In this paper, we establish the complete convergence and complete moment convergence of weighted sums for arrays of rowwise $\varphi$-mixing random variables, and the Baum-Katz-type result for arrays of rowwise $\varphi$-mixing random variables. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of $\varphi$-mixing random variables is obtained. We extend and complement the corresponding results of X. J. Wang, S. H. Hu (2012).

Let $\{X_{n,j},1\le j\le m\left(n\right),n\ge 1\}$ be an array of rowwise pairwise negative quadrant dependent mean 0 random variables and let $0<b_n\to \infty$. Conditions are given for ${\sum }_{j=1}^{m\left(n\right)}{X}_{n,j}/b_n\to 0$ completely and for ${max}_{1\le k\le m\left(n\right)}\left|{\sum }_{j=1}^k{X}_{n,j}\right|/b_n\to 0$ completely. As an application of these results, we obtain a complete convergence theorem for the row sums ${\sum }_{j=1}^{m\left(n\right)}{X}_{n,j}^{*}$ of the dependent bootstrap samples $\{\{X_{n,j}^{*},1\le j\le m\left(n\right)\},n\ge 1\}$ arising from a sequence of i.i.d. random variables $\{X_n,n\ge 1\}$.

Complete $f$-moment convergence is much more general than complete convergence and complete moment convergence. In this work, we mainly investigate the complete $f$-moment convergence for weighted sums of widely orthant dependent (WOD, for short) arrays. A general result on Complete $f$-moment convergence is obtained under some suitable conditions, which generalizes the corresponding one in the literature. As an application, we establish the complete consistency for the weighted linear estimator in nonparametric...

Let $\{Y_i,-\infty <i<\infty \}$ be a doubly infinite sequence of identically distributed $\varphi$-mixing random variables, and $\{a_i,-\infty <i<\infty \}$ an absolutely summable sequence of real numbers. We prove the complete $q$-order moment convergence for the partial sums of moving average processes $\left\{X_n={\sum }_{i=-\infty }^{\infty }{a}_i{Y}_{i+n},n\ge 1\right\}$ based on the sequence $\{Y_i,-\infty <i<\infty \}$ of $\varphi$-mixing random variables under some suitable conditions. These results generalize and complement earlier results.

We formulate two results on controllability properties of the 3D Navier–Stokes (NS) system. They concern the approximate controllability and exact controllability in finite-dimensional projections of the problem in question. As a consequence, we obtain the existence of a strong solution of the Cauchy problem for the 3D NS system with an arbitrary initial function and a large class of right-hand sides. We also discuss some qualitative properties of admissible weak solutions for randomly forced NS...

Given a probability space (Ω,, P) and a closed subset X of a Banach lattice, we consider functions f: X × Ω → X and their iterates $fⁿ:X\times {\Omega }^{\mathbb{N}}\to X$ defined by f¹(x,ω) = f(x,ω₁), $f^{n+1}(x,\omega )=f(fⁿ(x,\omega ),{\omega }_{n+1})$, and obtain theorems on the convergence (a.s. and in L¹) of the sequence (fⁿ(x,·)).

In this research, strong convergence properties of the tail probability for weighted sums of negatively orthant dependent random variables are discussed. Some sharp theorems for weighted sums of arrays of rowwise negatively orthant dependent random variables are established. These results not only extend the corresponding ones of Cai [4], Wang et al. [19] and Shen [13], but also improve them, respectively.

In this paper the authors study the convergence properties for arrays of rowwise pairwise negatively quadrant dependent random variables. The results extend and improve the corresponding theorems of T. C. Hu, R. L. Taylor: On the strong law for arrays and for the bootstrap mean and variance, Int. J. Math. Math. Sci 20 (1997), 375–382.