Statistica Sinica 16(2006), 1409-1422
EXACT CONVERGENCE RATE AND LEADING TERM IN
THE CENTRAL LIMIT THEOREM FOR U -STATISTICS
Qiying Wang and Neville C Weber
The University of Sydney
Abstract: The leading term in the normal approximation to the distribution of U statistics of degree 2 is derived. This result is applied to establish the exact rate
of convergence in the Central Limit Theorem for U -statistics and to obtain the
one-term Edgeworth expansion for the distribution function. Analogous results for
more general U -type statistics are also considered.
Key words and phrases: Berry-Esséen theorem, characterisation of rate of convergence, Edgeworth expansion, optimal moments, nonlattice condition, U -statistics,
L-statistics.
1. Introduction and Main Results
Let X, X1 , . . . , Xn be a sequence of independent and identically distributed
(i.i.d.) random variables. Let h(x, y) be a real-valued Borel measurable function,
symmetric in its arguments with Eh(X 1 , X2 ) = 0. For n ≥ 2, a U -statistic of
degree 2 with kernel h(x, y) is defined by
n −1 X
Un =
h(Xi , Xj ).
(1)
2
1≤i<j≤n
Write g(x) = Eh(x, X1 ) and φ(x, y) = h(x, y) − g(x) − g(y). The statistic U n may
be represented as
n
n −1
2X
Un =
g(Xj ) +
n
2
j=1
X
φ(Xi , Xj ) := U1n + U2n .
(2)
1≤i<j≤n
See, for example, Lee (1990, p.25).
Throughout we assume that Eg 2 (X1 ) = 1. This assumption implies that
√
nU1n /2 is a standard sum of non-degenerate iid random variables and its distribution may be approximated by a standard normal distribution Φ. Indeed,
the classical result (see Hall (1982, p.11), for example) shows that
√nU
1
1
1n
sup P
≤ x − Φ(x) + n− 2 δn + n− 2 .
(3)
2
x
1410
QIYING WANG AND NEVILLE C WEBER
Here and below we define
1
δn = Eg 2 (X1 )I(|g(X1 )|≥√n) + n− 2 Eg 3 (X1 )I(|g(X1 )|≤√n) +n−1 Eg 4 (X1 )I(|g(X1 )|≤√n) ,
and we say that two sequences of positive numbers {a n } and {bn } satisfy an bn
if 0 < lim inf n→∞ an /bn ≤ lim supn→∞ an /bn < ∞. Note that (3) yields concise
results about the rate of convergence in the Central Limit Theorem.
In past decades, there has been considerable interest in stating the accuracy
√
of the normal approximation to the distribution of nUn /2 in a manner that
is similar to (3). In increasing generality, the upper bound for the accuracy of
the normal approximation has been established in a number of papers. We mention only Bickel (1974), Chan and Wierman (1977), Callaert and Janssen (1978),
Borovskikh (1996, 2001), Alberink and Bentkus (2001, 2002) and Wang (2002).
The result given by Borovskikh (2001) (also see Alberink and Bentkus (2001)),
which is closest to the upper bound in (3), states that if E|h(X1 , X2 )|5/3 < ∞,
then
√nU
n
sup P
≤ x − Φ(x)
2
x
i
h
1
1
≤ A Eg 2 (X1 )I(|g(X1 )|≥√n) + n− 2 E|g(X1 )|3 I(|g(X1 )|≤√n) + O(n− 2 ), (4)
where A is an absolute positive constant.
In contrast to rich results on the upper bound, there are only a few papers concerned with the lower bound for the accuracy of the normal approxima√
tion to the distribution of nUn /2. Maesono (1988, 1991) obtained a lower
bound of order O(n−1/2 ) under the condition Eh4 (X1 , X2 ) < ∞. Only assuming the existence of Eh2 (X1 , X2 ), Wang (1992) derived a result for the
√
distribution of nUn /2 that is similar to (3). In a slightly different problem
Bentkus, Götze and Zitikis (1994) proved that the best bound of order O(n −1/2 )
in (4) cannot be obtained under E|h(X1 , X2 )|5/3− < ∞ for any > 0.
