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The well‐posedness and regularity properties of diffusion–aggregation equations, emerging from interacting particle systems, are established on the whole space for bounded interaction force kernels by utilizing a compactness convergence argument to treat the nonlinearity as well as a Moser iteration. Moreover, we prove a quantitative estimate in probability with arbitrary algebraic rate between the approximative interacting particle systems and the approximative McKean–Vlasov SDEs, which implies propagation of chaos for the interacting particle systems. [ABSTRACT FROM AUTHOR]

Eigenvalues of Wigner matrices has been a major topic of investigation. A particularly important subclass of such random matrices, useful in many applications, are what are known as sparse or diluted random matrices, where each entry in a Wigner matrix is multiplied by an independent Bernoulli random variable with mean p$p$. Alternatively, such a matrix can be viewed as the adjacency matrix of an Erdős–Rényi graph Gn,p$\mathcal {G}_{n,p}$ equipped with independent and identically distributed (i.i.d.) edge‐weights. An observable of particular interest is the largest eigenvalue. In this paper, we study the large deviations behavior of the largest eigenvalue of such matrices, a topic that has received considerable attention over the years. While certain techniques have been devised for the case when p$p$ is fixed or perhaps going to zero not too fast with the matrix size, we focus on the case p=dn$p = \frac{d}{n}$, that is, constant average degree regime of sparsity, which is a central example due to its connections to many models in statistical mechanics and other applications. Most known techniques break down in this regime and even the typical behavior of the spectrum of such random matrices is not very well understood. So far, results were known only for the Erdős–Rényi graph Gn,dn$\mathcal {G}_{n,\frac{d}{n}}$without edge‐weights and with Gaussian edge‐weights. In the present article, we consider the effect of general weight distributions. More specifically, we consider entry distributions whose tail probabilities decay at rate e−tα$e^{-t^\alpha }$ with α>0$\alpha >0$, where the regimes 0<α<2$0<\alpha < 2$ and α>2$\alpha > 2$ correspond to tails heavier and lighter than the Gaussian tail, respectively. While in many natural settings the large deviations behavior is expected to depend crucially on the entry distribution, we establish a surprising and rare universal behavior showing that this is not the case when α>2$\alpha > 2$. In contrast, in the α<2$\alpha < 2$ case, the large deviation rate function is no longer universal and is given by the solution to a variational problem, the description of which involves a generalization of the Motzkin–Straus theorem, a classical result from spectral graph theory. As a byproduct of our large deviation results, we also establish the law of large numbers behavior for the largest eigenvalue, which also seems to be new and difficult to obtain using existing methods. In particular, we show that the typical value of the largest eigenvalue exhibits a phase transition at α=2$\alpha = 2$, that is, corresponding to the Gaussian distribution. [ABSTRACT FROM AUTHOR]

Let { X , X n , n ≥ 1 } be a sequence of pairwise NQD identically distributed random variables and { b n , n ≥ 1 } be a sequence of positive constants. In this article, we study the strong laws of large numbers for the sequence { X , X n , n ≥ 1 } , under the general moment condition ∑ n = 1 ∞ P (| X | > b n / log n) < ∞ , which improve some known results. [ABSTRACT FROM AUTHOR]

We obtain strong laws of large numbers of Marcinkiewicz–Zygmund's type for weighted sums of pairwise positively quadrant dependent random variables stochastically dominated by a random variable X ∈ L p , 1 ⩽p < 2. We use our results to establish the strong consistency of estimators which emerge from regression models having pairwise positively quadrant-dependent errors. [ABSTRACT FROM AUTHOR]

In this paper, we prove an extension of the Jajte weak law of large numbers for exchangeable random variables. We make a simulation to illustrate the asymptotic behavior in the sense of convergence in probability for weighted sums of exchangeable weighted random variables. [ABSTRACT FROM AUTHOR]

