Zhu Journal of Inequalities and Applications (2015) 2015:103 DOI 10.1186/s13660-015-0626-3

d Journal of Inequalities and Applications

a SpringerOpen Journal

RESEARCH

Open Access

A generalization of almost sure local limit theorem of uniform empirical process

Chenglian Zhu*

Correspondence: hynuzcl@126.com Schoolof MathematicalScience, Huaiyin NormalUniversity, Huaian, Jiangsu 223300, People's Republic of China

Abstract

Let {Xn;n > 1} be a sequence of independent and identically distributed U[0,1]-distributed random variables. In this paper, we are concerned with the almost sure local central limit theorem of ||Fn|| and sup0<f<1 Fn(t), and some corresponding results are derived.

MSC: 60E15; 60F15

Keywords: almost sure central limit theorem; almost sure local central limit theorem; uniform empirical process

ft Spri

1 Introduction

Throughout this paper, let {Xn; n > 1} be a sequence of independent and identically distributed U[0,1]-distributed random variables and put Sn = ^ 1=1 Xk. Define the uniform empirical process Fn(t) = n-2 n=1(I{Xi<t} -t), 0 < t < 1, ||Fn|| = sup0<t<1 |Fn(t)|. It is well known that there has been recently a lively interest in probability theory concerning almost sure versions of classical limit theorems. The prototype of such a theorem is the almost sure central limit theorem (ASCLT), which has the simplest form as follows:

1 n 1 i S 1

lim -V-1\ — < A = $(x) a.s. (1.1)

n^œ log n^ k J

for all x e R; here and in the sequel, I{A} is the indicator function of the event A and stands for the standard normal distribution function. This result was firstly proved independently by Brosamler [1] and Schatte [2] under a stronger moment condition. Since then, this type of almost sure version, which mainly dealt with logarithmic average limit theorems, has been extended in various directions.

Especially, Fahrner and Stadtmuller [3] and Cheng et al. [4] extended this almost sure convergence for partial sums to the case of maxima of independent and identically distributed (i.i.d.) random variables. Under some suitable conditions, they proved the following:

lim _L y - A < x\ = G(x) a.s. (1.2)

n^œ log n 4—' k [ ak

ringer

© 2015 Zhu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

for all x e R, where ak >0 and bk e R satisfy

lim p(< x) = G(x) k^œ y ak j

k^TO y ak

for any continuity point x of G.

For Gaussian sequences, Csaki and Gonchigdanzan [5] presented the validity of (1.2) for maxima of stationary Gaussian sequences under some mild conditions. Furthermore, Chen and Lin [6] extended it to non-stationary Gaussian sequences. As for some other dependent random variables, Peligrad and Shao [7] and Dudzinski [8] derived some corresponding results about an almost sure central limit theorem. The almost sure central limit theorem in a joint version for log average in the case of independent and identically distributed random variables was obtained by Peng etal. [9], a joint version of almost sure limit theorem for log average of maxima and partial sums in the case of stationary Gaussian random variables was derived by Dudzinski [10]. In this direction, an extension of almost sure central limit theory was studied by Hormann [11].

Moreover, Wu [12-14] explored the almost sure limit theorem for product of partial sums, stable distribution and product of sums of partial sums, respectively. Zang [15] derived the almost sure limit theorem of random fields for more general weights than the usual logarithmic average. Recently, Zhang [16] established the almost sure central limit theorem for uniform empirical processes with logarithmic average. And then, under some regular conditions, a general result of almost sure central limit theorem for uniform empirical processes with general weights was derived by Zang [17] with the methodology of Hormann [11].

On the other hand, Chung and Erdos [18] proved the following result.

Theorem A Let Xi,X2,... be a sequence ofi.i.d. integer valued random variables with EX\ = 0. Assume that every integer a is a possible value of Skfor all sufficiently large k. Then

1 ^ P(Sk = a) lim - > -= 1 a.s.,

log Tn Tk

Tn = £ P(Sk = a).

Further, if the condition EXl = a2 < œ is satisfied, we have Tk ~ , therefore

i x^P(Sk = 0) 42 lim - > -—— = —— a.s.

n^œ logn k=i Vk a^n

This result may be called almost sure local central limit theorem (ASLCLT), while (1.1) may be called almost sure global central limit theorem.

A more general version of this theorem was proved by Csaki et al. [19] with finite third moment, they derived

1 A I{ak < Sk < bk} lim -} ---— = 1 a.s.

n^cx> logn pf kP{ak < Sk < bk}

n log k

y^ —-= 0(log n) as n ^to,

k=1 k2 P{ak < Sk < bk}

where {an, n > 1} and {bn, n > 1} are both real sequences such that an < 0 < bn for n > 1.

In this paper, under some mild conditions, we are concerned with the almost sure local central limit theorems of ||Fn|| and sup0<t<1 Fn(t), which is inspired by the above result, especially Csaki et al. [19] and the references therein concerning the almost sure local central limit theorem.

The rest of this paper is organized as follows. In Section 2, a generalization of almost sure local central limit theorem of uniform empirical process is formulated. In Section 3, proofs of our main results are established. In Section 4, the paper is concluded and some statistical applications for future research are outlined.

2 Main results

In this section, let the real-valued sequences {un, n > 1}, {vn, n > 1} be such that un > vn, and satisfy

A = °(log n), (2.1)

1<tT<nl2pkp'

pkpl=0

wherepk = P{vk < ||Fk|| < uk}. Set

(P-I{vk < ||Fk ||< uk}, Pk =0,

ak =\ Pk

11, Pk = 0

= J ¿I{vk < sup0<t<1 Fk(t) < uk}, qk = 0,

11, qk = 0,

where qk = P{vk < sup0<t<1 Fk(t) < uk} satisfies the corresponding condition (2.1).