In the present paper we give the leading term in a normal approximation
√
to the distribution of nUn /2. Using the leading term we derive the exact
convergence rate in the Central Limit Theorem for U -statistics, up to terms of
order O(n−1/2 ), under E|h(X1 , X2 )|5/3 < ∞. As mentioned above, to get the
terms of order O(n−1/2 ), the latter moment condition is the best possible. We
also show that, if in addition E|g(X 1 )|3 < ∞, the leading term transforms into
the conventional first term in an Edgeworth expansion of the distribution of
U -statistics.
Our main result is the following.
EXACT CONVERGENCE RATE AND LEADING TERM
Theorem 1.1. If E|h(X1 , X2 )|5/3 < ∞, then
√nU
1
n
sup P
≤ x − Φ(x) − L1n (x) + L2n (x) = o(δn ) + O(n− 2 ),
2
x
1411
(5)
where δn is defined as in (3),
n
1
g(X1 ) o
− Φ(x) − Φ(2) (x),
L1n (x) = n EΦ x − √
2
n
(3)
Φ (x) L2n (x) = √ E g(X1 )g(X2 )φ(X1 , X2 )I
3
.
2
)
(|φ(X
,X
)|≤n
1
2
2 n
If in addition g(X1 ) is nonlattice, then the right-hand side of (5) may be replaced
by o(δn + n−1/2 ).
As is well-known, δn → 0, as n → ∞, and
sup |L1n (x)| δn ,
(6)
x
(see, for example, Chapter 2 of Hall (1982)). We show in Section 3 that
1
sup |L2n (x)| = o(δn ) + O(n− 2 ).
x
(7)
Together, (5)−(7) give concise results about the rate of convergence in the Central
Limit Theorem for U -statistics. Indeed, if E|h(X 1 , X2 )|5/3 < ∞, then
√nU
1
1
n
≤ x − Φ(x) + n− 2 δn + n− 2 .
sup P
(8)
2
x
Note that (8) refines (4) even for the upper bound. One application of (8) is
to characterise the rate of convergence. The following theorem gives examples.
Generalizations of the examples are readily derived, refer to Theorems 2.9 and
2.10 of Hall (1982) for more details.
Theorem 1.2. Assume E|h(X1 , X2 )|5/3 < ∞. If 0 ≤ r < 1/2, then
∞
√nU
X
n
r−1
n
sup P
≤ x − Φ(x) < ∞
2
x
(9)
n=1
if and only if E|g(X1 )|2(r+1) < ∞. If 0 < r < 1/2, then
√nU
n
sup P
≤ x − Φ(x) = O(n−r )
2
x
if and only if Eg 2 (X1 )I(|g(X1 )|≥x) = o(x−2r ).
(10)
1412
QIYING WANG AND NEVILLE C WEBER
It is also interesting to note that the effect of L 2n (x) in (5) on the rate of
convergence appears only when δn = O(n−1/2 ). This can be easily seen from
(7) and the following corollary, which provides the main result of Jing and Wang
(2003), about an Edgeworth expansion of the distribution of U -statistics under
optimal conditions. Corollary 1.1 below also is an alternative to Theorem 5
of Borovskikh (1998) with m = 2, where the Edgeworth expansion is obtained
under lim sup|t|→∞ |Eeitg(X1 ) | < 1 instead of the condition that the distribution
of g(X1 ) is nonlattice. Borovskikh (1998) also uses a weaker moment condition
on the kernel h(x, y).
Corollary 1.1. Assume that E|h(X1 , X2 )|5/3 < ∞, E|g(X1 )|3 < ∞, and the
distribution of g(X1 ) is nonlattice. Then, as n → ∞,
√
1
nUn
sup P
(11)
≤ x − Fn (x) = o(n− 2 ),
2
x
√
where Fn (x) = Φ(x) − (Φ(3) (x)/(6 n)) Eg 3 (X1 ) + 3Eg(X1 )g(X2 )h(X1 , X2 ) .
The proof of all results will be given in Section 3. To conclude this section
we mention that the rate of convergence in the Central Limit Theorem for U statistics depends on the moment conditions for both h(X 1 , X2 ) and g(X1 ). If
only E|h(X1 , X2 )|p < ∞, where 4/3 < p < 5/3, the term of order O(n −1/2 ) in
(8) has to be replaced by a term of lower order. This follows from Theorem 2.1
in the next section, which gives an extension of Theorem 1.1 to U -type statistics. Throughout the paper we denote constants by A, A 1 , A2 , . . ., which may be
different at each occurrence.
