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

One of the sources of “invariance principle” is that the limit properties of the uniform empirical process coincide with that of a Brownian bridge. e 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, when ε 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. e method of strong approximation is used in this paper. e main theorems are proved by using the Brownian bridge B(t) to approximate the uniform empirical process αn(t). e relevant results for the uniform sample quantile process are also presented.


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]. e 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 in nity, 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. e solution of these problems must rely on the probability limit theorem.
Let X, X n ; n ≥ 1 be a sequence of independent and identically distributed (i.i.d.) random variables with the common distribution function F, and set S n n i 1 X i for n ≥ 1. Hsu and Robbins [6] introduced the following complete convergence.
is holds if ΕX 0, and ΕX 2 < ∞. e converse part was proved by Erdös [7]. e complete convergence is stronger than the almost sure convergence. Obviously, the sum in (1) tends to in nity as ε ↘ 0. e rst result on the convergence rate of this kind was given by Heyde [8]. It is proved that if ΕX 0, and ΕX 2 < ∞. Heyde [8], Alam [4] got general conclusions and termed them "precise asymptotics." e precise asymptotics for "S n " 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 U 1 , U 2 , · · · , U n be a sequence of i.i.d. U[0, 1]− distributed random variables. Define the uniform empirical process as α n (t) � n − 1/2 n i�1 (I U i ≤ t − t), 0 ≤ t ≤ 1. Denote the norm of a function f(t) on [0, 1] by ‖f‖ � sup 0≤t≤1 |f(t)|, and log x � ln(x∨e). e following is one conclusion provided by Zhang and Yang [11].
The proof of Theorem 1 is based on the classical method introduced by Gut and Spǎtaru [12]. In this paper, we consider the situation "ε↘c 0 " where c 0 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 � U (n) 0 ≤ U (n) 1 ≤ · · · ≤ U (n) n ≤ U (n) n+1 � 1 denote the order statistics of the random sample U 1 , U 2 , . . . , U n , for each n ≥ 1. Define the uniform quantile function as uniform sample quantile process should be defined as u n (y) � n 1/2 (U n (y) − y), 0 ≤ y ≤ 1. e following are our main results. and and Remark 1. We define the general empirical process as is a continuous distribution function since α n (F(x)) � β n (x) , the results for β n (x) can be obtained immediately from the uniform case. But we cannot handle the quantile process in the same way.

Proofs
e starting point of this paper is the empirical distribution function. e empirical distribution function plays a very important role in statistics [13][14][15][16][17][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. e empirical process is constructed on the basis of the empirical distribution function. e uniform empirical process is a special and important one [19][20][21].
We lay out some lemmas which will be used in the proofs later. Lemma 1 is well known (cf. [22]). Lemma 2 and 3 are from Csörgó and Révész [23,24].
In particular,

Lemma 2.
ere exists a sequence of Brownian bridges B n (t); 0 ≤ t ≤ 1 such that for all n and x we have (9) where K, L, λ are positive absolute constants.

□
Proof of eorem 2. Here, we only present the proof for (3) since the argument for (4) is similar. It is obvious, for On the other hand, since p < − 1/2, we have P ‖B‖ ≥ ������ 2 log n (ε + b n (ε)) ± (log n) p From (13) to (15), we get the result of eorem 2. □ Proof of eorem 3. In this part, we only present the outline of the proof for the uniform sample quantile process, so the arguments for eorem 2 and 3 are mutually complementary.
Follow the proof of Proposition 1 closely, we can get the following conclusion. For any 0 < θ < 1, there exist δ > 0 and n 0 such that for all n ≥ n 0 and ε ∈ ( Like (13), we have.

Conclusion
e empirical process theory plays an important role in large sample theory in statistics. e 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. e 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. e 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.
Data Availability e data set can be accessed upon request.

Conflicts of Interest
e author declares that there are no conflicts of interest.