The law of iterated logarithm for a class of random variables satisfying Rosenthal type inequality

<abstract><p>Let $ \{Y_n, n\geq 1\} $ be sequence of random variables with $ EY_n = 0 $ and $ \sup_nE|Y_n|^p &lt; \infty $ for each $ p &gt; 2 $ satisfying Rosenthal type inequality. In this paper, the law of the iterated logarithm for a class of random variable sequence with non-identical distributions is established by the Rosenthal type inequality and Berry-Esseen bounds. The results extend the known ones from i.i.d and NA cases to a class of random variable satisfying Rosenthal type inequality.</p></abstract>

Functional Central Limit Theorem for Empirical Processes

Here we discuss the Gaussian approximation for the empirical process under different kinds of dependence assumptions for the underlying stationary sequence. First, we state a general criterion to prove tightness of the empirical process associated with a stationary sequence of uniformly distributed random variables. This tightness criterion can be verified for many different dependence structures. For ρ‎-mixing sequences, by an application of a Rosenthal-type inequality, tightness is verified under the same condition leading to the usual CLT. For α‎-dependent sequences whose α‎-dependent coefficients decay polynomially to zero, it is shown to hold with the help of the Rosenthal inequality stated in Section 3.3. Since the asymptotic behavior of the finite-dimensional distributions of the empirical process is handled via the CLT developed in previous chapters, we then derive the functional CLT for the empirical process associated with the above-mentioned classes of stationary sequences. β‎-dependent sequences are also investigated by directly proving tightness of the empirical process.

Moment Inequalities via Martingale Methods

In this chapter we establish different kinds of moment inequalities for partial sums and the maximum of partial sums of a large class of random variables, including martingale sequences, mixingales, and other dependent structures. All the bounds involve the moments of the conditional expectations of either the partial sums or the individual random variables. In most of the proofs martingale approximations are used. This method allows us to use the moment inequalities for the martingale part developed in Chapter 2. We start with a dyadic scheme useful for analysis of the variance of partial sums in the stationary setting. Then, we obtain Burkholder-type inequalities via Maxwell–Woodroofe-type characteristics and an extension of Doob’s maximal inequality for adapted sequences. A Rosenthal-type inequality for stationary sequences is also provided with bounds using conditional expectations of the partial sums. Maximal exponential inequalities are established involving either Maxwell–Woodroofe-type characteristics or the projective operators.

Complete convergence for weighted sums of a class of random variables

Let {ani,1?i?n,n?1} be an array of real numbers and {Xn,n?1} be a sequence of random variables satisfying the Rosenthal type inequality, which is stochastically dominated by a random variable X. Under mild conditions, we present some results on complete convergence for weighted sums ?ni=1 aniXi of random variables satisfying the Rosenthal type inequality. The results obtained in the paper generalize some known ones in the literatures.

The Strong Consistency of the Estimator of Fixed-Design Regression Model under Negatively Dependent Sequences

We study the strong consistency of estimator of fixed design regression model under negatively dependent sequences by using the classical Rosenthal-type inequality and the truncated method. As an application, the strong consistency for the nearest neighbor estimator is obtained.

Complete Consistency of the Estimator of Nonparametric Regression Models Based onρ~-Mixing Sequences

We study the complete consistency for estimator of nonparametric regression model based onρ~-mixing sequences by using the classical Rosenthal-type inequality and the truncated method. As an application, the complete consistency for the nearest neighbor estimator is obtained.