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In this correspondence, we establish mean convergence theorems for the maximum of normed double sums of Banach space valued random elements. Most of the results pertain to random elements which are M-dependent. We expand and improve a number of particular cases in the literature on mean convergence of random elements in Banach spaces. One of the main contributions of the paper is to simplify and improve a recent result of Li, Presnell, and Rosalsky [Journal of Mathematical Inequalities, 16, 117–126 (2022)]. A new maximal inequality for double sums of M-dependent random elements is proved which may be of independent interest. The sharpness of the results is illustrated by four examples. [ABSTRACT FROM AUTHOR]

The convergence of a sequence of point processes with dependent points, defined by a symmetric function of iid high-dimensional random vectors, to a Poisson random measure is proved. This also implies the convergence of the joint distribution of a fixed number of upper order statistics. As applications of the result a generalization of maximum convergence to point process convergence is given for simple linear rank statistics, rank-type U-statistics and the entries of sample covariance matrices. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, the marginal distribution of second-order concomitants of order statistics (SCOSs) from the Cambanis family is determined. An advantageous recurrence relation between single moments of SCOSs and concomitants of order statistics (COSs) is established based on the Cambanis family. We also use this family to derive an elegant beneficial recurrence relation between single moments of the SCOSs. Some information measures, such as Shannon differential entropy, differential extropy, and negative cumulative extropy (NCEX) of SCOSs based on this family, are investigated. Furthermore, we present some numerical examples to back up the theoretical findings. In addition, a set of bivariate data from the real world was examined. Finally, an estimate of a marginal variable’s scale parameter in the copula Cambanis family is obtained. [ABSTRACT FROM AUTHOR]

We investigate the asymptotic behaviour of the sequence of forward type iterations of a given random-valued vector function on the state space being a separable and complete metric space. Assuming non-linear contraction in mean we prove that the considered sequence converges weakly to a random variable with a finite first moment and independent of the initial state. Moreover, we show that the speed of this convergence does not have to be geometric. We also present examples illustrating the result obtained. [ABSTRACT FROM AUTHOR]

In this correspondence, we prove the von Bahr–Esseen moment inequality for pairwise independent random vectors in Hilbert spaces. Our constant in the von Bahr–Esseen moment inequality is better than that obtained for the real-valued random variables by Chen et al. [The von Bahr–Esseen moment inequality for pairwise independent random variables and applications, J. Math. Anal. Appl. 419 (2014), 1290–1302], and Chen and Sung [Generalized Marcinkiewicz–Zygmund type inequalities for random variables and applications, J. Math. Inequal. 10(3) (2016), 837–848]. The result is then applied to obtain mean convergence theorems for triangular arrays of rowwise and pairwise independent random vectors in Hilbert spaces. Some results in the literature are extended. [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]

In this article, we investigate the complete moment convergence for a sequence of coordinatewise pairwise negatively quadrant dependent random vectors in Hilbert space, which is coordinatewise weakly upper bounded by a random vector under special case. [ABSTRACT FROM AUTHOR]

This paper presents several statistical convergence concepts of complex uncertain sequences based on a regular matrix of Fibonacci numbers: statistical Fibonacci convergence almost surely, statistical Fibonacci convergence in measure, statistical Fibonacci convergence in mean and statistical Fibonacci convergence uniformly almost surely. Furthermore, the interrelations between them are studied and depicted in the form of a diagram. [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]

A multivalued version of Cesàro mean convergence theorem is established. As a consequence, various convergence results of strong law of large numbers for pairwise independent Pettis integrable random sets in a Banach space without any additional geometric condition on the space are provided. [ABSTRACT FROM AUTHOR]