In this work, by using Marcinkiewicz-Zygmund type moment inequality of asymptotically negatively associated (ANA, in short) sequences, the strong law of large numbers of linear processes with random coefficients generated by ANA sequences is studied. The obtained results extend the convergence of linear processes with constant coefficients to the case of random coefficients. [ABSTRACT FROM AUTHOR]

The laws of large numbers are studied from the arrays of asymmetrical Cauchy random variables. Some exact laws of large numbers are obtained for the k-th order statistics, including a weak law and a strong law. It turns out that the weak law cannot extend to a strong law. Our results extend some existing works for independent and dependent cases. [ABSTRACT FROM AUTHOR]

Let be an array of martingale difference random vectors and be an array of m × d matrices of real numbers. In this paper, the Marcinkiewicz–Zygmund type weak law of large numbers for maximal weighted sums of martingale difference random vectors is obtained with not necessarily finite p-th (1 < p < 2) moments. Moreover, the complete convergence and strong law of large numbers are established under some mild conditions. An application to multivariate simple linear regression model is also provided. [ABSTRACT FROM AUTHOR]

We investigate barycenters of probability measures on Gromov hyperbolic spaces, toward development of convex optimization in this class of metric spaces. We establish a contraction property (the Wasserstein distance between probability measures provides an upper bound of the distance between their barycenters), a deterministic approximation of barycenters of uniform distributions on finite points, and a kind of law of large numbers. These generalize the corresponding results on CAT(0)-spaces, up to additional terms depending on the hyperbolicity constant. [ABSTRACT FROM AUTHOR]

The Čech and Rips constructions of persistent homology are stable with respect to perturbations of the input data. However, neither is robust to outliers, and both can be insensitive to topological structure of high-density regions of the data. A natural solution is to consider 2-parameter persistence. This paper studies the stability of 2-parameter persistent homology: we show that several related density-sensitive constructions of bifiltrations from data satisfy stability properties accommodating the addition and removal of outliers. Specifically, we consider the multicover bifiltration, Sheehy's subdivision bifiltrations, and the degree bifiltrations. For the multicover and subdivision bifiltrations, we get 1-Lipschitz stability results closely analogous to the standard stability results for 1-parameter persistent homology. Our results for the degree bifiltrations are weaker, but they are tight, in a sense. As an application of our theory, we prove a law of large numbers for subdivision bifiltrations of random data. [ABSTRACT FROM AUTHOR]

In the framework of sublinear expectation, we investigate the limit behavior of linear processes and derive a strong law of large numbers for them. It turns out that our theorem is a natural extension of the one in the classical linear case, and we can derive the corresponding strong law of large numbers for independent random variables under sublinear expectation from our result. [ABSTRACT FROM AUTHOR]

In information theory, the entropy ergodic theorem is a fundamental one well-known to all. In this article, we will study the generalized entropy ergodic theorem for Markov chains indexed by a spherically symmetric tree. First, we give the definition of Markov chains indexed by a spherically symmetric tree. Meanwhile, the generalized strong law of large numbers for Markov chains indexed by a spherically symmetric tree is proved. Finally, we obtain the generalized entropy ergodic theorem for Markov chains indexed by a spherically symmetric tree. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the difference between the deterministic and stochastic models of nonlocal diffusion. We use a nonlocal reaction-diffusion equation and a multi-dimensional jump Markov process to analyze these mathematical models. First, we demonstrate that the difference converges to 0 in probability with a supremum norm for a sizeable network. Next, we consider the rescaled difference and show that it converges to a stochastic process in distribution on the Skorokhod space. [ABSTRACT FROM AUTHOR]

Charles Pence's book, "The Rise of Chance in Evolutionary Theory: A Pompous Parade of Arithmetic," offers a well-documented and philosophical analysis of the role of chance, statistics, and probability in the development of Charles Darwin's theory of evolution. Pence challenges the traditional dichotomy between biometrics and Mendelism, arguing that the two approaches are interconnected and contributed to the Modern Synthesis. The book examines the contributions of key figures such as Francis Galton and Walter Frank Raphael Weldon, and highlights the importance of statistical analysis in understanding natural selection. While the book does not extensively explore the conceptual transformation of chance, it provides a valuable framework for further research in the field. [Extracted from the article]

