Tightness Criterion and Weak Convergence for the Generalized Empirical Process in D [ 0 , 1 ]

We prove Shao and Yu’s tightness criterion for the generalized empirical process in the space D[0, 1] with J 1 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.


Introduction
Let {  } ≥1 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   () =  −1 ∑  =1 [  ≤ ].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 {ℎ  } ≥1 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][2][3][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, F () can be easily updated with each new observation since it is computable recursively by where F0 () = 0.The authors discussed the asymptotic bias and quadratic-mean convergence and established the pointwise asymptotic normality of F () 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:   () = √ [  () −   ()] , where  ∈ R (4) 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.
Explicitly, we take a look onto the process we shall call from now on the generalized empirical process.
Let us pay attention to the fact that in the case of (i) ℎ , = ℎ  for  ∈ {1, . . ., },   () is the empirical process based on the kernel estimator of the d.f.; is the empirical process based on the recursive kernel estimator of the d.f.; (iii) () =  [0,∞] (),   () is the standard empirical process (based on the empirical d.f.).
It is well known that the crucial procedure in showing weak convergence for an empirical process is to verify tightness.In [8], Shao and Yu gave the following criterion: under which the standard empirical process based on stationary sequence of uniform [0, 1] r.v.'s is tight.It is stated there that the proof of that fact is an easy standard procedure parallel to the one presented in [9].It is the main aim of this paper to carry it in details but for the generalized empirical process defined by (6) and without assuming stationarity.Nevertheless, we will always return to stationarity assumption while establishing weak convergence.
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   and   has to decline at the right rate while  and  are growing apart.In this paper we lower the demanded rate of covariance decay using the covariance inequalities for associated (c.f.[10,11]) and multivariate totally positive of order 2 (MTP 2 ) (c.f.[12]) r.v.'s obtained in [13].
The paper is organized as follows.In Section 2 we present the proof of the Shao and Yu's tightness criterion formulated for our generalized empirical process.Sections 3 and 4 are devoted to application of the criterion to showing tightness and thus weak convergence of the specific types of empirical processes.Section 5 concerns weak convergence of the recursive kernel-type process for i.i.d.r.v.'s.
Proof.The proof boils down to showing that under the assumptions made in Theorem 1, conditions of Theorem 13.2 in [9] hold.Let us recall that in light of the above mentioned theorem, a process lim where ‖ ⋅ ‖ is the supremum norm, that is,           = sup 0≤≤1       ()     (11) and   1 (  , ) is the modulus of continuity of the function The infimum runs over all finite "-sparse" decompositions In other words, it runs over all choices of increasingly ordered points {  } 1≤≤] such that min 1≤≤] (  −  −1 ) > , where  0 = 0,  ] = 1.
Step 1.We need a moment inequality for the r.v.  () −   () involving the distance between the points  and .Let us then assume that for the constants , , Let us notice that for r.v.'s {  } ≥1 the conditions of Theorem 10.2 in [9] are satisfied with 4 = ,   =   ∀ <≤ and 2 = min{ 1 ,  2 +  3 }.Therefore, we are equipped with the following maximal inequality: for all  > 0.
Step 3. Let  = sup ∈[0,1] (), where  is the p.d.f. of r.v.'s {  } ≥1 .For fixed ,  > 0 let us take  > 0 such that and define   := ⌊/  ⌋.For sufficiently large  ∈ N we have   ≥ 1 and then we get Step 4. Our goal is to obtain an inequality which enables us to bound the supremum of the increment of the function   via the maximum of the increments of that function on some subintervals.To be more precise, we will find the upper bound for sup where Cai and Roussas in [1] showed that under assumptions made on the d.f. and the kernel function , we have Similarly, by Taylor expansion, it is easy to see that Thus, where  0 ∈ [+ 0 ,  + ( 0 + 1)] is a point at which the above supremum is attained and Step 5. Finally, we are in a position to obtain inequality (23), that is, We now successively make use the inequalities (29), ( 41), ( 27 where 4 + √ℎ 4  is arbitrarily small.Since condition (10) is checked, the proof is completed.