Theorem 2.1 Let {Xn; n > 1} be a sequence of independent and identically distributed (i.i.d.) random variables. Then

lim -V -k = 1 a.s. (2.2)

n^TO logn k

Theorem 2.2 Let {Xn; n > 1} be a sequence of independent and identically distributed (i.i.d.) random variables. Then

lim —1— ir ^ = 1 a.s. (2.3)

n^TO logn ^ k

Remark 2.3 We believe that condition (2.1) can be weakened through more complicated calculating procedures, so we will study it in the future work.

3 The proofs of the main results

In this section, we shall give some auxiliary lemmas which will be used to prove our main result. The first lemma comes from Gonchigdanzan [20].

Lemma 3.1 Assume that f2,... are random variables such that Efi = 1for k = 1,2,____

o^ e( £ -

lim „. . n^TO log n \ ^ k

Furthermore, if > 0 for k > 1 and

E f) ^ dog n)2

for some e >0 and large enough n, then

1 £ ak lim - > — = 1 a.s.

n^TO logn k

Here and in the sequel, an ^ bn denotes limsupn^TO |an/bn| < to.

Lemma 3.2 IfX1, X2,... are i.i.d. random variables with common distributed function F. Denote Dn = supx | n ^ n=11 X < x} - F (x) |, then there exist positive constants c0 and c such that, for all n, all F, and all positive r, one has

1- P(Dn < r/n1/2) < coe-cr2.

Proof This lemma is due to Kiefer and Wolfowitz [21]. □

Now, we give the proofs of our main results.

Proof of Theorem 2.1 Denote

Fk,l(t) = l-2 J2 fet} -1), 0 < t < 1,

l|Fk,l|| = sup \Fk,l(t)\.

Firstly, by Lemma 3.1, it only needs to verify

v^L j) «(l08")2-s.

Note that

w (x^ «A x Var(au) ^ Cov(ak, ai)

Var(£T)-£ u2 +2 £ —a—.

\ k-1 / k-1 1 <k<i<n

Furthermore, observe that if pk - 0, then

Var(ak) - 0; ifpk - 0, then

Var (ak) - ^ < 1, pk pk

and consequently

x Var (ak) x 1 x 1 / —r2— </ Tt— < > T2-«log n;

tr k2 t! k2Pk 1 <tr< nl2PkPi

ifPkPi - 0, then

Cov(ak, ai) - 0;

if PkPl - 0, we have

Cov(/ {vk < lift II < uk}, I {vi < ||Fi || < ui}) <|Cov(I{|FkII < uk},I{|Fi|| < ui})| + |Cov(l{|Fk||< uk},l{ ||Fi| < vi})| + |Cov(l{||Fk||< Vk},I{||Fi|| < ui})| + |Cov(l{|Fk||< Vk},I{||Fi|| < vi})| -: A1 + A2 + A3 + A4.

For A1, k < i, it follows that

A1 - |Cov(I{|Fk|| < uk},I{|Fi|| < ui})|

< |Cov(I{|Fk|| < uk},I{||Fi || < u} -1{||Fk,i || < u} + |Cov(I{|Fk|| < uk},I{||Fk,i|| < ui})|

« £|I {|Fi || < ui} -1 {||Fk,i || < ui}|

|Cov(I{ ||Fk ||< Uk},1{\\Fki ||<ui)

=: An + Ai2.

Making use of Lemma 3.2 and Fubini's theorem, we have

Au « E|||Fi|| - \\Fv |

sup |Fi(t)| - sup (t)|

0<t<l 0<t<l

« E sup ||Fi(t)|-|Fktl(t)||

« E sup |Fi(t)-Fki(t)|

= E sup I-2

= ( k) 2E sup k-2

I / 0<t<l

Efe) -1) k

E^ <t) -1)

p\ sup k-2 V(I{X,<t) - t) l0<t<l i=l

e-cx2 dx

> x\ dx

«, f2

In virtue of the independence of {Xn; n > l), we have Al2 = 0. Then

Furthermore, by similar reasoning, we have

Ai « -, i = 2,3,4,

and therefore,

A2 + A3 + A4 « k.

Thus, according to our assumptions, we can derive Cov(I{vk < Rk < uk), I{vi < Ri < ui))

l<k<l<n

l<k<l<n

kipkpi

Ai + A2 + A3 + A4 kipkpi

_j-f,ktp,pAU

l<k<i<n

= O(log n).

Hence, Var(J]n=l k) « (log n). Further, we complete the proof of Theorem 2.l. □

Proof of Theorem 2.2 The proof is similar to the above procedures, so we omit it here.

4 Concluding remarks

In this paper, we are concerned with the limit theory of uniform empirical process. A generalization of almost sure local central limit theorem of uniform empirical process has been established.

Some statistical applications related to our main result deserve further investigation. By virtue of being a new approach of testing based on ASCLT, Thangavelu [22] investigated hypothesis testing via the estimation of quantiles of the distribution of the concerned statistics. Based on the theorem on ASCLT for rank statistics, he also proposed a small-sample approximation for the two-sample nonparametric Behrens-Fisher problem. These statistical applications concerning our work will be discussed in the future work.

Competing interests

The author declares that they have no competing interests. Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11326174 and 11401245), the Natural Science Foundation of Jiangsu Province (Grant No. BK20130412),the Natural Science Research Project of Ordinary Universities in Jiangsu Province (Grant No. 12KJB110003).

Received: 31 July 2014 Accepted: 9 March 2015 Published online: 18 March 2015 References

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