Let $(X_n,n\ge 1),({\tilde{X}}_n,n\ge 1)$ be two sequences of i.i.d. random vectors with values in ${ℝ}^k$ and $S_n=X_1+\cdots +X_n$, ${\tilde{S}}_n={\tilde{X}}_1+\cdots +{\tilde{X}}_n$, $n\ge 1$. Assuming that $E{X}_1=E{\tilde{X}}_1$, $E|X_1{|}^2<\infty$, $E|{\tilde{X}}_1{|}^{k+2}<\infty$ and the existence of a density of ${\tilde{X}}_1$ satisfying the certain conditions we prove the following inequalities: $$v(S_n,{\tilde{S}}_n)\le cmax\{v(X_1,{\tilde{X}}_1),{\zeta }_2(X_1,{\tilde{X}}_1)\},n=1,2,\cdots ,$$ where $v$ and ${\zeta }_2$ are the total variation and Zolotarev’s metrics, respectively.

The aim of this paper is to compare various criteria leading to the central limit theorem and the weak invariance principle. These criteria are the martingale-coboundary decomposition developed by Gordin in Dokl. Akad. Nauk SSSR188 (1969), the projective criterion introduced by Dedecker in Probab. Theory Related Fields110 (1998), which was subsequently improved by Dedecker and Rio in Ann. Inst. H. Poincaré Probab. Statist.36 (2000) and the condition introduced by Maxwell and Woodroofe in Ann. Probab.28...

Let $\left\{X_{ij}\right\}$, $i,j=\cdots$, be a double array of independent and identically distributed (i.i.d.) real random variables with $E{X}_{11}=\mu$, $E|X_{11}{-\mu |}^2=1$ and $E|X_{11}{|}^{4}lt;\infty$. Consider sample covariance matrices (with/without empirical centering) $𝒮=\frac{1}{n}{\sum }_{j=1}^n({𝐬}_j-\overline{𝐬}){({𝐬}_j-\overline{𝐬})}^T$ and $𝐒=\frac{1}{n}{\sum }_{j=1}^n{𝐬}_j{𝐬}_j^T$, where $\overline{𝐬}=\frac{1}{n}{\sum }_{j=1}^n{𝐬}_j$ and ${𝐬}_j={𝐓}_n^{1/2}{(X_{1j},...,X_{pj})}^T$ with ${\left({𝐓}_n^{1/2}\right)}^2={𝐓}_n$, non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of $𝒮$ and $𝐒$ are different as $n\to \infty$ with $p/n$ approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the average behavior...

In the present paper, we have established the complete convergence for weighted sums of pairwise independent random variables, from which the rate of convergence of moving average processes is deduced.

The rate of moment convergence of sample sums was investigated by Chow (1988) (in case of real-valued random variables). In 2006, Rosalsky et al. introduced and investigated this concept for case random variable with Banach-valued (called complete convergence in mean of order $p$). In this paper, we give some new results of complete convergence in mean of order $p$ and its applications to strong laws of large numbers for double arrays of random variables taking values in Banach spaces.

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.