Xinghui Wang

The consistency for estimator of nonparametric regression model based on NOD errors

By using some inequalities for NOD random variables, we give its application to investigate the nonparametric regression model based on these errors. Some consistency results for the estimator of g(x) are presented, including the mean convergence, uniform convergence, almost sure convergence and convergence rate. We generalize some related results and as an example of designed assumptions for weight functions, we give the nearest neighbor weights.AMS Mathematical Subject Classification 2000: 62G05; 62G08.

On Complete Convergence for an Extended Negatively Dependent Sequence

In this article, the Rosenthal-type maximal inequality for extended negatively dependent (END) sequence is provided. By using the Rosenthal type inequality, we present some results of complete convergence for weighted sums of END random variables under mild conditions.

On Complete Convergence of Weighted Sums for Arrays of Rowwise Asymptotically Almost Negatively Associated Random Variables

Let be an array of rowwise asymptotically almost negatively associated (AANA, in short) random variables. The complete convergence for weighted sums of arrays of rowwise AANA random variables is studied, which complements and improves the corresponding result of Baek et al. (2008). As applications, the Baum and Katz type result for arrays of rowwise AANA random variables and the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of AANA random variables are obtained.

Convergence Rates in the Strong Law of Large Numbers for Martingale Difference Sequences

We study the complete convergence and complete moment convergence for martingale difference sequence. Especially, we get the Baum-Katz-type Theorem and Hsu-Robbins-type Theorem for martingale difference sequence. As a result, the Marcinkiewicz-Zygmund strong law of large numbers for martingale difference sequence is obtained. Our results generalize the corresponding ones of Stoica (2007, 2011).

On the Rate of Complete Convergence for Weighted Sums of Arrays of Rowwise ϕ-Mixing Random Variables

Let be an array of rowwise ϕ-mixing random variables. A rate of complete convergence for weighted sums of arrays of rowwise ϕ-mixing random variables is obtained without assumption of identical distribution. The techniques used in the paper are the Rosenthal type inequality and the truncated method. As an application, the Baum and Katz type result for arrays of rowwise ϕ-mixing random variables is obtained.

A Note on the Inverse Moment for the Non Negative Random Variables

Let {Zn} be a sequence of non negative random variables satisfying a Rosenthal-type inequality and , where {Mn} is a sequence of positive real numbers. By using the Rosenthal-type inequality, the inverse moment E(a + Xn)− α can be asymptotically approximated by (a + EXn)− α for all a > 0 and α > 0. Furthermore, we show that E[f(Xn)]− 1 can be asymptotically approximated by [f(EXn)]− 1 for a function f( · ) satisfying certain conditions. Our results generalize and improve some corresponding results, which can allow immediate applications to compute the inverse moments for the non negative random variables whose distributions are such as Binomial distribution, Poisson distribution, Gamma distribution, etc.

Complete moment convergence for randomly weighted sums of martingale differences

In this article, we obtain the complete moment convergence for randomly weighted sums of martingale differences. Our results generalize the corresponding ones for the nonweighted sums of martingale differences to the case of randomly weighted sums of martingale differences.MSC:60G50, 60F15.

Complete convergence for weighted sums of arrays of rowwise ˜ ρ-mixing random variables

Let {Xni,i≥1,n≥1} be an array of rowwise ρ˜-mixing random variables. Some sufficient conditions for complete convergence for weighted sums of arrays of rowwise ρ˜-mixing random variables are presented without assumptions of identical distribution. As applications, the Baum and Katz type result and the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of ρ˜-mixing random variables are obtained.MSC:60F15.

Convergence properties for weighted sums of NSD random variables

Abstract Let {Xn, n ⩾ 1} be a sequence of negatively superadditive dependent (NSD, in short) random variables and {bni, 1 ⩽ i ⩽ n, n ⩾ 1} be an array of real numbers. In this article, we study the strong law of large numbers for the weighted sums ∑ni = 1bniXi without identical distribution. We present some sufficient conditions to prove the strong law of large numbers. As an application, the Marcinkiewicz-Zygmund strong law of large numbers for NSD random variables is obtained. In addition, the complete convergence for the weighted sums of NSD random variables is established. Our results generalize and improve some corresponding ones for independent random variables and negatively associated random variables.

The Convergence of Double-Indexed Weighted Sums of Martingale Differences and Its Application

We investigate the complete moment convergence of double-indexed weighted sums of martingale differences. Then it is easy to obtain the Marcinkiewicz-Zygmund-type strong law of large numbers of double-indexed weighted sums of martingale differences. Moreover, the convergence of double-indexed weighted sums of martingale differences is presented in mean square. On the other hand, we give the application to study the convergence of the state observers of linear-time-invariant systems and present the convergence with probability one and in mean square.