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We prove Shao and Yu's tightness criterion for the generalized empirical process in the space

Let

Recently, Li et al. proposed in [

In this paper, however, we will focus on the empirical process built on an estimator

Yu [

In this paper, we shall study the empirical process

Explicitly, we take a look onto the process

It is well known that the crucial procedure in showing weak convergence for an empirical process is to verify tightness. In [

In order to obtain tightness, one has to assume appropriate covariance structure of the underlying r.v.’s, that is, the covariance of a pair of r.v.’s

The paper is organized as follows. In Section

We start with the key point of the paper.

Let

If there exist constants

The proof boils down to showing that under the assumptions made in Theorem

Let us first show that condition (

We shall now proceed to checking condition (

Let us recall that

Let us recall that

Since condition (

In this section, we deal with the standard empirical process built on an associated sequence of uniformly

While proving tightness of the empirical process, we will use the criterion proved in the first section as well as some of our covariance inequalities.

We shall start with the fact known under the name of multinomial theorem. We recall it in the following lemma.

For natural numbers

In particular,

If

We shall now focus on estimating

Let us now introduce the following notation:

Let

Yu assumed stationarity of

Louhichi, in [

In the final analysis, our result’s advantage is the absence of the stationarity assumption and the rate of decay for

Unfortunately, with a view to obtaining weak convergence of the process in question, that is also convergence of finite-dimensional distributions, we do not know how to manage without the assumption of stationarity. Therefore, we conclude with the following corollary.

Let

Then, if

If the r.v.’s

It remains to establish convergence of finite-dimensional distributions repeating the procedure from [

In this section we shall weaken assumption imposed on the covariance structure of r.v.’s

It turns out that it suffices to have

On the way to prove it, we will also take use of a Rosenthal-type inequality due to Shao and Yu (see Theorem 2 in [

Let

As we can see, the lemma assumes association, but it works for negatively associated r.v.’s as well, since in the proof, it reaches back the result of Newman (see Proposition 15 in [

Let us recall that

We now arrive at the following inequality:

In light of the above,

Let

The remaining covergence of finite-dimensional distributions of

The aim of this section is to show weak convergence of the empirical process:

We will prove that

It turns out that the tightness criterion (

Proceeding like Cai and Roussas in [

Firstly, we observe that each summand converges to

Secondly, applying Toeplitz lemma, we get

We summarize the result of that section in the following theorem.

Let

The author would like to thank the referees for the careful reading of the paper and helpful remarks.