Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 972324, 9 pages
doi:10.1155/2010/972324
Research Article
Second Moment Convergence Rates for Uniform
Empirical Processes
You-You Chen and Li-Xin Zhang
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
Correspondence should be addressed to You-You Chen, cyyooo@gmail.com
Received 21 May 2010; Revised 3 August 2010; Accepted 19 August 2010
Academic Editor: Andrei Volodin
Copyright q 2010 Y.-Y. Chen and L.-X. Zhang. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Let {U1 , U2 , . . . , Un } be a sequence of independent and identically distributed
U0, 1-distributed
random variables. Define the uniform empirical process as αn t n−1/2 ni1 I{Ui ≤ t} − t, 0 ≤
t ≤ 1, αn sup0≤t≤1 |αn t|. In this paper, we get the exact convergence rates of weighted infinite
series of Eαn 2 I{αn ≥ εlog n1/β }.
1. Introduction and Main Results
Let {X, Xn ; n ≥ 1} be a sequence of independent and identically distributed i.i.d. random
variables with zero mean. Set Sn ni1 Xi for n ≥ 1, and log x lnx ∨ e. Hsu and Robbins
1 introduced the concept of complete convergence. They showed that
∞
P {|Sn | ≥ εn} < ∞,
ε>0
1.1
n1
if EX 0 and EX2 < ∞. The converse part was proved by the study of Erdös in 2. Obviously,
the sum in 1.1 tends to infinity as ε 0. Many authors studied the exact rates in terms of ε
cf. 3–5. Chow 6 studied the complete convergence of E{|Sn | − εnα } , ε > 0. Recently, Liu
and Lin 7 introduced a new kind of complete moment convergence which is interesting,
and got the precise rate of it as follows.
2
Journal of Inequalities and Applications
Theorem A. Suppose that {X, Xn ; n ≥ 1} is a sequence of i.i.d. random variables, then
∞
1 1
ES2n I{|Sn | ≥ εn} 2σ 2
2
ε0 − log ε
n
n1
lim
1.2
holds, if and only if EX 0, EX2 σ 2 , and EX2 log |X| < ∞.
Other than partial sums, many authors investigated precise rates in some different
cases, such as U-statistics cf. 8, 9 and self-normalized sums cf. 10, 11. Zhang and
Yang 12 extended the precise asymptotic results to the uniform empirical process. We
suppose U1 , U2 , · · · , Un is the sample of U0, 1 random variables and En t is the empirical
√
distribution function of it. Denote the uniform empirical process by αn t nEn t − t,
0 ≤ t ≤ 1, and the norm of a function ft on 0, 1 by f sup0≤t≤1 |ft|. Let Bt, t ∈ 0, 1
be the Brownian bridge. We present one result of Zhang and Yang 12 as follows.
Theorem B. For any δ > −1, one has
lim ε
ε0
2δ2
∞ log n δ EB2δ2
P αn ≥ ε log n .
n
δ1
n1
1.3
Inspired by the above conclusions, we consider second moment convergence rates for
the uniform empirical process in the law of iterated logarithm and the law of the logarithm.
Throughout this paper, let C denote a positive constant whose values can be different from
one place to another. x will denote the largest integer ≤ x. The following two theorems are
our main results.
Theorem 1.1. For 0 < β ≤ 2, δ > 2/β − 1, one has
lim ε
βδ1−2
ε0
∞ log n δ−2/β
1/β βEBβδ1
Eαn 2 I αn ≥ ε log n
.
n
βδ 1 − 2
n2
1.4
Theorem 1.2. For 0 < β ≤ 2, δ > 2/β − 1, one has
lim ε
ε0
βδ1−2
∞ log log n δ−2/β
1/β βEBβδ1
Eαn 2 I αn ≥ ε log log n
.
n log n
βδ 1 − 2
n3
1.5
Journal of Inequalities and Applications
3
k1 −2k2 x2
e
, x > 0 see Csörgő and
Remark 1.3. It is well known that P {B ≥ x} 2 ∞
k1 −1
Révész 13, page 43. Therefore, by Fubini’s theorem we have
EBβδ1 βδ 1
∞
xβδ1−1 P {B ≥ x}dx
0
2βδ 1
∞
xβδ1−1
0
∞
−1k1 e−2k
2 2
x
dx
k1
1.6
∞
βδ 1Γ βδ 1/2 −1k1 k−βδ1 .
2βδ1/2
k1
Consequently, explicit results of 1.4 and 1.5 can be calculated further.
2. The Proofs
In order to prove Theorem 1.1, we present several propositions first.
