# Precise Asymptotics for the Uniform Empirical Process and the Uniform Sample Quantile Process

Authors: Youyou Chen; Zaichen Qian
Journal: Scientific Programming
 (2022)

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## Abstract

One of the sources of “invariance principle” is that the limit properties of the uniform empirical process coincide with that of a Brownian bridge. The deep discussion of limit theorem of the uniform empirical process gathered wild interest of the researchers. In this paper, the precise convergence rate of the uniform empirical process is considered. As is well-known, whenε tends to 0, the precise asymptotic theorems can be demonstrated by referring to the classical method of Gut and Spǎtaru, by using some nice probability inequalities and so on. However, if ε tends to a positive constant, other powerful methods and tools are needed. The method of strong approximation is used in this paper. The main theorems are proved by using the Brownian bridge Bt to approximate the uniform empirical process αnt. The relevant results for the uniform sample quantile process are also presented.


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## Body

## 1. Introduction and Main Results

Random phenomena exist in almost every branch of science and engineering and permeate every aspect of ordinary people’s modern life [1, 2]. Probability theory is a subject that studies the quantitative regularity of random phenomena everywhere. Probability is a method of thinking about the world [3].Probability limit theory is one of the main branches of probability theory [4, 5]. The famous probability scientists Kolmogorov and Gnedenko once said, “the epistemological value of probability theory can be revealed only through the limit theorem. Without the limit theorem, it is impossible to understand the real meaning of the basic concepts of probability theory.” Probability limit theory is also an important basis of statistical large sample theory [4]. People are very concerned about whether the estimator approximates the real parameter when the sample size tends to infinity, that is, the so-called consistency in statistical large sample theory. Furthermore, we need to consider the speed at which the estimator approximates the real parameters and how to solve these statistical large sample problems. The solution of these problems must rely on the probability limit theorem.LetX,Xn;n≥1 be a sequence of independent and identically distributed (i.i.d.) random variables with the common distribution function F, and set Sn=∑i=1nXi for n≥1. Hsu and Robbins [6] introduced the following complete convergence.(1)∑n=1∞PSn≥εn<∞,ε>0,This holds ifΕX=0, and ΕX2<∞. The converse part was proved by Erdös [7]. The complete convergence is stronger than the almost sure convergence. Obviously, the sum in (1) tends to infinity as ε ↘ 0.The first result on the convergence rate of this kind was given by Heyde [8]. It is proved that(2)limε↘0ε2∑n=1∞PSn≥εn=EX2,if ΕX=0, and ΕX2<∞. Heyde [8], Alam [4] got general conclusions and termed them “precise asymptotics.”The precise asymptotics for “Sn” have been extensively studied. One can refer to Zhang [9], Huang [10], and so on. Now, we consider the relevant results for the uniform empirical process. Let U1,U2,⋯,Un be a sequence of i.i.d. U0,1− distributed random variables. Define the uniform empirical process as αnt=n−1/2∑i=1nIUi≤t−t, 0≤t≤1. Denote the norm of a function ft on [0, 1] by f=sup0≤t≤1ft, and logx=lnx∨e. The following is one conclusion provided by Zhang and Yang [11].Theorem 1.
LetBt;0≤t≤1 be a Brownian bridge, and for anyδ>−1, we havelimε↘0ε2δ+2∑n=1∞lognδ/nPαn≥εlogn=EB2δ+2/δ+1.
The proof of Theorem1 is based on the classical method introduced by Gut and Spǎtaru [12]. In this paper, we consider the situation “ε↘c0” where c0 is a positive constant, and the classical argument for the case of “ε ↘ 0” does not work anymore. We will use more powerful tools, such as strong approximation. Besides the uniform empirical process, we also consider the uniform sample quantile process. Let 0=U0n≤U1n≤⋯≤Unn≤Un+1n=1 denote the order statistics of the random sample U1,U2,…,Un , for each n≥1. Define the uniform quantile function as Uny=Uknifk−1/n<y≤k/n,k=1,2,⋯,n0ify=0. The uniform sample quantile process should be defined as uny=n1/2Uny−y, 0≤y≤1. The following are our main results.Theorem 2.
Leta>−1, b>−1, andbnεbe a function ofεsuch thatbnεlogn⟶τ as n⟶∞,ε↘a+1/2. Then,(3)limε↘a+1/24ε2−a+1b+1∑n=1∞nalognbPαn≥2lognε+bnε=2exp−4τa+1Γb+1,and(4)limε↘a+1/24ε2−a+1b+1∑n=1∞nalognbPun≥2lognε+bnε=2exp−4τa+1Γb+1,Theorem 3.
Leta>−1,b>−1, anddnεbe a function ofεsuch thatdnεloglogn⟶τ as n⟶∞,ε↘a+1/2. Then,(5)limε↘a+1/24ε2−a+1b+1∑n=1∞lognaloglognbnPαn≥2loglognε+dnε=2exp−4τa+1Γb+1,and(6)limε↘a+1/24ε2−a+1b+1∑n=1∞lognaloglognbnPun≥2loglognε+dnε=2exp−4τa+1Γb+1.Remark 1.
We define the general empirical process asβnx=nFnx−Fx, −∞<x<∞, where Fnx=1/n∑i=1nI−∞,xXi. If F⋅ is a continuous distribution function since αnFx=βnx , the results for βnx can be obtained immediately from the uniform case. But we cannot handle the quantile process in the same way.