In this paper, we investigate two types of the conditional strong law of large numbers with a new notion of conditionally independent random variables under G-expectation which are related to the symmetry G-function. Our limit theorem demonstrates that the cluster points of empirical averages fall within the bounds of the lower and upper conditional expectations with lower probability one. Moreover, for conditionally independent random variables with identical conditional distributions, we show the existence of two cluster points of empirical averages that correspond to the essential minimum and essential maximum expectations, respectively, with G-capacity one. [ABSTRACT FROM AUTHOR]

Abstract.In this article, under some proper and sufficient conditions, we gave complete convergence for weighted sums and maximal weighted sums of arrays of rowwise mn-extended negatively dependent (rowwise mn-END) random variables, which is a new dependent structure. In addition, a relationship between {mn,n≥1} and moment condition for convergence is revealed in a sense. The results obtained in the article generalize some corresponding ones for independent and some dependent random variables. As an application, the strong consistency for the weighted estimator in a non parametric regression model is established. [ABSTRACT FROM AUTHOR]

In a vast area of probabilistic limit theorems for dynamical systems with chaotic behaviors always only functional form (exponential, power, etc.) of the asymptotic laws and of convergence rates were studied. However, for basically all applications, e.g., for computer simulations, development of algorithms to study chaotic dynamical systems numerically, as well as for design and analysis of real (e.g., in physics) experiments, the exact values (or at least estimates) of constants (parameters) of the functions, which appear in the asymptotic laws and rates of convergence, are of primary interest. In this paper, we provide such estimates of constants (parameters) in the central limit theorem, large deviations principle, law of large numbers and the rate of correlations decay for strongly chaotic dynamical systems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some sufficient conditions of Chung type almost sure convergence and strong law of large numbers of double array of arbitrarily dependent random variables. In the proofs, the techniques developed by Ghosal and Chandra are employed. [ABSTRACT FROM AUTHOR]

We study two problems. First, we consider the large deviation behavior of empirical measures of certain diffusion processes as, simultaneously, the time horizon becomes large and noise becomes vanishingly small. The law of large numbers (LLN) of the empirical measure in this asymptotic regime is given by the unique equilibrium of the noiseless dynamics. Due to degeneracy of the noise in the limit, the methods of Donsker and Varadhan [Comm. Pure Appl. Math. 29 (1976), pp. 389–461] are not directly applicable and new ideas are needed. Second, we study a system of slow-fast diffusions where both the slow and the fast components have vanishing noise on their natural time scales. This time the LLN is governed by a degenerate averaging principle in which local equilibria of the noiseless system obtained from the fast dynamics describe the asymptotic evolution of the slow component. We establish a large deviation principle that describes probabilities of divergence from this behavior. On the one hand our methods require stronger assumptions than the nondegenerate settings, while on the other hand the rate functions take simple and explicit forms that have striking differences from their nondegenerate counterparts. [ABSTRACT FROM AUTHOR]

Integral tests are found for the convergence of two Spitzer-type series associated with a class of weighted averages introduced by Jajte [On the strong law of large numbers, Ann. Probab., 31(1):409–412, 2003]. Our main theorems are valid for a large family of dependent random variables that are not necessarily identically distributed. As a byproduct, we improve the Marcinkiewicz–Zygmund strong law of large numbers for asymptotically almost negatively associated sequences due to Chandra and Ghosal [Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables Acta Math. Hung., 71(4):327–336, 1996]. We also complement two limit theorems recently derived by Anh et al. [TheMarcinkiewicz–Zygmund-type strong law of large numbers with general normalizing sequences, J. Theor. Probab., 34(1):331–348, 2021] and Thành [On a new concept of stochastic domination and the laws of large numbers, Test, 32(1):74–106, 2023]. The obtained results are new even when the summands are independent. [ABSTRACT FROM AUTHOR]