2. Extensions to U -type Statistics and L-statistics
Let α(x) and β(x, y) be some real-valued Borel measurable functions of
x and y. Furthermore, let Vn ≡ Vn (X1 , . . . , Xn ) be real-valued functions of
{X1 , . . . , Xn }. Define a U -type statistic by
1
Tn = n − 2
n
X
j=1
3
α(Xj ) + n− 2
X
β(Xi , Xj ) + Vn .
(12)
i6=j
In this section we derive the leading term in a normal approximation to the
distribution of Tn under mild conditions, which gives an extension of Theorem
1.1.
Theorem 2.1. Assume that
(a) Eα(X1 ) = 0 and Eα2 (X1 ) = 1;
(b) E[β(X1 , X2 )|Xi ] = 0, i = 1, 2, and E|β(X1 , X2 )|p < ∞ for 4/3 < p ≤ 5/3;
EXACT CONVERGENCE RATE AND LEADING TERM
1413
(c) P (|Vn | ≥ C0 n−1/2 ) ≤ C1 n−1/2 for some constants C0 > 0 and C1 > 0.
Then, as n → ∞,
sup P Tn ≤ x − Φ(x) − L̃1n (x) + L̃2n (x)
x
= o(δ1n + n
4−3p
2
1
) + O(n− 2 ),
(13)
where
1
δ1n = Eα2 (X1 )I(|α(X1 )|≥√n) + n− 2 Eα3 (X1 )I(|α(X1 )|≤√n) +n−1 Eα4 (X1 )I(|α(X1 )|≤√n) ,
α(X1 ) 1
L̃1n (x) = n EΦ x − √
− Φ(x) − Φ(2) (x),
n
2
io
n
h
(3)
Φ (x)
L̃2n (x) = √ E α(X1 )α(X2 ) β(X1 , X2 )I
.
3
3 + β(X2 , X1 )I
(|β|≤n 2 )
(|β|≤n 2 )
2 n
If the condition (c) is replaced by (c 0 ) P |Vn | ≥ o(n−1/2 ) ≤ o(n−1/2 ), and in
addition α(X1 ) is nonlattice, then the right-hand side of (13) may be replaced by
o(δn + n(4−3p)/2 ).
Note that the U -type statistic Tn defined by (12) is quite general. We next
consider an application to L-statistics. Let X 1 , . . . , Xn be i.i.d. real random
variables with distribution function F . Define F n to be the empirical distribution,
P
i.e., Fn (x) = n−1 nj=1 I{Xi ≤ x}, where I{·} isR the indicator function. Let J(t)
be a real-valued function on [0, 1] and T (G) = xJ(G(x)) dG(x). The statistic
T (Fn ) is called an L-statistic (see Chapter 8 of Serfling (1980)). Write
ZZ
2
2
σ ≡ σ (J, F ) =
J (F (s)) J (F (t)) F (min{s, t}) [1 − F (max{s, t})] dsdt,
and define the distribution function of the standardized L-statistic T (F n ) by
√
Hn (x) = P nσ −1 (T (Fn ) − T (F )) ≤ x .
As is well-known, Hn (x) converges to Φ(x) uniformly in x provided E|X 1 |2 <
∞, σ 2 > 0, and some smoothness conditions on J(t) hold, see Serfling (1980) and
Helmers, Janssen and Serfling (1990) for example. The upper bounds for the
rate of convergence to normality were investigated by Helmers (1977) van Zwet
(1984), Helmers, Janssen and Serfling (1990), Wang, Jing and Zhao (2000) and
Wang (2002).
As a consequence of Theorem 2.1, the following theorem derives the exact
convergence rate (two-sided bound) in the Central Limit Theorem for L-statistics,
up to terms of order O(n−1/2 ), under mild conditions.
Theorem 2.2. Assume that
1414
QIYING WANG AND NEVILLE C WEBER
(a) |J(s) − J(t)| ≤ K|s − t|, 0 < s < t < 1, for some K > 0;
(b) EX12 < ∞ and σ 2 > 0.