Proposition 2.1. For β > 0, δ > −1, one has
lim ε
βδ1
ε0
∞ log n δ 1/β EBβδ1
P B ≥ ε log n
.
n
δ1
n2
2.1
Proof. We calculate that
lim ε
βδ1
ε0
∞ log n δ 1/β P B ≥ ε log n
n
n2
lim ε
βδ1
∞ ε0
β
∞
2
log y
y
δ 1/β dy
P B ≥ ε log y
2.2
tβδ1−1 P {B ≥ t}dt
0
EBβδ1
.
δ1
Proposition 2.2. For β > 0, δ > −1, one has
lim ε
ε0
βδ1
∞ log n δ 1/β 1/β − P B ≥ ε log n
0.
P αn ≥ ε log n
n
n2
2.3
4
Journal of Inequalities and Applications
Proof. Following 4, set Aε expM/εβ , where M > 1. Write
∞ log n δ 1/β 1/β − P B ≥ ε log n
P αn ≥ ε log n
n
n2
n≤Aε
δ
log n 1/β 1/β − P B ≥ ε log n
P αn ≥ ε log n
n
2.4
δ
log n 1/β 1/β − P B ≥ ε log n
P αn ≥ ε log n
n
n>Aε
: I1 I2 .
d
It is wellknown that αn · → B· see Csörgő and Révész 13, page 17. By continuous
d
mapping theorem, we have αn → B. As a result, it follows that
Δn : sup|P {αn ≥ x} − P {B ≥ x}| −→ 0,
as n −→ ∞.
x
2.5
Using the Toeplitz’s lemma see Stout 14, pages 120-121, we can get limε0 εβδ1 I1 0. For
I2 , it is obvious that
I2 ≤
n>Aε
δ log n δ log n
1/β 1/β P B ≥ ε log n
P αn ≥ ε log n
n
n
n>Aε
2.6
: I3 I4 .
Notice that Aε − 1 ≥
ε
βδ1
Aε, for a small ε. Via the similar argument in 4 we have
I3 ≤ ε
βδ1
≤C
∞
n>Aε
M/21β
δ log n
1/β P B ≥ ε log n
n
yβδ1−1 P B ≥ y dy −→ 0,
2.7
as M −→ ∞.
From Kiefer and Wolfowitz 15, we have
P {αn ≥ x} ≤ Ce−Cx .
2
2.8
Journal of Inequalities and Applications
5
Therefore,
ε
βδ1
I4 ≤ Cε
βδ1
n>Aε
≤ Cε
≤C
βδ1
δ
log n
2/β exp −Cε2 log n
n
∞
√
∞
CM/22/β
Aε
log x
x
δ
2/β exp −Cε2 log x
dx
yβδ1/2−1 e−y dy −→ 0,
2.9
as M −→ ∞.
From 2.6, 2.7, and 2.9, we get limε0 εβδ1 I2 0. Proposition 2.2 has been proved.
Proposition 2.3. For β > 0, δ > 2/β − 1, one has
lim ε
βδ1−2
ε0
∞ log n δ−2/β ∞
2EBβδ1
2yP
≥
y
dy
.
B
1/β
n
δ 1 βδ 1 − 2
εlog n
n2
2.10
Proof. The calculation here is analogous to 2.1, so it is omitted here.
Proposition 2.4. For 0 < β ≤ 2, δ > 2/β − 1, one has
∞
∞ log n δ−2/β ∞
lim εβδ1−2
2yP αn ≥ y dy −
2yP B ≥ y dy 0.
1/β
1/β
ε0
n
εlog n
εlog n
n2
2.11
Proof. Like 4 and Proposition 2.2, we divide the summation into two parts,
∞
∞ log n δ−2β ∞
2yP αn ≥ y dy −
2yP B ≥ y dy
εlog n1/β
n
εlog n1/β
n2
n≤Aε
δ−2β ∞
∞
log n
2yP αn ≥ y dy −
2yP B ≥ y dy
εlog n1/β
n
εlog n1/β
n>Aε
: J1 J2 .
δ−2/β ∞
∞
log n
2yP αn ≥ y dy −
2yP B ≥ y dy
εlog n1/β
n
εlog n1/β
2.12
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Journal of Inequalities and Applications
First, consider J1 ,
J1 ≤
δ−2/β ∞
log n
2yP αn ≥ y − P B ≥ y dy
n
εlog n1/β
n≤Aε
≤
δ ∞
log n
1/β 1/β − P B ≥ x ε log n
2x εP αn ≥ x ε log n
dx
n
0
n≤Aε
≤
δ log n−1/β Δ−1/4
n
log n
1/β 2x εP αn ≥ x ε log n
n
0
n≤Aε
1/β −P B ≥ x ε log n
dx
:
n≤Aε
∞
1/β dx
2x εP B ≥ x ε log n
log n−1/β Δ−1/4
n
∞
log n−1/β Δ−1/4
n
1/β dx
2x εP αn ≥ x ε log n
δ
log n
J11 J12 J13 .
