Tightness Criterion and Weak Convergence for the Generalized Empirical Process in

We prove Shao and Yu's tightness criterion for the generalized empirical process in the space with topology. Covariance inequalities are used in applying the criterion to particular types of the empirical processes. We weaken the assumptions imposed on the covariance structure as well as the properties of the underlying sequence of r.v.'s, under which presented processes converge weakly. 1. Introduction Let be a sequence of absolutely continuous identically distributed (i.d.) random variables (r.v.’s) with an unknown distribution function (d.f.) and probability density function (p.d.f.) . The empirical distribution function, based on the first r.v.’s, is defined by . It is well known, however, that this estimate does not make use of the smoothness of , that is, the existence of the p.d.f. . Therefore, the kernel estimate has been proposed, where the kernel function is a known d.f. and is a sequence of positive constants descending at an appropriate rate. Such estimator has been deeply studied in the last two decades mainly by Cai and Roussas in [1–4], Li and Yang in [5] and others. Asymptotic normality, Berry-Essen bounds for smooth estimator are only examples of their fruitful results. Recently, Li et al. proposed in [6] the so-called recursive kernel estimator of the d.f. as follows: The seemingly tiny modification they introduced to the formula of the typical kernel estimator has an important advantage. Namely, in the case of a large size of a sample, can be easily updated with each new observation since it is computable recursively by where . The authors discussed the asymptotic bias and quadratic-mean convergence and established the pointwise asymptotic normality of under relevant assumptions. In this paper, however, we will focus on the empirical process built on an estimator of the d.f. rather than itself. Let us recall that the following process: is called the empirical process built on an estimator . Yu [7] studied the case when is a standard empirical d.f. and showed weak convergence of to the Gaussian process assuming stationarity and association of the underlying r.v.’s. Cai and Roussas [1] obtained a similar result in the case when is the kernel estimator of the d.f. built on a stationary sequence of negatively associated r.v.’s. In this paper, we shall study the empirical process generated by the generalized kernel estimator of the d.f. given by the formula A1: is a sequence of absolutely continuous i.d. r.v.’s taking values in and having twice differentiable d.f. with first and second derivative bounded;A2: is a kernel function such that