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Abstract.In this article, by means of Rosenthal-type inequality, we study the complete convergence and strong law of large numbers for weighted sums of m-widely accpetable random variables under sub-linear expectations, which extend some existing results in classical probability space. As applications, we investigate the complete consistency and strong consistency of weighted estimators in nonparametric regression models under sub-linear expectations. In addition, we provide some simulation studies to verify the finite sample performance of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, strong consistency of tail value-at-risk (TVaR) estimator under widely orthant dependent (WOD) samples is established, and a numerical simulation is performed to verify the validity of the theoretical results. To reveal the essence of the result, theoretical discussion on complete and complete moment convergence corresponding to the Baum–Katz law, as well as the Marcinkiewicz–Zygmund type strong law of large numbers (MZSLLN) for maximal weighted sums and maximal product sums of widely orthant dependent (WOD) random variables are investigated. The results obtained in the context extend the corresponding ones for independent and some dependent random variables. [ABSTRACT FROM AUTHOR]

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]

AbstractIn this article, we investigate the trajectory fitting estimation for the integrated Ornstein-Uhlenbeck process driven by the Lévy process based on continuous observations. The strong consistency and asymptotic distributions of the estimator are obtained by the strong law of large numbers and the inner clock property of the α-stable stochastic integral. [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]

Abstract.In this work, we deal with some Bayesian inference problems in the presence of right censored data. First, we propose a dual ϕ−divergence Bayes type estimators for parametric models and we establish their asymptotic normality. To establish this result, we need a uniform strong law of large numbers that we prove as well. Then, we consider the problem of prior distributions construction for model selection using ϕ−divergences. Finally, we consider the problem of the predictive density estimation on the basis of ϕ−divergences. We apply an expansion result of the generalized Bayesian predictive density on two parametric models widely used in survival analysis, namely the Weibull and the inverse Weibull model, under right censoring. We also check the performances of our proposed methods through simulations and real data applications. The results of these studies show that our proposed dual ϕ−divergence Bayes type estimators are more robust than other Bayesian estimators. Moreover, the generalized Bayesian predictive density performs better than the classical estimative density especially for the inverse Weibull model. [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]

Most insurance contracts are inherently linked to financial markets, be it via interest rates, or—as hybrid products like equity‐linked life insurance and variable annuities—directly to stocks or indices. However, insurance contracts are not for trade except sometimes as surrender to the selling office. This excludes the situation of arbitrage by buying and selling insurance contracts at different prices. Furthermore, the insurer uses private information on top of the publicly available one about financial markets. This paper provides a study of the consistency of insurance contracts in connection with trades in the financial market with explicit mention of the information involved. By defining strategies on an insurance portfolio and combining them with financial trading strategies, we arrive at the notion of insurance–finance arbitrage (IFA). In analogy to the classical fundamental theorem of asset pricing, we give a fundamental theorem on the absence of IFA, leading to the existence of an insurance–finance‐consistent probability. In addition, we study when this probability gives the expected discounted cash‐flows required by the EIOPA best estimate. The generality of our approach allows to incorporate many important aspects, like mortality risk or general levels of dependence between mortality and stock markets. Utilizing the theory of enlargements of filtrations, we construct a tractable framework for insurance–finance consistent valuation. [ABSTRACT FROM AUTHOR]

Using elementary probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, we prove that for every infinite compact spaces K and L the product K\times L admits a sequence \langle \mu _n\colon n\in \mathbb {N}\rangle of normalized signed measures with finite supports which converges to 0 with respect to the weak* topology of the dual Banach space C(K\times L)^*. Our approach is completely constructive—the measures \mu _n are defined by an explicit simple formula. We also show that this result generalizes the classical theorem of Cembranos [Proc. Amer. Math. Soc. 91 (1984), pp. 556–558] and Freniche [Math. Ann. 267 (1984), pp. 479–486] which states that for every infinite compact spaces K and L the Banach space C(K\times L) contains a complemented copy of the space c_0. [ABSTRACT FROM AUTHOR]

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]

In this paper, we establish the complete f-moment convergence and the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of extended negatively dependent (END, for short) random variables under sub-linear expectations, which extend and improve corresponding ones in sub-linear expectation space. As applications of the main results, the complete consistency and strong consistency of weighted estimators in nonparametric regression models under sub-linear expectations are also obtained. [ABSTRACT FROM AUTHOR]

We study general random dynamical systems of continuous maps on some compact metric space. Assuming a local contraction condition and proximality, we establish probabilistic limit laws such as the (functional) central limit theorem, the strong law of large numbers, and the law of the iterated logarithm. Moreover, we study exponential synchronization and synchronization on average. In the particular case of iterated function systems on S 1 , we analyze synchronization rates and describe their large deviations. In the case of C 1 + β -diffeomorphisms, these deviations on random orbits are obtained from the large deviations of the expected Lyapunov exponent. [ABSTRACT FROM AUTHOR]

In this article, the complete convergence and the Kolmogorov strong law of large numbers for weighted sums of (α,β)\left(\alpha ,\beta )-mixing random variables are presented. An application to simple linear errors-in-variables model is provided. Simulation studies are also carried out to support the theoretical results.