## 2. Proofs

The starting point of this paper is the empirical distribution function. The empirical distribution function plays a very important role in statistics [13–18]. Although it is not a beautiful piecewise function, as a nonparametric estimation of the distribution function, it is unbiased, consistent, and asymptotically obeys the normal distribution. The empirical process is constructed on the basis of the empirical distribution function. The uniform empirical process is a special and important one [19–21].We lay out some lemmas which will be used in the proofs later. Lemma1 is well known (cf. [22]). Lemma 2 and 3 are from Csörgó and Révész [23, 24].Lemma 1.
LetBt;0≤t≤1 be a Brownian bridge. Then, for allx>0,(7)PBt≥x=2∑k=1∞−1k+1e−2k2x2.In particular,(8)PBt≥x∼2e−2x2asx⟶+∞.Lemma 2.
There exists a sequence of Brownian bridgesBnt;0≤t≤1such that for all n and x we have(9)Psup0≤t≤1αnt−Bnt>n−1/2Klogn+x≤Le−λx,where K, L, λ are positive absolute constants.Lemma 3.
There exists a sequence of Brownian bridgesBnt;0≤t≤1such that for eachn=1,2,⋯, and for allz<cnandc>0, we have(10)Psup0≤t≤1unt−Bnt>n−1/2Alogn+z≤Be−Cz,where A, B, C, c are positive absolute constants.
First, we obtain the conclusion for the Brownian bridgeBt;0≤t≤1.Proposition 1.
Leta>−1, b>−1, andbnεbe a function ofεsuch that(11)bnεlogn⟶τasε↘a+12.
Then,limε↘a+1/24ε2−a+1b+1∑n=1∞nalognbPB≥2lognε+bnε=2exp−4τa+1Γb+1.Proof.
By Lemma1 and (11), we have PB≥2lognε+bnε∼2exp−4lognε+bnε2∼2exp−4ε2lognexp−8εbnεlogn as n⟶∞, uniformly in ε∈a+1/2,a+1/2+δ for some δ>0. Therefore, for any 0<θ<1, there exist δ>0 and n0 such that for all n≥n0 and ε∈a+1/2,a+1/2+δ,(12)2exp−4ε2lognexp−4a+1τ−θ≤PB≥2lognε+bnε≤2exp−4ε2lognexp−4a+1τ+θ.
We calculate thatlimε↘a+1/24ε2−a+1b+1∑n=1∞nalognb⋅exp−4ε2logn=limε↘a+1/24ε2−a+1b+1∫e∞xalognbexp−4ε2logxdx=limε↘a+1/24ε2−a+1b+1∫1∞ybexp−4ε2−a+1ydy=limε↘a+1/2∫4ε2−a+1∞zbe−zdz=Γb+1.
From (12), and noting that θ is arbitrary, we get the proposition immediately.Proof of Theorem 2.
Here, we only present the proof for (3) since the argument for (4) is similar. It is obvious, for p<−1/2,(13)Psup0≤t≤1Bt≥2lognε+bnε+lognp−Psup0≤t≤1αnt−Bt≥lognp≤Psup0≤t≤1αnt≥2lognε+bnε≤Psup0≤t≤1Bt≥2lognε+bnε−lognp+Psup0≤t≤1αnt−Bt≥lognp.
From Lemma2, we have Psup0≤t≤1αnt−Bt≥lognp≤Psup0≤t≤1αnt−Bt≥Klogn+a+2logn/λ/n≤Le−a+2logn=Ln−a+2, and then ∑n=1∞nalognbPsup0≤t≤1αnt−Bt≥lognp≤L∑n=1∞n−2lognb<∞. Furthermore, it follows(14)limε↘a+1/24ε2−a+1b+1∑n=1∞nalognbPsup0≤t≤1αnt−Bt≥lognp=0.
On the other hand, sincep<−1/2, we have PB≥2lognε+bnε±lognp=PB≥2lognε+bnε±1/2lognp−1/2∼2exp−4lognε+bnε±1/2lognp−1/22∼2exp−4ε2lognexp−8εbnεlogn∼PB≥2lognε+bnε, as. n⟶∞
With Proposition1, it follows(15)limε↘a+1/24ε2−a+1b+1∑n=1∞nalognbPsup0≤t≤1Bt≥2lognε+bnε±lognp=2exp−4τa+1Γb+1.