n
2.13
Since n ≤ Aε means ε < M/ log n1/β , it follows
2/β
2/β
J11 ≤ log n
log n
log n−1/β Δ−1/4
n
2x εΔn dx
0
−1/β −1/4 2/β −1/β 1/β 2
≤ log n
Δn log n
Δn log n
M
2.14
2
1/β 1/2
≤ Δ1/4
M
Δ
−→ 0, as n −→ ∞.
n
n
By Lemma 2.1 in Zhang and Yang 12, we have P {B ≥ x} ≤ 2e−2x . For J12 , it is easy to get
2
2/β
2/β
log n
J12 ≤ log n
≤C
∞
Δ−1/4
n
∞
−2/β
· 2yP B ≥ y dy
log n
εlog n1/β Δ−1/4
n
2y exp −2y2 dy
≤ C exp −2Δ−1/2
−→ 0,
n
2.15
as n −→ ∞.
Journal of Inequalities and Applications
7
In the same way, by the inequality P {αn ≥ x} ≤ Ce−Cx , we can get
2
2/β
−→ 0,
log n
J13 ≤ C exp −CΔ−1/2
n
as n −→ ∞.
2.16
Put the three parts together, we get that log n2/β J11 J12 J13 → 0 uniformly in ε as
n → ∞. Using Toeplitz’s lemma again, we have limε0 εβδ1−2 J1 0.
In the sequel, we verify limε0 εβδ1−2 J2 0. It is easy to see that
J2 ≤
n>Aε
δ−2/β ∞
log n
2xP {B ≥ x}dx
n
εlog n1/β
n>Aε
δ−2/β ∞
log n
2xP {αn ≥ x}dx
n
εlog n1/β
2.17
: J21 J22 .
We estimate J22 first, by noticing 0 < β ≤ 2 and 2.8, it follows
J22 ≤
n>Aε
≤C
δ−2/β ∞
log n
1/β 1/β ε 1/β−1
2ε log y
P αn ≥ ε log y
dy
log y
n
βy
n
∞
Aε
≤C
∞
Aε
≤ Cε
2
δ−2/β ∞ 2 2/β−1
ε log y
log x
2/β exp −Cε2 log y
dy dx
x
y
x
2/β−1
ε2 log y
2/β δ−2/β1
exp −Cε2 log y
dy
log y
y
∞
Aε
≤ Cε2
log y
y
logδ Aε
AεCε
2
δ
2.18
exp −Cε2 log y dy
≤ Cε2−βδ .
Therefore, we get limε0 εβδ1−2 J22 0. So far, we only need to prove lim0 εβδ1−2 J21 0.
2
Use the inequality P {B ≥ x} ≤ 2e−2x again and follow the proof of J22 , we can get this
result. The proof of the proposition is completed now.
Proof of Theorem 1.1. According to Fubini’s theorem, it is easy to get
EXI{X ≥ a} aP {X ≥ a} ∞
a
P {X ≥ x}dx,
2.19
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Journal of Inequalities and Applications
for a > 0. Therefore, we have
2/β 1/β 1/β ε2 log n
P αn ≥ ε log n
Eαn 2 I αn ≥ ε log n
∞
εlog n
1/β
2yP αn ≥ y dy.
2.20
From Proposition 2.1– 2.4, we have
lim ε
βδ1−2
ε0
lim ε
∞ log n δ−2/β
1/β Eαn 2 I αn ≥ ε log n
n
n2
βδ1
ε0
∞ log n δ 1/β P αn ≥ ε log n
n
n2
∞ log n δ−2/β ∞
βδ1−2
lim ε
2yP αn ≥ y dy
1/β
ε0
n
εlog n
n2
a
2.21
βEBβδ1
.
βδ 1 − 2
Proof of Theorem 1.2. From 2.19, we have
εβδ1−2
∞ log log n δ−2/β
1/β Eαn 2 I αn ≥ ε log log n
n log n
n3
ε
βδ1−2
ε
∞ log log n δ−2/β ∞
2yP αn ≥ y dy
n log n
εlog log n1/β
n3
βδ1
2.22
∞ log log n δ 1/β .
P αn ≥ ε log log n
n log n
n3
Via the similar argument in Proposition 2.1 and 2.2,
lim ε
ε0
βδ1
∞ log log n δ 1/β EBβδ1
P αn ≥ εlog log n
.
n log n
δ1
n3
2.23
Also, by the analogous proof of Proposition 2.3 and 2.4,
lim εβδ1−2
ε0
∞ log log n δ−2/β ∞
2EBβδ1
2yP αn ≥ y dy .
n log n
δ 1 βδ 1 − 2
εlog log n1/β
n3
2.24
Combine 2.22, 2.23, and 2.24together, we get the result of Theorem 1.2.
Journal of Inequalities and Applications
9
Acknowledgment
This work was supported by NSFC No. 10771192 and ZJNSF No. J20091364.
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