Interactive computer simulations are commonly used as pedagogical tools to support students' statistical reasoning. This paper examines whether and how these simulations enable their intended effects. We begin by contrasting two theoretical frameworks—dual processes and grounded cognition—in the context of people's conceptions about statistical sampling, setting the stage for the potential benefits of simulations in learning such conceptions. Then, we continue with reviewing the educational literature on statistical sampling simulations. Our review tentatively suggests benefits of the simulations for building statistical habits of mind. However, challenges seem to persist when more specific concepts and skills are investigated. With and without simulations, students have difficulty forming an aggregate view of data, interpreting sampling distributions, showing a process-based understanding of the law of large numbers, making statistical inferences, and context-independent reasoning. We propose that grounded cognition offers a framework for understanding these findings, highlighting the bidirectional relationship between perception and conception, perceptual design features, and guided perceptual routines for supporting students' meaning making from simulations. Finally, we propose testable instructional strategies for using simulations in statistics education. Significance: Interactive computer simulations are popularly used to teach statistical sampling and inference. A substantial body of classroom-based design research has emerged over the last two decades on this topic, paralleling the interest of cognitive psychologists in statistical reasoning. This review bridges the gap by synthesizing diverse literature, from laboratory-based cognitive research to classroom-based design research, to investigate people's reasoning about statistical sampling with interactive computer simulations. We organize the commonly occurring findings from these studies under a grounded cognition framework. Using this framework, we also identify instructional design strategies that future empirical researchers can test and statistics and data science practitioners can adopt. First, we highlight the importance of repeated exposure to simulations in a way that fosters creating perception–action routines aligned with mathematical principles. Second, we argue that intuitive representations ground students' meaning making from simulations, and idealized representations help generalize learning. Third, we recommend that visual routines be guided during activities with simulations. Fourth, we note the separate affordances of simulations and verbal materials. Lastly, we propose that statistical processes depicted in the simulations should be reified as foundations for more advanced concepts and practices. Overall, the paper contributes to the learning theories and instructional design in the context of simulation-based learning in statistics. [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]

This paper provides sets of sufficient conditions for mean convergence theorems for arrays of dependent random variables. We expand and improve a number of particular cases in the literature including Theorem 2.1 in Sung (Appl Math Lett 26(1):18–24, 2013), Theorems 3.1–3.3 in Wu and Guan (J Math Anal Appl 377(2):613–623, 2011), and Theorem 3 in Lita da Silva (Results Math 74(1):1–11, 2019), among others. The proof is different from those in the aforementioned papers and the main results can be applied to obtain mean convergence results for arrays of functions of non-homogeneous Markov chains and dependent bootstrap. [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]

We consider weighted sums of independent random variables regulated by an increment sequence and provide operative conditions that ensure a strong law of large numbers for such sums in both the centred and non-centred case. The existing criteria for the strong law are either implicit or based on restrictions on the increment sequence. In our setup we allow for an arbitrary sequence of increments, possibly random, provided the random variables regulated by such increments satisfy some mild concentration conditions. In the non-centred case, convergence can be translated into the behaviour of a deterministic sequence and it becomes a game of mass when the expectation of the random variables is a function of the increment sizes. We identify various classes of increments and illustrate them with a variety of concrete examples. [ABSTRACT FROM AUTHOR]

In the classical two-armed bandit (TAB) problems, the goal of maximizing the total return specifies the optimal decision rule and fosters the development of abundant practicable methods, such as $ \epsilon $-greedy and upper confidence bound for TAB procedures. However, the existence of an arched reward function for the total return breaks the previously updated algorithms, like traditional myopic strategy, and prompts the generation of novel computation and theory for maximizing its expectation. Here we develop a special class of Bayesian two-armed bandit problems by imposing a prior probability on which arm has a greater expectation of the returns and propose a myopic strategy to specify the decision rule. We prove that the proposed myopic strategy is optimal under the structure of the arched reward function by exploring the dynamic programming principle. This article also demonstrates that arched functions are extensions of the monotone functions including the classical linear functions. Meanwhile, we establish a corresponding law of large numbers for our myopic strategy. Simulation studies pose supportive evidence that the newly proposed strategy performs well. [ABSTRACT FROM AUTHOR]