Numerical solutions of stochastic problems involve approximations of the random functions in their definitions by deterministic functions of time and/or space and finite numbers of random variables, referred to as finite dimensional (FD) random functions. Since the solutions of these problems are complex functionals of random functions, the resulting numerical solutions are informative only if target random functions and their FD representations have similar samples. We establish conditions under which extremes and other functionals of real-valued continuous processes X(t) can be estimated from samples of FD processes X d (t) . Four types of FD processes are developed for stationary and nonstationary processes X(t). Numerical examples are presented to illustrate the construction of these FD processes and assess their capabilities to characterize extremes of X(t). [ABSTRACT FROM AUTHOR]

This paper establishes mean convergence theorems and weak laws of large numbers for the maximum normed partial sums from a d-dimensional array of random elements taking values in a real separable Banach space, irrespective of their joint distributions. The main results extend and improve several ones in the literature. The sharpness of the results is illustrated by three examples. [ABSTRACT FROM AUTHOR]

The aim of this treatise is to introduce the concept of strongly almost convergence in complex uncertain sequences. Also, we investigate the strongly almost convergence in almost surely (in shortly a.s.), convergence in measure, convergence in mean, convergence in distribution and convergence in uniformly almost surely of complex uncertain sequences and study the relationships among them. [ABSTRACT FROM AUTHOR]

In this paper, the marginal distributions of concomitants of k-record values based on Sarmanov family of bivariate distributions are obtained, as an extension of several recent papers. Besides, we derive the joint distribution of concomitants of k-record values for this family. Furthermore, some new and useful properties of information measures, namely, the extropy, Shannon entropy, inaccuracy measure, cumulative entropy, cumulative residual entropy, and cumulative residual Fisher information are studied. Finally, we offered various examples accompanied by numerical investigations that backed up the theoretical findings. [ABSTRACT FROM AUTHOR]

Dynamic mode decomposition (DMD) has attracted much attention in recent years as an analysis method for time series data. In this paper, we perform asymptotic analysis on the DMD to prove strong consistency in the statistical sense. More specifically, we first give a statistical model of random noise for data with observation noise. Among many variants of the DMD, the total least squares DMD (TLS-DMD) is known as a robust method for the random noise in the observation. We focus on consistency analysis of the TLS-DMD in the statistical sense and aim to generalize the analysis for projected methods. This paper gives a general framework for designing projection methods based on efficient dimensionality reduction analogously to the proper orthogonal decomposition under the statistical model and proves its strong consistency of the estimation. [ABSTRACT FROM AUTHOR]

In this paper, we consider a special class of dependent random variables (APND) that contains negative dependent random variables classes(ND,PND,NUOD,NA,NLOD,) and some classes of positive dependent random variables. Then we generalize Kolmogrov–Feller weak law of large numbers for i.i.d. random variables to APND random variables. we further give corresponding forms of dependence for random elements taking values in separable Banach space. This development will be of use for obtaining Banach space weak law of large numbers. [ABSTRACT FROM AUTHOR]

In 1971, Chow and Teicher presented a probability approximation Theorem via summability methods. The goal of this article is to extend Chow and Teicher results to higher dimension. To achieve this we consider multidimensional totally monotonic and independent identically random variables. Using these notions we present a series of approximation type theorems. [ABSTRACT FROM AUTHOR]

Economic globalisation and the development of financial trade liberalisation lead to a higher probability of financial crises. At the same time, the occurrence of financial crises has a particular risk of contagion. Based on this research background, this paper constructs a dynamic Copula model. It demonstrates the application of this model in financial market risk management based on the correlation changes between the US stock market and the Chinese stock market before and after the financial crisis. The results show that the Standard & Poor’s Index and China before the crisis broke out There is a specific correlation between the stock markets, which shows that the financial crisis has affected both the Chinese and American stock markets. Therefore, risks in the financial market are contagious.