The convergence theorems play an important role in the theory of probability and statistics and in its application. In recent times, we studied three types of convergence of intuitionistic fuzzy observables, i.e., convergence in distribution, convergence in measure and almost everywhere convergence. In connection with this, some limit theorems, such as the central limit theorem, the weak law of large numbers, the Fisher–Tippet–Gnedenko theorem, the strong law of large numbers and its modification, have been proved. In 1997, B. Riečan studied an almost uniform convergence on D-posets, and he showed the connection between almost everywhere convergence in the Kolmogorov probability space and almost uniform convergence in D-posets. In 1999, M. Jurečková followed on from his research, and she proved the Egorov's theorem for observables in MV-algebra using results from D-posets. Later, in 2017, the authors R. Bartková, B. Riečan and A. Tirpáková studied an almost uniform convergence and the Egorov's theorem for fuzzy observables in the fuzzy quantum space. As the intuitionistic fuzzy sets introduced by K. T. Atanassov are an extension of the fuzzy sets introduced by L. Zadeh, it is interesting to study an almost uniform convergence on the family of the intuitionistic fuzzy sets. The aim of this contribution is to define an almost uniform convergence for intuitionistic fuzzy observables. We show the connection between the almost everywhere convergence and almost uniform convergence of a sequence of intuitionistic fuzzy observables, and we formulate a version of Egorov's theorem for the case of intuitionistic fuzzy observables. We use the embedding of the intuitionistic fuzzy space into the suitable MV-algebra introduced by B. Riečan. We formulate the connection between the almost uniform convergence of functions of several intuitionistic fuzzy observables and almost uniform convergence of random variables in the Kolmogorov probability space too. [ABSTRACT FROM AUTHOR]

In this paper, by establishing a Borel–Cantelli lemma for a capacity which is not necessarily continuous, and a link between a sequence of independent random variables under the sub-linear expectation and a sequence of independent random variables on ℝ∞ under a probability, we give the sufficient and necessary conditions of the strong law of large numbers for independent and identically distributed random variables under the sub-linear expectation, and the sufficient and necessary conditions for the convergence of an infinite series of independent random variables, without the assumption on the continuity of the capacities. A purely probabilistic proof of a weak law of large numbers is also given. [ABSTRACT FROM AUTHOR]

To improve the detection precision of the transient electromagnetic method, dense observation points and windows are necessary. However, to keep the number of samples in every acquisition window unchanged, the sampling rate of principal transient electromagnetic instruments decreases with transmitter frequency, and the increased windows reduce the original signal-to-noise ratio at the cost of decreasing the number of samples participating in the statistical average. Taking the V8 receiver from Phoenix Geophysics as an example, this study analyzes the main steps in the signal detection of a transient electromagnetic instrument. Following the Wiener-Khinchin law of large numbers, herein, a fixed, high-sampling-rate receiver scheme, which does not decrease the number of samples participating in the statistical average while increasing the windows, is proposed. Starting from the Nyquist sampling theorem, we provide the definition of detection precision with the dimension of length. Then, according to the necessary conditions for reconstructing a geoelectric structure, we determine the relationship between a survey grid space and a detection depth space corresponding to window interval as well as the minimum period in a geoelectric signal spectrum. The exploration of the multilayer, water-filled goaf of the Dazigou Coal Mine in Datong shows the aliasing caused by undersampling and the fine geoelectric section obtained using additional conditions under a geophysical premise. The necessary conditions for the reconstruction of the geoelectric structure indicate that satisfying the Nyquist sampling theorem may be insufficient in reconstructing the geoelectric structure, but the structure must not be reconstructed completely without not satisfying this theorem. [ABSTRACT FROM AUTHOR]

In this article, we investigate complete convergence and strong laws of large numbers for weighted sums of coordinatewise pairwise negative quadrant dependent random variables taking values in Hilbert spaces. As an application, the complete convergence and the almost sure convergence of degenerate von Mises-statistics are investigated. [ABSTRACT FROM AUTHOR]

