Precise Asymptotics on Second-Order Complete Moment Convergence of Uniform Empirical Process

and Applied Analysis 3 Proof. A well known fact in Billingsley [10] reveals that the uniform empirical process converges weakly to Brownian bridge, F n ⇒ B. By continuous mapping theorem, we have ‖F n ‖ ⇒ ‖B‖. Thus, as n → ∞, Δ n = sup x∈R 󵄨󵄨󵄨P { 󵄩󵄩󵄩Fn 󵄩󵄩󵄩 ≥ x} − P (‖B‖ ≥ x) 󵄨󵄨󵄨 󳨀→ 0. (13) Let A(ε) = g(Mε), where g(x) is the inverse function of g(x) andM is an arbitrary positive number; then there exists a positive constant C such that ∑ n≤A(ε) g 󸀠 (n) ≤ C∫ A(ε) 1 g 󸀠 (x) dx ≤ Cg (A (ε)) = CMε −2 ; (14) thus ε 2 ∑ n≤A(ε) g 󸀠 (n) ≤ C < ∞. (15) By Toeplitz Lemma listed in Appendix, we know lim ε↘0 ε 2 ∑ n≤A(ε) g 󸀠 (n) {P ( 󵄩󵄩󵄩Fn 󵄩󵄩󵄩 ≥ ε√g (n))


Introduction and Main Result
Let {,   ,  ≥ 1} be a sequence of iid random variables and   = ∑  =1   .Hsu and Robbins [1] introduced the concept of complete convergence and obtained that ∑ ∞ =1 (|  | ≥ ) < ∞,  > 0, whenever E = 0 and E 2 < ∞.The result is extended by Baum and Katz [2], who obtained that, for 0 <  < 2,  ≥ , ∑ ∞ =1  (/)−2 (|  | ≥  1/ ) < ∞,  > 0, if and only if E||  < ∞, and when  ≥ 1, E = 0. Since then, some researchers concern the convergence of the series where () and () are all positive functions defined on [1, ∞), and ∑ ∞ =1 () = ∞.Because of the fact that the series tends to infinity when  ↘ 0, 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 lim whenever E = 0 and E 2 < ∞.Chow [4] studied the similar result on complete moment convergence of Later, Liu and Lin [5] extended the result to the  ( < 2) order complete moment convergence, which states that when E = 0, E 2 =  2 , and 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, selfnormalized sums, order statistics, eigenvalue statistics, and random fields.For the details on this topic, one can refer to Gut and Steinebach [6].
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  > 0, 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
Based on the existing results above, we will add a general precise asymptotic result on the second-order complete moment convergence of   ().
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.

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 lim  → ∞ (  ( + 1)/  ()) = 1, we can find that, for any 0 <  0 < 1, there exists a sufficient large number  = ( 0 ), such that   ( + 1)/  () ≤ 1 +  0 and   ( + 1)/  () ≥ 1 −  0 for all  ≥ .Thus we have At the same time, by making a change of variables and integration by parts, for any  ≥ 0, 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  0 → 0.
Proof.A well known fact in Billingsley [10] Let () =  −1 ( −2 ), where  −1 () is the inverse function of () and  is an arbitrary positive number; then there exists a positive constant  such that By the result of Kiefer and Wolfowitz [11], there exists  0 > 0, such that Then, by letting  ↘ 0 and then  → ∞, we can get lim By Lemma 2.1 in Zhang and Yang [7], for any  ∈ R, Then, a similar argument in (19) can deduce that lim 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.