Then, as n → ∞,
1
1
sup |Hn (x) − Φ(x)| + n− 2 δ1n + n− 2 ,
x
where α(X1 ) = −σ −1
Theorem 2.1.
R
(14)
J(F (t))(I(X1 ≤ t) − F (t)) dt, and δ1n is defined as in
3. Proofs
Proof of Theorem 1.1. The result is an immediate consequence of Theorem
2.1.
Proof of Theorem 2.1. Without loss of generality we assume that β(x, y) is
symmetric. Otherwise it is enough to replace β(X i , Yj ) by β(Xi , Yj ) + β(Xj , Yi ).
The proof is along the lines of Jing and Wang (2003). Write
β̃(Xi , Xj ) = β(Xi , Xj )I
3 ,
(|β(Xi ,Xj )|≤n 2 )
2(n − 1) ∗
α∗ (Xj ) = E β̃(Xi , Xj ) | Xj , α∗∗ (Xj ) =
α (Xj )I(|α∗ (Xj )|≤√n) ,
n
n
1 X
3 X
β̃(Xi , Xj ) − α∗ (Xi ) − α∗ (Xj )
Tn∗ = n− 2
(α(Xj ) + α∗∗ (Xj )) + 2n− 2
i<j
j=1
+Vn .
Noting Eβ(X1 , X2 ) = 0, it is easily seen that
|α∗ (Xj )| = E β(Xi , Xj )I(|β(Xi ,Xj )|≤n3/2 ) |Xj ≤ E |β(Xi , Xj )| I(|β(Xi ,Xj )|≥n3/2 ) |Xj ,
(15)
and, as in (2.24)−(2.25) of Jing and Wang (2003),
sup |P (Tn ≤ x) − P (Tn∗ ≤ x)|
x
≤ nP (|α∗ (X1 )| ≥
4−3p
√
3
n) + n2 P |β(X1 , X2 )| ≥ n 2
≤ 2n 2 E|β(X1 , X2 )|p I
3
(|β(X1 ,X2 )|≥n 2 )
4−3p .
=o n 2
(16)
We further let, m0 = ([10 log n]+1)/b, where b > 0 is a constant to be chosen
EXACT CONVERGENCE RATE AND LEADING TERM
1415
later,
ηj = α(Xj ) + α∗∗ (Xj ) − Eα∗∗ (Xj ),
i
h
γij = 2 β̃(Xi , Xj ) − α∗ (Xi ) − α∗ (Xj ) + E β̃(Xi , Xj ) ,
1
Sn = n − 2
3
∆m = n − 2
n
X
ηj ,
j=1
n
X
γmj
j=m+1
∆n,m =
n−1
X
∆k
for 1 ≤ m ≤ n − 1,
if 0 < m < n,
and ∆n,m = 0
k=m
if m ≥ n.
√
It follows immediately that Tn∗ = Sn + ∆n,m0 + Ṽn + nEα∗∗ (X1 ) − (n −
P 0 −1
1)n−1/2 E β̃(X1 , X2 ), where Ṽn = Vn + m
m=1 ∆m . Note that for any fixed
1 ≤ m < k ≤ n and 1 ≤ q ≤ 2,
3q
E|∆n,m − ∆n,k |q ≤ 8n− 2 +1 (k − m)E|γ12 |q ;
(17)
see Theorem 2.1.3 in Koroljuk and Borovskich (1994). It follows from (17) with
q = p and E|γ12 |p < ∞ (see (21) below) that for 4/3 ≤ p ≤ 5/3,
1
0 −1
4−3p mX
n− 2 ≤ An1−p log1+p n E|γ12 |p = o n 2
.
P ∆m ≥
log n
(18)
m=1
√
In terms of the condition (c) (or (c0 )), (18) and the fact that | nEα∗∗ (X1 ) −
√
(n − 1)n−1/2 E β̃(X1 , X2 )| ≤ 3 nE|β(X1 , X2 )|I(|β|≥n3/2 ) = o(n(4−3p)/2 ), routine
calculations show that, to prove (13), it suffices to prove
In := sup P Sn + ∆n,m0 ≤ x − Φ(x) − L̃1n (x) + L̃2n (x)
x
= o(δ1n + n
4−3p
2
1
) + O(n− 2 )
(19)
and, if in addition α(X1 ) is nonlattice, then the right-hand side of (19) may be
replaced by o(δn + n(4−3p)/2 ).