In this paper we study the Marcinkiewicz–Zygmund-type strong law of large numbers with general normalizing sequences under sublinear expectation. Specifically, we establish complete convergence in the Marcinkiewicz–Zygmund-type strong law of large numbers for sequences of negatively dependent and identically distributed random variables under certain moment conditions. We also give results for sequences of independent and identically distributed random variables. The moment conditions in this paper are based on a class of slowly varying functions that satisfy some convergence properties. Moreover, some special examples and comparisons to existing results are also given. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the asymptotic behavior of the capacity of the range of symmetric random walks on finitely generated groups. We show the corresponding strong law of large numbers and central limit theorem. [ABSTRACT FROM AUTHOR]

In this paper, the complete convergence for maximum of weighted sums of widely negative orthant dependent (WNOD) random variables are investigated. Some sufficient conditions for the convergence are provided and a relationship between the weight and the boundary function is revealed. Additionally, a Marcinkiewicz-Zygmund type strong law of large number for weighted sums of WNOD random variables is obtained. The results obtained in this paper generalize some corresponding ones for independent and some dependent random variables. As an application, the strong consistency for the weighted estimator in a non-parametric regression model is established. MR(2010) Subject Classification: 60F15; 62G05. [ABSTRACT FROM AUTHOR]

Hyperdiverse tropical forest tree communities illustrate a fundamental problem in ecology: How can many species coexist given relatively few limiting resources? Neutral theory provides a solution by positing that species have equal fitness and hence drift to extinction slowly. However, neutral theory seriously under‐predicts temporal changes in species abundances. This can be remedied by breaking neutrality and adding temporal environmental stochasticity (TES), but the mechanisms mediating the effects of TES on species richness remain unclear. Here, we make progress by analysing a local community model with species competing for a common resource under TES, to derive formulae partitioning species richness according to different mechanisms.By applying our formulae to generic parameter sets for tropical forest tree communities, we found that when the autocorrelation time of TES was short, the dominant mechanism driving species richness was non‐linear averaging of the interspecific competition term over time, which reduced the typical strength of interspecific competition and boosted richness relative to the neutral case. However, greater immigration to the community resulted in more species and hence weaker non‐linear averaging due to the law of large numbers. In contrast, when the autocorrelation time of TES was long, the dominant mechanism driving species richness was strong selection between changes in environmental conditions, which increased the typical strength of interspecific competition and reduced richness relative to the neutral case.By applying our formulae to a specific parameter set for a tropical forest tree community in Panama, we found that TES had minor effects on species richness because (i) the immigration rate was sufficiently large for non‐linear averaging of the interspecific competition term to be weak and (ii) the autocorrelation time was sufficiently short to suppress the effects of selection.Synthesis. We provide a novel mechanistic explanation of how TES affects tree species richness in tropical forests, in particular how TES often has minor effects on richness despite having substantial effects on temporal changes in species abundances. This provides a deeper insight into why a neutral model with added TES can accurately capture static and dynamic aspects of tree community diversity in the tropics. [ABSTRACT FROM AUTHOR]

• A simple and easy-to-understand seismic risk assessment method for cultural artifacts was proposed. • The Law of Large Numbers was applied to achieve the seismic risk assessment. • Three types of common porcelain vases were taken as case studies for seismic risk assessment. • A comparison was conducted between the developed method and an existing method. Earthquakes can cause significant damage to cultural artifacts, which often hold significant historical or cultural value. Seismic risk assessments can contribute to a more targeted approach by museum staff to the preventive conservation of cultural artifacts. The primary objective of this research is to suggest a new approach for assessing the seismic risk of cultural artifacts. This innovative method is founded on the Law of Large Numbers and aims to provide a more user-friendly way of evaluating the likelihood of potential seismic threat to cultural artifacts. The proposed method takes into account the statistical distribution of seismic ground motion and the seismic response characteristics of artifacts, and its accuracy and practicality are demonstrated by case studies. Compared with existing methods, the proposed method has the advantages of theoretical simplicity, low computational effort, and easier to be understood and mastered by museum staff. Furthermore, the impact of sample size on the assessment results was investigated. The findings demonstrate that the proposed method represents a valuable tool for cultural heritage risk decision-makers to evaluate the seismic risk of artifacts. By using this method, they can more effectively assess the potential damage caused by seismic effects and design suitable mitigation measures accordingly. [Display omitted] [ABSTRACT FROM AUTHOR]