Sarmanov family of bivariate distributions, which was suggested by Sarmanov (Mathematical models in hydrology symposium, 1974) as a new mathematical model of hydrological processes, is considered one of the most flexible and efficient extended families of the traditional FGM family. Despite its manifest advantages, it was never investigated in the literature. In this paper, we revisit this family by revealing several new prominent statistical properties. The distribution theory of concomitants of order statistics from this family is investigated. Besides, some aspects of information measures, namely the Shannon entropy, inaccuracy measure, and Fisher information number, are theoretically and numerically studied. Two bivariate real-world data sets have been analyzed for illustrative purposes, and the performance is quite satisfactory. [ABSTRACT FROM AUTHOR]

We prove an analogue of Khinchin's theorem for weakly correlated random elements with values in the spaces lp, 1 ⩽ p < ∞. [ABSTRACT FROM AUTHOR]

In this paper we show that the set of Gibbs measures for models with countable set of spin values is convex and compact. Also, by using analogue of Kolmogorov’s extension theorem, we prove the set of Gibbs measures for the models is nonempty. [ABSTRACT FROM AUTHOR]

Assume (Ω , A , P) is a probability space, X is a compact metric space with the σ -algebra B of all its Borel subsets and f : X × Ω → X is B ⊗ A -measurable and contractive in mean. We consider the sequence of iterates of f defined on X × Ω N by f 0 (x , ω) = x and f n (x , ω) = f (f n - 1 (x , ω) , ω n) for n ∈ N , and its weak limit π . We show that if ψ : X → R is continuous, then for every x ∈ X the sequence 1 n ∑ k = 1 n ψ (f k (x , ·)) n ∈ N converges almost surely to ∫ X ψ d π . In fact, we are focusing on the case where the metric space is complete and separable. [ABSTRACT FROM AUTHOR]

We prove the existence of an intermediate Banach space between the space where the Gaussian measure lives and its RKHS, thus extending what happens with Wiener measure, where the intermediate space can be chosen as a space of Hölder paths. From this result, it is very simple to deduce a result of exponential tightness for Gaussian probabilities. [ABSTRACT FROM AUTHOR]

This note establishes the weak laws of large numbers with general normalizing sequences for weighted sums of dependent identically distributed random vectors taking values in real separable Hilbert spaces. The dependent structures include pairwise and coordinatewise negative dependence and coordinatewise negative association. The sharpness of the result for the case where the random vectors are coordinatewise negatively associated is illustrated by an example. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to give a vector lattice version of a Theorem by Burkholder about convergence of martingales. The proof is based on a vector lattice analogue of Austin's sample function theorem, proved recently by Grobler, Labuschagne and Marraffa and on a new characterization of elements of the sup-completion of a universally complete vector lattice do not belong to the space. [ABSTRACT FROM AUTHOR]

The paper consists of two parts. At first, assuming that (Ω, A, P) is a probability space and (X, ϱ) is a complete and separable metric space with the σ-algebra 𝒝 of all its Borel subsets we consider the set 𝒭c of all 𝒝 ⊗ 𝒜-measurable and contractive in mean functions f : X × Ω → X with finite integral ∫ Ω ϱ (f(x, ω), x) P (dω) for x ∈ X, the weak limit π f of the sequence of iterates of f ∈ 𝒭c, and investigate continuity-like property of the function f ↦ πf, f ∈ 𝒭c, and Lipschitz solutions φ that take values in a separable Banach space of the equation

Given a sequence (X n) of symmetrical random variables taking values in a Hilbert space, an interesting open problem is to determine the conditions under which the series ∑ n = 1 ∞ X n is almost surely convergent. For independent random variables, it is well known that if ∑ n = 1 ∞ E (‖ X n ‖ 2) < ∞ , then ∑ n = 1 ∞ X n converges almost surely. This has been extended to some cases of dependent variables (namely negatively associated random variables), but in the general setting of dependent variables, the problem remains open. This paper considers the case where each variable X n is given as a linear combination a n , 1 Z 1 + ⋯ + a n , n Z n where (Z n) is a sequence of independent symmetrical random variables of unit variance and (a n , k) are constants. For Gaussian random variables, this is the general setting. We obtain a sufficient condition for the almost sure convergence of ∑ n = 1 ∞ X n which is also sufficient for the almost sure convergence of ∑ n = 1 ∞ ± X n for all (non-random) changes of sign. The result is based on an important bound of the mean of the random variable sup (‖ X 1 + ⋯ + X k ‖ : 1 ≤ k ≤ n) which extends the classical Lévy's inequality and has some independent interest. [ABSTRACT FROM AUTHOR]

