Exponential Inequality for a Class of NOD Random Variables and Its Application

In this paper,an exponential inequality for weighted sums of identically distributed NOD （negatively orthant dependent） random variables is established,by which we obtain the almost sure convergence rate of which reaches the available one for independent random variables in terms of Berstein type inequality. As application,we obtain the relevant exponential inequality for Priestley-Chao estimator of nonparametric regression estimate under NOD samples,from which the strong consistency rate is also obtained.

AN EXPONENTIAL INEQUALITY FOR AUTOREGRESSIVE PROCESSES IN ADAPTIVE TRACKING

A wide range of literature concerning classical asymptotic properties for linear models with adaptive control is available, such as strong laws of large numbers or central limit theorems. Unfortunately, in contrast with the situation without control, it appears to be impossible to find sharp asymptotic or nonasymptotic properties such as large deviation principles or exponential inequalities. Our purpose is to provide a first step towards that direction by proving a very simple exponential inequality for the standard least squares estimator of the unknown parameter of Gaussian autoregressive process in adaptive tracking.

Exponential Inequalities and Complete Convergence for Extended Negatively Dependent Random Variables

Some exponential inequalities and complete convergence are established for extended negatively dependent(END) random variables. The inequalities extend and improve the results of Kim and Kim(On the exponential inequality for negative dependent sequence.Communications of the Korean Mathematical Society, 2007, 22(2): 315-321) and Nooghabi and Azarnoosh(Exponential inequality for negatively associated random variables. Statistical Papers, 2009, 50(2): 419-428). We also obtain the convergence rate O(n-1/2ln1/2n) for the strong law of large numbers, which improves the corresponding ones of Kim and Kim,and Nooghabi and Azarnoosh.

STRONG LIMIT THEOREMS FOR EXTENDED INDEPENDENT RANDOM VARIABLES AND EXTENDED NEGATIVELY DEPENDENT RANDOM VARIABLES UNDER SUB-LINEAR EXPECTATIONS

Limit theorems for non-additive probabilities or non-linear expectations are challenging issues which have attracted a lot of interest recently.The purpose of this paper is to study the strong law of large numbers and the law of the iterated logarithm for a sequence of random variables in a sub-linear expectation space under a concept of extended independence which is much weaker and easier to verify than the independence proposed by Peng[20].We introduce a concept of extended negative dependence which is an extension of the kind of weak independence and the extended negative independence relative to classical probability that has appeared in the recent literature.Powerful tools such as moment inequality and Kolmogorov’s exponential inequality are established for these kinds of extended negatively independent random variables,and these tools improve a lot upon those of Chen,Chen and Ng[1].The strong law of large numbers and the law of iterated logarithm are also obtained by applying these inequalities.

Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications

In this paper, an exponential inequality for the maximal partial sums of negatively superadditive-dependent （NSD, in short） random variables is established. By uSing the exponen- tial inequality, we present some general results on the complete convergence for arrays of rowwise NSD random variables, which improve or generalize the corresponding ones of Wang et al. [28] and Chen et al. [2]. In addition, some sufficient conditions to prove the complete convergence are provided. As an application of the complete convergence that we established, we further investigate the complete consistency and convergence rate of the estimator in a nonparametric regression model based on NSD errors.

PROBABILITY INEQUALITIES FOR SUMS OF INDEPENDENT UNBOUNDED RANDOM VARIABLES

The tail probability inequalities for the sum of independent unbounded random variables on a probability space (Omega, T, P) were studied and a new method was proposed to treat the sum of independent unbounded random variables by truncating the original probability space (Omega, T, P). The probability exponential inequalities for sums of the results, some independent unbounded random variables were given. As applications of interesting examples were given. The examples show that the method proposed in the paper and the results of the paper are quite useful in the study of the large sample properties of the sums of independent unbounded random variables.

Global stability of interval recurrent neural networks

The robust global exponential stability of a class of interval recurrent neural networks（RNNs） is studied,and a new robust stability criterion is obtained in the form of linear matrix inequality.The problem of robust stability of interval RNNs is transformed into a problem of solving a class of linear matrix inequalities.Thus,the robust stability of interval RNNs can be analyzed by directly using the linear matrix inequalities（LMI） toolbox of MATLAB.Numerical example is given to show the effectiveness of the obtained results.

Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm

Kolmogorov's exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper establishes the Kolmogorov type exponential inequalities of the partial sums of independent random variables as well as negatively dependent random variables under the sub-linear expectations. As applications of the exponential inequalities, the laws of the iterated logarithm in the sense of non-additive capacities are proved for independent or negatively dependent identically distributed random variables with finite second order moments.For deriving a lower bound of an exponential inequality, a central limit theorem is also proved under the sublinear expectation for random variables with only finite variances.

Exponential inequalities for associated random variables and strong laws of large numbers

Some exponential inequalities for partial sums of associated random variables are established. These inequalities improve the corresponding results obtained by Ioannides and Roussas (1999), and Oliveira (2005). As application, some strong laws of large numbers are given. For the case of geometrically decreasing covariances, we obtain the rate of convergence n-1/2(log log n)1/2(logn) which is close to the optimal achievable convergence rate for independent random variables under an iterated logarithm, while Ioannides and Roussas (1999), and Oliveira (2005) only got n-1/3(logn)2/3 and n-1/3(logn)5/3, separately.

Some Exponential Inequalities for Negatively Ort han t Dependent Random Variables

In the paper,we establish some exponential inequalities for non-identically distributed negatively orthant dependent(NOD,for short)random variables.In addition,we also establish some exponential inequalities for the partial sum and the maximal partial sum of identically distributed NOD random variables.As an application,the Kolmogorov strong law of large numbers for identically distributed NOD random variables is obtained.Our results partially generalize or improve some known results.

On the laws of the iterated logarithm under sub-linear expectations

In this paper,we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space,where the random variables are not necessarily identically distributed.Exponential inequalities for the maximum sum of independent random variables and Kolmogorov’s converse exponential inequalities are established as tools for showing the law of the iterated logarithm.As an application,the sufficient and necessary conditions of the law of the iterated logarithm for independent and identically distributed random variables under the sub-linear expectation are obtained.In the paper,it is also shown that if the sub-linear expectation space is rich enough,it will have no continuous capacity.The laws of the iterated logarithm are established without the assumption on the continuity of capacities.

The Law of Logarithm for Arrays of Random Variables under Sub-linear Expectations

Under the framework of sub-linear expectation initiated by Peng,motivated by the concept of extended negative dependence,we establish a law of logarithm for arrays of row-wise extended negatively dependent random variables under weak conditions.Besides,the law of logarithm for independent and identically distributed arrays is derived more precisely and the sufficient and necessary conditions for the law of logarithm are obtained.

Output Feedback Stabilization of an Unstable Wave Equation with Observations Subject to Time Delay

This paper focuses on boundary stabilization of a one-dimensional wave equation with an unstable boundary condition,in which observations are subject to arbitrary fixed time delay.The observability inequality indicates that the open-loop system is observable,based on which the observer and predictor are designed:The state of system is estimated with available observation and then predicted without observation.After that equivalently the authors transform the original system to the well-posed and exponentially stable system by backstepping method.The equivalent system together with the design of observer and predictor give the estimated output feedback.It is shown that the closed-loop system is exponentially stable.Numerical simulations are presented to illustrate the effect of the stabilizing controller.