We study finite particle systems on the one-dimensional integer lattice, where each particle performs a continuous-time nearest-neighbour random walk, with jump rates intrinsic to each particle, subject to an exclusion interaction which suppresses jumps that would lead to more than one particle occupying any site. We show that the particle jump rates determine explicitly a unique partition of the system into maximal stable sub-systems, and that this partition can be obtained by a linear-time algorithm using only elementary arithmetic. The internal configuration of each stable sub-system possesses an explicit product-geometric limiting distribution, and the location of each stable sub-system obeys a strong law of large numbers with an explicit speed; the characteristic parameters of each stable sub-system are simple functions of the rate parameters for the corresponding particles. For the case where the entire system is stable, we provide a central limit theorem describing the fluctuations around the law of large numbers. Our approach draws on ramifications, in the exclusion context, of classical work of Goodman and Massey on partially-stable Jackson queueing networks. [ABSTRACT FROM AUTHOR]

We introduce a class of Markov models to describe the bid–ask price dynamics in the presence of liquidity fluctuations. In a highly competitive regime, the spread evolution belongs to a class of Markov processes known as a population process with uniform catastrophes. Our mathematical analysis focuses on establishing the law of large numbers, the central limit theorem, and large deviations for this catastrophe-based model. Large deviation theory allows us to illustrate how huge deviations in the spread and prices can occur in the model. Moreover, our research highlights how these local trends and volatility are influenced by the typical values of the bid–ask spread. We calibrated the model parameters using available high-frequency data and conducted Monte Carlo numerical simulations to demonstrate its ability to reasonably replicate key phenomena in the presence of liquidity fluctuations. [ABSTRACT FROM AUTHOR]

Abstract. In this article, we establish complete convergence for randomly weighted sums of negatively orthant-dependent random variables. We also establish complete convergence for the maximums of randomly weighted sums of negatively associated random variables. As a corollary, a Marcinkiewicz-Zygmund-type strong law for randomly weighted sums of negatively associated random variables is obtained. The results can be applied to the bootstrap sample means, and an open problem in a convergence rate of the bootstrap sample means posed by Csörgő (2004) is solved completely. [ABSTRACT FROM AUTHOR]

Let λ = (λ1,...,λn) be β-Jacobi ensembles with parameters p1, p2, n and β while β varying with n. Set γ = lim n → ∞ n p 1 and σ = lim n → ∞ p 1 p 2 . In this paper, supposing lim n → ∞ log n β n = 0 , we prove that the empirical measures of different scaled λ converge weakly to a Wachter distribution, a Marchenko–Pastur law and a semicircle law corresponding to σγ > 0, σ = 0 or γ = 0, respectively. We also offer a full large deviation principle with speed βn2 and a good rate function to precise the speed of these convergences. As an application, the strong law of large numbers for the extremal eigenvalues of β-Jacobi ensembles is obtained. [ABSTRACT FROM AUTHOR]

Quantum trajectories are Markov processes modeling the evolution of a quantum system subjected to repeated independent measurements. Under purification and irreducibility assumptions, these Markov processes admit a unique invariant measure — see Benoist et al. (2019). In this article we prove finer limit theorems such as Law of Large Numbers (LLN), Functional Central Limit Theorem, Law of Iterated Logarithm and Moderate Deviation Principle. The proof of the LLN is based on Birkhoff's ergodic theorem and an analysis of harmonic functions. The other theorems are proved using martingale approximation of empirical sums. [ABSTRACT FROM AUTHOR]