Tightness of the Standard Empirical Process
In this section, we deal with the standard empirical process built on an associated sequence of uniformly where  ∈ [0, 1].We shall relax the restrictions imposed on the process by Yu in [7] to obtain tightness.Precisely, we do not need stationarity any more due to the technique drawn from [14], and we lower the assumed rate at which the covariance tends to zero.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.Lemma 2. For natural numbers ,  and  1 ,  2 , . . .,   ∈ R where In particular, which implies for   := ∑  =1   , that Let us now introduce the following notation due to Doukhan and Louhichi (see [14]) for a sequence of centered r.v.'s where the supremum runs over all divisions of the group composed of  r.v.'s into two subgroups, such that the distance between the highest index of the r.v.'s in the first group and the lowest index of the r.v.'s from the second group is equal to ,  ∈ {1, 2, . ..}.For  = 0, we shall define  0, = 1.Let us then put We will now estimate the summands in (48  ,4 ) . (54)

ISRN Probability and Statistics
The terms obtained in (54) may further be bounded from above in the following way: We thus get the inequality We shall now focus on estimating  ,2 and  ,4 .Since   1 and   2 are associated uniformly [0, 1] distributed r.v.'s,   1 and   2 -as monotone functions of these r.v.'s-are associated as well.In order to bound let us notice that from Schwarz inequality On the other hand, invoking inequalities from [13,15], where for the sake of simplicity,  1 ,  2 , and  3 are, respectively, the free coefficient, the expression with , and the expression with  2 .Using the Lebowitz inequality (see [15] for instance) and inequalities obtained in [13,15] Eventually, we have where Let us now introduce the following notation: and assume it decays powerly at rate  in the following way: Let us get back to inequality (56), we can now carry on.At first, for associated r.v.'s, where  and  1 are constants and () , >9  ( ln ) ,  = 9  ( 4−/3 ) , 3 <  < 9.
(69) It is worth mentioning, that in the last inequality of (68), we used the estimate (70) At the same time, in the case of MTP 2 r.v.'s, we get where  2 is constant and has the fourth moment estimated-in the case of associated r.v.'s-by and in the case of MTP 2 r.v.'s by In light of the Shao and Yu's criterion, our process is tight for associated r.v.'s when  > 6 and for MTP 2 r.v.'s when  > 2.
Let us sum up this result in the following theorem.
Louhichi, in [16], proposed a different tightness criterion involving the so-called bracketing numbers.She managed to enhance Yu's result-even more than Shao and Yu in [8]-since she proved that it suffices to take  > 4 to get tightness of the empirical process based on the associated r.v.'s.Nevertheless, she kept the assumption of stationarity valid.
In the final analysis, our result's advantage is the absence of the stationarity assumption and the rate of decay for   remains (up to the author's knowledge) unimproved for MTP 2 r.v.'s.
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.
Proof.It remains to establish convergence of finite-dimensional distributions repeating the procedure from [7].

Tightness of the Kernel-Type Empirical Process
In this section we shall weaken assumption imposed on the covariance structure of r.v.'s {  } ≥1 by Cai and Roussas in [1] for the kernel estimator of the d.f.
They deal with a stationary sequence of negatively associated r.v.'s (c.f.[17]) and need the same condition as Yu [7], that is, to get tightness of the smooth empirical process (see condition (A4) in [1]).
It turns out that it suffices to have where  > 4 is a positive constant taken from the tightness criterion (8).It is easy to see that asymptotically we get the rate 3.
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 [8]) we shall recall in the following lemma.Lemma 5. Let  > 2 and  be a real valued function bounded by 1 with bounded first derivative.Suppose that {  } ≥1 is a sequence of stationary and associated r.v.'s, such that for  ∈ N Cov ( 1 ,   ) =  ( − ) , for some  >  − 1. (83) Then, for any  > 0 there exists some positive constant   independent of the function , for which ). (84) 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 [18]), where both types of association are allowed. Let where  2 is a constant relevant to the covariance inequality for negatively associated r.v.'s (see [13]).
in order to obtain where (⋅) is the zero mean Gaussian process with covariance structure defined by and ,  ∈ [0, 1].
Proof.The remaining covergence of finite-dimensional distributions of   () is established in [1].

Weak Convergence of the Recursive Kernel-Type Empirical Process under I.I.D. Assumption
The aim of this section is to show weak convergence of the empirical process: where the second equality follows from assumed independence of r.