We consider a class of weakly interacting particle systems of mean-field type. The interactions between the particles are encoded in a graph sequence, i.e. two particles are interacting if and only if they are connected in the underlying graph. We establish a law of large numbers for the empirical measure of the system that holds whenever the graph sequence is convergent to a graphon. The limit is the solution of a non-linear Fokker–Planck equation weighted by the (possibly random) graphon limit. In contrast with the existing literature, our analysis focuses on both deterministic and random graphons: no regularity assumptions are made on the graph limit and we are able to include general graph sequences such as exchangeable random graphs. Finally, we identify the sequences of graphs, both random and deterministic, for which the associated empirical measure converges to the classical McKean–Vlasov mean-field limit. [ABSTRACT FROM AUTHOR]

In this paper, we prove an extension of the Jajte strong law of large numbers for exchangeable random variables, we make a simulation study for the asymptotic behavior in the sense of convergence almost surly for weighted sums of exchangeable weighted random variables. [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]

This paper provides sharp sufficient conditions for almost sure summability of the maximal normed partial sums of m-dependent random elements in stable type p Banach spaces, complementing recent results of Li, Qi, and Rosalsky (Trans Amer Math Soc 368(1):539–561, 2016) and Thành (Math Nachr 296(1):402–423, 2023). The main theorems are new even when the underlying random elements are independent. Our proof is direct and can be extended to other dependence structures. Two illustrative examples are also presented. [ABSTRACT FROM AUTHOR]

Motivated by modeling data transmission with optimization consideration, we investigate in this paper a class of stochastic recursive systems from the point of view of stochastic decision. By proving a law of large numbers for paths of the systems, we show that the control and optimization problem are closely linked to the solution of a differential equation. This differential equation may involve discontinuous function. Applications to a simple model of data transmission with optimal strategy is illustrated. [ABSTRACT FROM AUTHOR]

We study a d-dimensional branching Brownian motion (BBM) among Poissonian obstacles, where a random trap field in Rd is created via a Poisson point process. In the soft obstacle model, the trap field consists of a positive potential which is formed as a sum of a compactly supported bounded function translated at the atoms of the Poisson point process. Particles branch at the normal rate outside the trap field; and when inside the trap field, on top of complete suppression of branching, particles are killed at a rate given by the value of the potential. Under soft killing, the probability that the entire BBM goes extinct due to killing is positive in almost every environment. Conditional on ultimate survival of the process, we prove a law of large numbers for the total mass of BBM among soft Poissonian obstacles. Our result is quenched, that is, it holds in almost every environment with respect to the Poisson point process. [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]

High-frequency financial data are becoming increasingly available and need to be analyzed under the current circumstances for the market prices of stocks, currencies, risk analysis, portfolio management and other financial instruments. An emblematic challenge in econometrics is estimating the integrated volatility for financial prices, i.e., the quadratic variation of log prices. Following this point, in this paper, we study the estimation of integrated self-weighted volatility, i.e., the generalized style of integrated volatility, by using intraday high-frequency data with noise. In order to reduce the effect of noise, the "pre-averaging" technique is used. Both the law of large numbers and the central limit theorem of the estimator of integrated self-weighted volatility are established in this paper. Meanwhile, a studentized version is also given in order to make some statistical inferences. At the end of this article, the simulation results obtained to evaluate the accuracy of approximating the sampling distributions of the estimator are displayed. [ABSTRACT FROM AUTHOR]

A necessary and sufficient condition for the iterated function system { f (· , ω) | ω ∈ Ω } with probability P to have exactly one invariant measure μ ∗ with μ ∗ ((0 , 1)) = 1 is given. The main novelty lies in the fact that we only require the transformations f (· , ω) to be increasing homeomorphims, without any smoothness condition, neither we impose conditions on the cardinality of Ω . In particular, positive Lyapunov exponents conditions are replaced with the existence of solutions to some functional inequalities. The stability and strong law of large numbers of the considered system are also proven. [ABSTRACT FROM AUTHOR]

Random monotone operators are stochastic versions of maximal monotone operators which play an important role in stochastic nonsmooth optimization. Several stochastic nonsmooth optimization algorithms have been shown to converge to a zero of a mean operator defined as the expectation, in the sense of the Aumann integral, of a random monotone operator. In this note, we prove a strong law of large numbers for random monotone operators where the limit is the mean operator. We apply this result to the empirical risk minimization problem appearing in machine learning. We show that if the empirical risk minimizers converge as the number of data points goes to infinity, then they converge to an expected risk minimizer. [ABSTRACT FROM AUTHOR]