We first establish five lemmas before the proof of (19). The proofs of these
lemmas will be omitted. The details can be found in Wang and Weber (2004),
on which the present paper is based.
1416
QIYING WANG AND NEVILLE C WEBER
Lemma 3.1. Write α̂(X1 ) = α∗∗ (X1 ) − Eα∗∗ (X1 ). We have
λ+2−3p λ
∗∗
λ
E|α̂(X1 )| ≤ 2E|α (X1 )| = o n 2
, for 1 ≤ λ ≤ 2,
E|γ12 |p ≤ 16|β̃(X1 , X2 )|p < ∞,
3(q−p) E|γ12 |q ≤ 16|β̃(X1 , X2 )|q = o n 2
,
4−3p
.
|Eη12 − 1| = o δ1n + n 2
(20)
(21)
for p < q ≤ 2,
(22)
(23)
Lemma 3.2. We have
Eγ12 e
it(η1 +η2 )
√
n
=−
o
2t2 n
E α(X1 )α(X2 )β̃(X1 , X2 )
n
4−3p
1
n− 2 (t2 + |t|3 ),
+ o δ1n + n 2
it(η1 +η2 ) 4(p−1)
1
t
√
−1
2
2
3
Eγ12 e
≤ A min ( √ ) p , n (Eγ12 ) 2 (t + |t| ) .
n
n
Next define, f (t) = Eeitη1 /
n(g(t) − 1) + t2 /2).
√
n,
g(t) = Eeitα(X1 )/
√
n,
and gn (t) = e−t
(24)
(25)
2 /2
(1 +
Lemma 3.3. There exists a constant c 0 > 0 such that for all |t| ≤ c0 n1/2 and
all sufficiently large n,
t2
t2
|g(t)| ≤ e− 4n ,
|f (t)| ≤ e− 8n ,
4−3p
t2 t2
n
f (t) − e− 2 ≤ A δ1n + o(n 2 ) (t2 + t4 )e− 16 ,
4−3p
t2
n
(t2 + t8 )e− 16 .
f (t) − gn (t) = o δ1n + n 2
(26)
(27)
(28)
If in addition α(X1 ) is nonlattice, then there exist constants b > 0 and n → ∞
√
such that for c0 ≤ |t|/ n ≤ n ,
b
|f (t)| ≤ e− 2 and |g(t)| ≤ e−b .
(29)
√
Lemma 3.4. For any |t| ≤ c0 n, where c0 is defined as in Lemma 3.3,
t2
o t2 e− 2 n
itSn
√
e
+
E
α(X
)α(X
)
β̃(X
,
X
)
+
β̃(X
,
X
)
E∆n,m0
1
2
1
2
2
1
2 n
4−3p
t2
(t2 + t6 )e− 16 .
= o δ1n + n 2
(30)
To introduce the next lemma we first define some notation. As in (2.13) and
EXACT CONVERGENCE RATE AND LEADING TERM
1417
(2.16) of Jing and Wang (2003), we have
Zn (t) := Eeit(Sn +∆n,m0 ) − EeitSn − itE∆n,m0 eitSn
h
i
(1)
(2)
= Zn1 (t) + it Zn2 (t) + Zn2 (t) ,
where, lm,k = n−3/2
and
Zn1 (t) =
Pn
j=k+1 γmj ,
n−1
X
m=m0
(1)
Zn2 (t)
=
Eeit(Sn +∆n,m+1 ) eit∆m − 1 − it∆m ,
n−1
X j(m)
X
m=m0 j=1
(2)
Zn2 (t) =
j(m) is the largest integer such that mj(m) < n
n−1
X j(m)
X
m=m0 j=1
Elm,jm eitSn eit∆n,jm+1 − eit∆n,(j+1)m+1 ,
E lm,jm − lm,(j+1)m eitSn eit∆n,(j+1)m+1 − 1 .
Lemma 3.5. For 4/3 < p ≤ 5/3, we have
Z
4−3p
1
2
,
|Z
(t)|
dt
=
o
δ
+
n
n1
1n
√
|t|≤c0 n |t|
Z
4−3p
(2)
(1)
2
(t)|
dt
=
o
δ
+
n
(t)|
+
|Z
|Z
,
1n
n2
n2
√
(31)
(32)
|t|≤c0 n
where c0 is defined as in Lemma 3.3.
