretical issues. In this paper, we give theoretical proofs of these assumptions under a set of simple sufficient conditions, thus, providing a theoretical justification for likelihood based inference on the regression parameters. We also provide an outline of the proof extending the asymptotic theory to the data obtained by Dorfman retesting, where all subjects in a positive group are retested. • This article provides a rigorous proof of identifiability in a parametric group testing regression model. • It also provides mild conditions under which asymptotic theory for MLE's is valid in such models, for both equal and unequal group sizes. • Develops asymptotic theory for MLE in such models, for equal and unequal group sizes. • Finally, the article considers Dorfman re-testing, and provides conditions under which standard asymptotic theory for MLE's can be developed. • Provides conditions for extending standard asymptotic theory for Dorfman re-testing. [ABSTRACT FROM AUTHOR]

A centroid node in a tree is a node for which the sum of the distances to all other nodes attains its minimum, or equivalently a node with the property that none of its branches contains more than half of the other nodes. We generalise some known results regarding the behaviour of centroid nodes in random recursive trees (due to Moon) to the class of very simple increasing trees, which also includes the families of plane-oriented and d-ary increasing trees. In particular, we derive limits of distributions and moments for the depth and label of the centroid node nearest to the root, as well as for the size of the subtree rooted at this node. [ABSTRACT FROM AUTHOR]

Abstract We introduce logarithmic generalized Maxwell distribution motivated by Vodă (Math. Rep. 11:171-179, 2009), which is an extension of the generalized Maxwell distribution. Some interesting properties of this distribution are studied and the asymptotic distribution of the partial maximum of an i.i.d. sequence from the logarithmic generalized Maxwell distribution is gained. The expansion of the limit distribution from the normalized maxima is established under the optimal norming constants, which shows the rate of convergence of the distribution for normalized maximum to the extreme limit.

We consider the tessellation induced by a stationary Poisson hyperplane process in d‐dimensional Euclidean space. Under a suitable assumption on the directional distribution, and measuring the k‐faces of the tessellation by a suitable size functional, we determine a limit distribution for the shape of the typical k‐face, under the condition of given size and this tending to zero. The limit distribution is concentrated on simplices. This extends a result of Gilles Bonnet. [ABSTRACT FROM AUTHOR]

We consider configuration graphs with N vertices. The degrees of vertices are independent identically distributed random variables having the power-law distribution with positive parameter τ. $\tau.$ We study properties of random graphs such that the sum of vertex degrees does not exceed n and the parameter is a random variable uniformly distributed on the interval [ a , b ] , 0 < a < b < ∞. $\left[ a,\,\,b \right],0

Abstract We discuss limit distributions for hitting-time functions of certain "exceptional" families of asymptotically rare events for ergodic probability preserving transformations. The abstract core is an inducing argument. The latter applies, for example, to shrinking intervals around periodic points (both uniformly expanding and neutral) of certain finite measure preserving interval maps. In particular, we give a complete answer to a question raised in Freitas et al. (2016). [ABSTRACT FROM AUTHOR]

We consider various probability distributions {Gn,n≥1} concentrated on the interval [-1,1]⊂R and investigate basic properties of the limit distribution Γ of the Banach random walk in a Banach space B generated by {Gn,n≥1}. In particular, we describe assumptions ensuring that the support of Γ is equal to the unit sphere in B and, on the other hand, we find conditions under which every ball of radius smaller than 1 has a positive measure Γ. [ABSTRACT FROM AUTHOR]