The given document is an abstract of talks given at the 7th International Conference on Stochastic Methods, held in Russia in June 2022. The conference featured talks from leading scientists from various countries, focusing on probability theory and mathematical statistics. The abstracts cover topics such as image classification techniques, spike trains analysis, computing system operating time, investment portfolio management, and nonlinear parabolic equations. The document also includes abstracts on hydrodynamics, dynamics of motion, interpolating deflators, statistical methods for fibrous materials, matrix substitutions, and conflict flows management. The text discusses mathematical models and theorems related to stochastic processes and differential equations, providing insights into their behavior and properties. The references cited offer further sources for exploration. Additionally, the document includes summaries of research papers on double barrier options pricing, branching random walks, series expansion of stochastic integrals, viscoelastic rope oscillations, and grid equations in computing systems. Each summary provides an overview of the research topic and main findings without expressing personal opinions or judgments. [Extracted from the article]

We prove the result above in the specific case of PCA-spectral cut then HT ht for HT ht (then HT ht ). Lemma 1 Let HT ht be the Karhunen-Loeve expansion of HT ht given at page 334. [Extracted from the article]

In this paper, some results on complete moment convergence of moving average process generated by m-WOD random variables are established. The results improve and extend some corresponding ones in the literature. As corollaries, two new results on Marcinkiewicz-Zygmund type strong law of large numbers are established for moving average processes generated by m-WOD random variables and respectively, m-END random variables. [ABSTRACT FROM AUTHOR]

This paper develops a Gaussian model for an open network of queues having a path-dependent net-input process, whose evolution depends on its early history, and satisfies a non-ergodic law of large numbers. We show that the Gaussian model arises as the heavy-traffic limit for a sequence of open queueing networks, each with a multivariate generalization of a Polya arrival process. We show that the net-input and queue-length processes for the Gaussian model satisfy non-ergodic laws of large numbers with tractable distributions. [ABSTRACT FROM AUTHOR]

We provide tools for identification and exploration of data with very large variability having power law tails. Such data describe extreme features of processes such as fire losses, flood, drought, financial gain/loss, hurricanes, population of cities, among others. Prediction and quantification of extreme events are at the forefront of the current research needs, as these events have the strongest impact on our lives, safety, economics, and the environment. We concentrate on the intuitive, rather than rigorous mathematical treatment of models with heavy tails. Our goal is to introduce instructors to these important models and provide some tools for their identification and exploration. The methods we provide may be incorporated into courses such as probability, mathematical statistics, statistical modeling or regression methods. Our examples come from ecology and census fields. for this article are available online. [ABSTRACT FROM AUTHOR]

In this article, the complete convergence and complete moment convergence for weighted sums of asymptotically negatively associated random vectors are investigated. The results obtained in this article extend the corresponding ones for random variables in Chen et al. to H-valued random vectors. Weighted version of strong law of large numbers and complete integral convergence result are also presented. [ABSTRACT FROM AUTHOR]

We investigate the convergence of higher order Cesàro means in Banach spaces. The main results of this paper are: (1) The proof of mean and Birkhoff-type ergodic theorems for higher order Cesàro means. (2) The existence of a one-to-one correspondence between convergent Cesàro means of different orders. (3) The proof of strong laws of large numbers for higher order sums of independent and identically distributed random elements. (4) A characterization of the ergodicity of measure preserving maps in terms of higher order mixing properties. To deal with higher order Cesàro means, one needs a notion of weighted mean more general than the one usually considered in the literature on weighted ergodic theorems. In this context, we also prove a characterization of generalized weighted means preserving Cesàro convergence. [ABSTRACT FROM AUTHOR]

