Let be a sequence of iid U[0, 1]-distributed random variables, and define the uniform empirical process , . When the nonnegative function satisfies some regular monotone conditions, it proves that .

1. Introduction and Main Result

Let be a sequence of iid random variables and . Hsu and Robbins [1] introduced the concept of complete convergence and obtained that , whenever and . The result is extended by Baum and Katz [2], who obtained that, for , , if and only if , and when , . Since then, some researchers concern the convergence of the series where and are all positive functions defined on , and . Because of the fact that the series tends to infinity when , one of the interesting problems is to examine the rate when it occurs; then we need to find a suitable normalizing rate function such that it, multiplied by the series, has a nontrivial limit. The research on this topic is usually called “precise asymptotics.” Heyde [3] first proved that whenever and . Chow [4] studied the similar result on complete moment convergence of . Later, Liu and Lin [5] extended the result to the order complete moment convergence, which states that when , , and , then

In addition to the partial sums of iid random variables, there are some corresponding precise asymptotic results on other subjects, such as uniform empirical process, self-normalized sums, order statistics, eigenvalue statistics, and random fields. For the details on this topic, one can refer to Gut and Steinebach [6].

The paper will focus on the precise asymptotic of the uniform empirical process. Let be a sequence of independent -distributed random variables; we can define the uniform empirical process ,  . Consider to be a Brownian bridge on and write . Zhang and Yang [7] established some precise asymptotics on the complete convergence of the uniform empirical process; one of their main results can be stated as follows.

Lemma 1. For , , then

Zang and Huang [8] obtained some results on the first-order complete moment convergence of . If the nonnegative function satisfies some regular monotone conditions, they proved the following.

Lemma 2. For any , one has

Chen and Zhang [9] further got some precise asymptotic result on the second-order complete moment convergence of it. A typical result in their work can be listed as follows.

Lemma 3. For any , , one has

Based on the existing results above, we will add a general precise asymptotic result on the second-order complete moment convergence of .

Theorem 4. Assume that the real-valued function satisfies the following conditions.(A1) is differentiable on the interval , which is nonnegative and strictly increasing to .(A2)The differentiable function is nonnegative and the function is monotone. If is monotone nondecreasing, we assume that .
One has

Remark 5. The assumptions on are rather mild; in fact, there are lots of functions satisfying them, such as ,  , and with suitable parameters .

The main proofs are presented in the next section. Throughout the paper, denotes an absolutely positive constant whose value can be different from one place to another.

2. The Proof

We first give some propositions, which will play a key role in the proof of Theorem 4.

Proposition 6. Under the assumptions of Theorem 4, one has

Proof. If is monotone nonincreasing, by the assumptions of Theorem 4, we can see that is also nonincreasing; thus If is monotone nondecreasing, by the assumption that , we can find that, for any , there exists a sufficient large number , such that and for all . Thus we have
At the same time, by making a change of variables and integration by parts, for any , we have
By relations (9)–(11) and the fact that the result of the proposition will remain unchanged when we add or subtract some finite sums on the left hand of it, we can complete the proof by taking .

Proposition 7. Under the assumptions of Theorem 4, one has

Proof. A well known fact in Billingsley [10] reveals that the uniform empirical process converges weakly to Brownian bridge, . By continuous mapping theorem, we have . Thus, as ,
Let , where is the inverse function of and is an arbitrary positive number; then there exists a positive constant such that thus By Toeplitz Lemma listed in Appendix, we know
On the other hand, by the similar argument in (11), we have By the result of Kiefer and Wolfowitz [11], there exists , such that Then, by letting and then , we can get
By Lemma 2.1 in Zhang and Yang [7], for any , Then, a similar argument in (19) can deduce that
By combining (16), (19), and (21) and using the triangular inequality, we can complete the proof.

Proposition 8. Under the assumptions of Theorem 4, one has

Proof. Similar to the argument in Proposition 6, no matter whether the function is monotone nonincreasing or monotone nondecreasing, we can deduce the following relations by applying the change of variables and the L'Hôpital's rule: Thus, the proof is completed.

Proposition 9. Under the assumptions of Theorem 4, one has

Proof. If we denote , where is the inverse function of , then we can write
For the term , via the change of variable , we have where and is defined by (13).
Since implies , then we can see that
By relation (18), when , we have
By using relation (20), the similar argument can prove that
Note that . A combination of (28)–(30) and Toeplitz Lemma can lead to that which indicates that
For the term , using relation (18) again, the same argument in Proposition 8 can deduce that which implies that
At last, by using relation (20) again, the similar argument as in the discussion on can yield that
Combining (25) and (32)–(35), we can complete the proof.

Proof of Theorem 4. According to the fact that, for any random variable and , we have By Propositions 6 and 7, we know which shows that
By Propositions 8 and 9, we can get that
A final combination of the above two relations can reveal that From Csörgö and Révèsz [12], we know , which yields that This concludes the proof of Theorem 4.


Lemma A.1 (Toeplitz Lemma in Loève [13]). Let , be numbers such that, for every fixed , , and for all , . If , then In particular, if and , then

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


The work is supported by the National Natural Science Foundation of China (nos. 11326173 and 11401169) and Foundation of Henan Educational Committee (nos. 13A110087 and 2014JSJYYB-011). The authors would like to thank the referee for some valuable comments and suggestions.