From (13) to (15), we get the result of Theorem 2.Proof of Theorem 3.
In this part, we only present the outline of the proof for the uniform sample quantile process, so the arguments for Theorem2 and 3 are mutually complementary.
Follow the proof of Proposition1 closely, we can get the following conclusion. For any 0<θ<1, there exist δ>0 and n0 such that for all n≥n0 and ε∈a+1/2,a+1/2+δ, 2exp−4ε2loglognexp−4τa+1−θ≤PB≥2loglognε+dnε≤2exp−4ε2loglognexp−4τa+1+θ
On the other hand,limε↘a+1/24ε2−a+1b+1∑n=1∞loglognaloglognb/n⋅exp−4ε2loglogn=Γb+1.
Therefore, we have(16)limε↘a+1/24ε2−a+1b+1∑n=1∞lognaloglognbnPB≥2loglognε+dnε=2exp−4τa+1Γb+1.
Like (13), we have. Psup0≤t≤1Bnt≥2loglognε+dnε+loglognp−Psup0≤t≤1unt−Bnt≥loglognp≤Psup0≤t≤1unt≥2loglognε+dnε≤Psup0≤t≤1Bnt≥2loglognε+dnε−loglognp+Psup0≤t≤1unt−Bnt≥loglognp
By lemma3, let z=logn/λ, we can get. ∑n=1∞lognaloglognb/nPsup0≤t≤1unt−Bnt≥loglognp≤∑n=1∞lognaloglognb/nPsup0≤t≤1unt−Bnt≥Clogn+logn/λ/n≤∑n=1∞lognaloglognb/n⋅Le−logn<∞
The rest of the proof is similar to Theorem2 and is omitted here.


## 3. Conclusion

The empirical process theory plays an important role in large sample theory in statistics. The researchers are very much interested in the large sample properties of the statistical estimator. As long as the sample size tends to infinity, the estimator converges to the true value of the parameter. In the procedure of demonstration of large sample properties, especially for the estimators in the semiparameter models, this study on convergence rates for the uniform empirical process and the uniform sample quantile process can provide a series of effective methods and tools.The limitation of this study may lie in the lack of consideration of the exact asymptotic properties of uniform empirical processes; in addition, in the study of the convergence rate of the uniform empirical process and uniform sample quantile process, the influence of the exact asymptotic behavior of self-regularity and logarithmic law on the convergence rate should also be taken into account.Due to the needs of practical applications, dependent random samples are often of more interest to statisticians. Positive and negative concomitants also widely exist in real life and engineering, such as reliability testing, statistical mechanics, and so on. The limit properties of the sequences of associated random variables, such as the law of iterated logarithm and the law of large numbers of the sequences of associated random variables, will be a hot topic in the future. In the future, the asymptotic properties of the test statistics of the model and parameters will be studied by parameter estimators.



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