Given a Gromov‐hyperbolic group G$G$ endowed with a finite symmetric generating set, we study the statistics of counting measures on the spheres of the associated Cayley graph under linear representations of G$G$. More generally, we obtain a weak law of large numbers for subadditive functions, echoing the classical Fekete lemma. For strongly irreducible and proximal representations, we prove a counting central limit theorem with a Berry–Esseen type error rate and exponential large deviation estimates. Moreover, in the same setting, we show convergence of interpolated normalized matrix norms along geodesic rays to Brownian motion and a functional law of iterated logarithm, paralleling the analogous results in the theory of random matrix products. Our counting large deviation estimates address a question of Kaimanovich–Kapovich–Schupp. In most cases, our counting limit theorems will be obtained from stronger almost sure limit laws for Patterson–Sullivan measures on the boundary of the group. [ABSTRACT FROM AUTHOR]

The main aim of this work is to investigate small noise limiting behavior of multi-scale McKean-Vlasov stochastic dynamical systems, where we allow the coefficients depend on the distributions of both slow and fast components. Firstly, the strong convergence in the functional law of large numbers is established by the time discretization scheme. Secondly, in order to characterize the probability of deviations away from the averaged limit, we prove the large deviation principle by the weak convergence approach for McKean-Vlasov equations. [ABSTRACT FROM AUTHOR]

A discrete quantum process is represented by a sequence of quantum operations, which are completely positive maps that are not necessarily trace preserving. We consider quantum processes that are obtained by repeated iterations of a quantum operation with noise. Such ergodic quantum processes generalize independent quantum processes. An ergodic theorem describing convergence to equilibrium for a general class of such processes has been recently obtained by Movassagh and Schenker. Under irreducibility and mixing conditions, we obtain a central limit type theorem describing fluctuations around the ergodic limit. [ABSTRACT FROM AUTHOR]

In this work, suppose that {Xn; n ≥ 1}is a sequence of asymptotically negatively associated random variables and {ani; 1 ≤ i ≤ n, n ≥ 1} is an array of real numbers such that ∑ i = 1 n | a n i | q = O (n) for some q > max { α p − 1 α − 1 / 2 , 2 } with αp > 1 and α > 1 2 . Let l (x) > 0 be a slowly varying function at infinity. We establish some equivalent conditions of the complete convergence for weighted sums of this form ∑ n = 1 ∞ n α p − 2 l (n) P ( max 1 ≤ j ≤ n | ∑ i = 1 j a n i X i | > ε n α) < ∞ for all ε > 0. As applications, some strong laws of large numbers for weighted sums of asymptotically negatively associated random variables are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak convergence and convergence rate in the weak law of large numbers for weighted sums of a class of random variables satisfying the Rosenthal type inequality. The necessary and sufficient conditions for the convergence rates in the weak law of large numbers under some mild conditions are provided. Moreover, the main results that we established are applied to simple linear errors-in-variables regression models and nonparametric regression models based on a class of random errors. Finally, we present some numerical simulations to assess the finite sample performance of the theoretical results. [ABSTRACT FROM AUTHOR]

On examples of the Chuk-a-luck three-dice game of chance and several card games, the probabilities, expectations, variances, and standard deviations of random variables and their sum are considered. Part I "Games of Chance" consists of 17 chapters on probability approach, about which the famous physicist R. Feynman said that "The theory of probability is a system for making better guesses" (p. 3). The book presents mathematical explanation of problems related to playing games of chance, combinatorial and strategic games, with descriptions of their historical perspectives and recreational aspects. [Extracted from the article]

Abstract Motivated by risk measure, super-hedge pricing, and modeling uncertainty in finance, Shige Peng established the theory of sub-linear expectation. In this article, we derive two results of laws of large numbers in the framework of sub-linear expectations. One is the strong law of large numbers for the array of random variables, which satisfies non identical distributed and exponential negatively dependent under sub-linear expectation. The other is the weak law of large numbers for the array of random variables, which satisfies non identical distributed and Φ -negatively dependent under sub-linear expectation. These results include and extend some existing results. [ABSTRACT FROM AUTHOR]