We are now ready to prove (19). We continue to use the notation defined in
2
Lemmas 3.1−3.5. Furthermore write ϕ n (t) = −t2 Bn e−t /2 , where
io
n
h
1
√
Bn =
3
.
E α(X1 )α(X2 ) β(X1 , X2 )I
3 + β(X2 , X1 )I
(|β|≤n 2 )
(|β|≤n 2 )
2 n
Using Lemmas 3.3−3.5 we have
Z
1 it(Sn +∆n,m )
0 − g (t) − itϕ (t) dt
J1 (n) :=
Ee
n
n
√ |t|
|t|≤c0 n
Z
Z
1
1 n
≤
|Zn (t)| dt +
|f (t) − gn (t)| dt
√ |t|
√ |t|
|t|≤c0 n
|t|≤c0 n
Z
E∆n,m eitSn − ϕn (t) dt
+
0
√
= o(δ1n + n
4−3p
2
|t|≤c0 n
).
(33)
1418
QIYING WANG AND NEVILLE C WEBER
R∞
Note that −∞ eitx d Φ(x) + L̃1n (x) − L̃2n (x) = gn (t) + itϕn (t). It follows from
Esseen’s smoothing lemma and (33) that
Z
A
1 it(Sn +∆n,m )
0 − g (t) − itϕ (t) dt + √
In ≤
Ee
n
n
√ |t|
n
|t|≤c0 n
= o(δ1n + n
4−3p
2
1
) + O(n− 2 ).
(34)
This proves the first part of (19).
If α(X1 ) is nonlatice, it follows from the fact that ∆ n,m0 only depends on
Xm0 +1 , . . . , Xn , and (29), that for any n → ∞,
Z
1 it(Sn +∆n,m )
0
Ee
−
g
(t)
−
itϕ
(t)
J2 (n) :=
dt
n
n
√
√ |t| c0 n≤|t|≤n n
Z
Z
1
1
m0
≤
|f (t)| dt + √
|g (t) + itϕn (t)| dt
√
√ |t|
√ |t| n
c0 n≤|t|≤n n
c0 n≤|t|≤n n
= o(δ1n + n
4−3p
2
).
(35)
Using (33), (35) and Esséen’s smoothing lemma again, we obtain for 4/3 ≤ p ≤
5/3,
Z
A
1 it(Sn +∆n,m )
0
In ≤
− gn (t) − itϕn (t) dt + √
√ |t| Ee
n n
|t|≤n n
1
≤ J1 (n) + J2 (n) + o(n− 2 )
= o(δ1n + n
4−3p
2
).
This implies the second part of (19) and hence the proof of (19).
The proof of Theorem 2.1 is now complete.
Proof of Theorem 2.2. Write ηj (t) = I{Xj ≤ t} − F (t),
Z
Z
α(Xj ) = −σ −1 J(F (t))ηj (t)dt, β(Xi , Xj ) = Kσ −1 ηi (t)ηj (t)dt.
As in (29) of Wang (2002), we have
1
n− 2
n
X
j=1
3
α(Xj ) − n− 2
X
i6=j
β(Xi , Xj ) − Vn
√
n (T (Fn ) − T (F ))
≤
σ
n
X
3 X
1
α(Xj ) + n− 2
≤ n− 2
β(Xi , Xj ) + Vn ,
j=1
i6=j
(36)
EXACT CONVERGENCE RATE AND LEADING TERM
where Vn = n−3/2
Pn
1419
−1 η 2 (t)dt. It is readily seen
j
j=1 Z(Xj ) with Z(Xj ) = Kσ
2
0, Eα (X1 ) = 1, E(β(Xi , Xj )Xi ) = 0, i 6= j, and similar to the
R
that Eα(X1 ) =
proof of Lemma A in Serfling (1980, p.288),
|α(Xj )| + |β(Xi , Xj )| + Z(Xj ) ≤ Aσ −1 (|Xj | + E|X1 |).
(37)
In terms of these facts, (14) follows easily from Theorem 2.1. We omit the details.