We investigate the asymptotic normality of distributions of the sequence ∑ k ∈ ℤ u n , k ⁢ X k {\sum_{k\in\mathbb{Z}}u_{n,k}X_{k}} , n ∈ ℕ {n\in\mathbb{N}} , where ( X k , k ∈ ℤ) {(X_{k},k\in\mathbb{Z})} either is a sequence of i.i.d. random elements or constitutes a linear process with i.i.d. innovations in a separable Hilbert space. The weights (u n , k) {(u_{n,k})} are in general a family of linear bounded operators. This model includes operator weighted sums of Hilbert space valued linear processes, operator-wise discounted sums in a Hilbert space as well some extensions of classical summation methods. [ABSTRACT FROM AUTHOR]

The G-pencil law under the G-Lindeberg condition for a random matrix is proven. [ABSTRACT FROM AUTHOR]

For a double array { V m , n , m ≥ 1 , n ≥ 1 } of random elements taking values in a real separable Banach space, we provide sufficient conditions for the double series ∑ i = 1 m ∑ j = 1 n V i , j to converge almost certainly to a random element S as min { m , n } → ∞. For a convergent double series, we study the rate of convergence to a random element S by studying the rate at which the corresponding tail series { T m , n , m ≥ 1 , n ≥ 1 } converges almost certainly to 0 as min { m , n } → ∞ where T m , n = S − ∑ i = 1 m ∑ j = 1 n V i , j , m ≥ 1 , n ≥ 1. [ABSTRACT FROM AUTHOR]

We derive the existence of infinite level GREM-like K-processes by taking the limit of a sequence of finite level versions of such processes as the number of levels diverges. The main step in the derivation is obtaining the convergence of the sequence of underlying finite level clock processes. This is accomplished by perturbing these processes so as to turn them into martingales, and resorting to martingale convergence to obtain convergence for the perturbed clock processes; nontriviality of the limit requires a specific choice of parameters of the original process; we conclude the step by showing that the perturbation washes away in the limit. The perturbation is done by inserting suitable factors into the expression of the clocks, as well as rescaling the resulting expression suitably; the existence of such factors is itself established through martingale convergence. [ABSTRACT FROM AUTHOR]

In this correspondence, for an array of rowwise independent random elements { V n , k , 1 ≤ k ≤ k n , n ≥ 1 , k n → ∞ } taking values in a real separable Rademacher type p (1 ≤ p ≤ 2) Banach space and a sequence of positive constants { c n , n ≥ 1 } , the main result provides conditions for the complete convergence result ∑ n = 1 ∞ c n P ( max 1 ≤ k ≤ k n | | ∑ i = 1 k V n , i | | > ε) < ∞ for all ε > 0 to hold. The complete convergence does not necessary hold if the Rademacher type p hypothesis is dispensed with. Corollaries of the main result are obtained and illustrative examples are presented. [ABSTRACT FROM AUTHOR]

It is known in the literature that in the RNP Banach space the set valued uniformly integrable martingale is a regular martingale. In this paper by using a selector approach we provide a weaker condition than uniform integrability of a set valued Aumann–Pettis integrable martingale to be a set valued Aumann–Pettis integrable regular martingale. The Converse is also established. As an application of the aforementioned results, a new convergence result of set valued Pettis integrable martingales in Slice topology is provided. [ABSTRACT FROM AUTHOR]

This paper contains the proof of the positivity of the top Lyapunov exponent for the multiplicative stochastic integrals related to the diffusion processes on the Lee algebra of the matrices with zero trace. [ABSTRACT FROM AUTHOR]