Proof of Corollary 1.1. Equation (11) follows easily from Theorem 1.1, the
classical result
1
Φ(3) (x)
sup L1n (x) + √ Eg 3 (X1 ) = o(n− 2 ),
6 n
x
and by Hölder’s inequality, that
Φ(3) (x)
sup L2n (x) − √ E {g(X1 )g(X2 )φ(X1 , X2 )} 2 n
x
n
− 12
≤ A n E |g(X1 )g(X2 )| |φ(X1 , X2 )|I
3
(|φ(X1 ,X2 )|≥n 2 )
≤ An
− 21
E|g(X1 )|
5
2
4
5
5
3
E|φ(X1 , X2 )| I
Proof of (7). It suffices to show that
n
E g(X1 )g(X2 )φ(X1 , X2 )I
3
(|φ(X1 ,X2 )|≤n 2 )
o
3
3
5
(|φ(X1 ,X2 )|≥n 2 )
1
= o(n− 2 ).
o √
1
( n)−1 = o(δn ) + O(n− 2 ). (38)
Write φ̃(X1 , X2 ) = φ(X1 , X2 )I(|φ(X1 ,X2 )|≤n3/2 ) . It is readily seen that
n
o
E g(X1 )g(X2 )φ̃(X1 , X2 )
n
o
= E g(X1 )I(|g(X1 )|≥√n) g(X2 )φ̃(X1 , X2 )
n
o
+E g(X1 )I(|g(X1 )|<√n) g(X2 )I(|g(X2 )|≥√n) φ̃(X1 , X2 )
o
n
+E g(X1 )I(|g(X1 )|<√n) g(X2 )I(|g(X2 )|<√n) φ̃(X1 , X2 )
:= I6n + I7n + I8n .
(39)
By Hölder’s inequality, and similar to (22), we have
1
|I6n | + |I7n |
φ̃(X1 , X2 )2
2
√
E
≤ 2 Eg(X1 )2 I(|g(X1 )|≥√n)
n
n
i1
1 h
1
1
2
≤ 2 δn2 o(n− 2 ) = o δn + n− 2 .
!1
2
(40)
1420
QIYING WANG AND NEVILLE C WEBER
Similarly, by noting E|φ̃(X1 , X2 )|5/3 < ∞,
√
|I8n |/ n ≤ n
− 21
5
2
I(|g(X1 )|<√n)
4 5
E|g(X1 )|
E|φ̃(X1 , X2 )|
4
5
1
5
≤ An− 2 E|g(X1 )| 2 I(|g(X1 )|<√n) .
5
3
3
5
(41)
In terms of (41), if E|g(X1 )|5/2 I(|g(X1 )|<√n) < ∞, then
1
|I8n |
√ = O(n− 2 ).
n
(42)
We show that if E|g(X1 )|5/2 I(|g(X1 )|<√n) = ∞, then
√
5
nE|g(X1 )| 2 I(|g(X1 )|<√n) ≤ A E|g(X1 )|4 I(|g(X1 )|<√n) ,
(43)
and hence it follows from (41), that
√
5
1
|I8n |/ n = o(1) n− 2 E|g(X1 )| 2 I(|g(X1 )|<√n)
= o(1) n−1 E|g(X1 )|4 I(|g(X1 )|<√n) = o(δn ).
(44)
Combining (39)−(40), (42) and (44), we obtain the proof of (38).
We next prove (43). Write lτ (x) = E|g(X1 )|τ I(|g(X1 )|<x) . Note that l4 (x) is
a non-decreasing function and l4 (x) ≤ A x2 . It follows from Proposition 2.2.1 of
Bingham, Goldie and Teugels (1987) that lim supx l4 (2x)/l4 (x) < ∞. Now (43)
follows easily from question 34 on page 289 of Feller (1971) ((also see Feller (1969)
or Theorem 2.6.6 of Bingham, Goldie and Teugels (1987)).
The proof of (7) is now complete.
Acknowledgements
The authors would like to thank two referees and an associate editor for their
detailed reading of this paper and valuable comments. The research is partially
supported by ARC discovery grant DP0451722 and Sesqui NSSS grant 2004.
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1422
QIYING WANG AND NEVILLE C WEBER
School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia.
E-mail: [email protected]
School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia.
E-mail: [email protected]
(Received April 2004; accepted October 2004)