Internat. J. Math. & Math. Sci.
(1988) 187-200
187
VOL. ii NO.
ON ASYMPTOTIC NORMALITY FOR
M-DEPENDENT U-STATISTICS
WANSOO T. RHEE
Faculty of Management Sciences
The Ohio State University
Columbus, Ohio 43210
(Received April 21, 1986)
ABSTRACT.
Let (X)
be a sequence of m-dependent random variables, not necessarily
n
equally distributed. We give a Berry-Esseen estimate of the convergence to normality
’of a suitable normalization of a U-statistic of the (X). This bound holds under
n
moment assumptions quite weaker than the existence of third moments for the kernel.
Since we obtain the sharpest bound, the order of the bound can not be improved.
KEY WORD? AND PHRASES.
U-Statistics, m-dependent random variables, Berry-Esseen
bound.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
INTRODUCTION.
Let (X ,X
2
P60F05.
X n) be random variables (r.v.). A very important and common
class of statistics is the class of U-statistics, of the type
[
U
<i
<... <ikn
(X
Xk)
where h is a measurable symmetric function which is called a kernel. (For notational
convenience, we do not consider the average of h but the sum of its values. This
will not make any difference since we will normalize U latter.) In the case of an
independent identically distributed (i.l.d.) sequence
(Xi)
it has been shown by
various authors that the distribution function of normalized U-statlstlcs converges
-I/2
to standard normal distribution function with the rate of n
the latest and
sharpest result being due to R. Helmets and W. R. Van Zwet [I]. These authors
studied U-statistics of order 2, viz.,
o
.<j <k <n
h(Xj, Xk).
But as they pointed out, their results can be extended to any order and to the
mult i-sample case.
In this work, we relax the independent and equi-distrlbuted assumptions about
i) of r.v.’s. Consider a m-dependent sequence (X i) of r.v.’s, i.e.
the sequence (X
W.T. RHEE
188
.< s < n-m,
for" each
the sequence
(Xi)[<.s
(Xt)i>s+ m
and
are independent of each
other, and which is not necessarily equi-distributed. Then we obtain a universal
Berry-Esseen bound for the convergence of the suitably normalized U-statistic to
standard normal. This bound involves the same moments as in the Helmers and Van
Zwet’s result, and leads the best rate of convergence for the independent case. We
deal only with the one sample case of order two; but there is no doubt that the same
methods could extend to the general case.
RESULTS.
2.
measurable function h: R
2
XI,...,X n
r.v.’s,
Consider a fixed sequence of m-dependent
R. We want to study U
.<j <-k <n
the field generated by X
by F
<.
that for
< n-m,
s
Then the sequence F
i.
V F. and
the two 0-fields
h(Xj, Xk).
Let us denote
is m-dependent, in the sense
F
V
<s-1
and a fixed
are independent (where
i>s+m
ieI). It is conceptually more elegant
V F. denote the e-field generated by the F
--i
h(Xj, X k) as a function hj, k which is Fj V F k
(Fi)1<i<n of m-dependent o-fields and consider for
Fj V Fk measurable. (Here we allow the possibility
and also more convenient to look at
measurable. So we fix a sequence
<.
<.
j
<.
k
n
r.v.h.j,k
which are
k since the proof will be the same and since it is convenient and allows to
extend the result to V-statlstics.)
For j < k, let
E(hj,klFk)
gj,k
Let f
[ gj ,i
J
_G
2. We make the following assumptions:
For any
< j
.<
k < n,
For any
< j
<
k < n,
For any
<
<
< j
k <
V
l-Jl
.>
that if k
G
Let
E(hj,klFj)"
gk,j
<j.<n
For
Let
<
3/2 < p
-2
2
j
n,
n, let
fi
Yj,k
Fi, _G2
-
E(hj, k)
hj,k has
V
2F_i
gj,k
V
Fi, we have
1{j-l,j,j+1,j+2}
I.
It is easily checked, and fundamental,
E(Yj ,j+1
G)
Z fi’ o2 ES2
I-I
L
O. Moreover, if k
We set
0
n
S
(2-3)
3.
Suppose now m
gk,j"
E(Yj,kI_G I) E(Yj,kI_G2)
we have
(2-2)
a moment of order p.
has a moment of order
hj,k
l-kl
(2-I)
0
n
o -3
Z
i=l
EII 3
For a e {I, 3/2, p}, we set
Ma
N
o
-3/2
0
-a
Z EIYj,I a
j<k
.
M
0<k-j3
M’a
(M3/2)2/3,
EIYj
I 3/2,
o
Z
a
j<k
(M/2)2/3
M’
N’
-a
o
-3/2
Z Elhj ,k
0<k_j< 3
3/2
j
and
ASYMPTOTIC NORMALITY FOR M-DEPENDENT U-STATISTICS
Let
189
denote the distribution of the standard normal law. The main result is as
follows:
THEOREM A.
Sup
I. There is a universal constant K such that
Assume m
--I
IP(U <
M’L 2/3
<. K{L
(t)
t)
NL I/3
(log L
I)p/eM’
P
t
M’L 5/3-p
MI’L 2}
p
Let us investigate this theorem in the case that h
(Eh(XI,X2)IX2)
are i.i.d. If g
(resp. [I, p])
we set b
K{n-I/2a3b 3
c
For p
possible order O(n
i’
Xj)
and the X
and for t e
[I, 3]
we get the bound
log
p
3/2, this bound is O(n
-I/2
Elhl,2 It,
t
(Eg)
h(X
n-I/2a3(b3c3/2)2/3 n-2/3(c/2)2/3931/3a5/2
(I/2
n
a3b3)P/2n2-3p/2aPc
I/3-p/2a5-2Pc 5/3-p n-1 /2a7c b}.
p
log
n
2 -I/2
and if we set a
Elgl t,
t
i,j
p
b
1/41og3/4n)
and for p
5/3, it is of the best
). A bound of the same order is obtained in the tationary
case.
It is possible with our method to find many other bounds of the ame type. An
example .f possible variation is given in section 6. More importantly, the term
M’L 5/3-p can be replaced by a term of the form M’L -p for any Y < 2 (but the
P
P
constant K will grow very fast when
gets close to 2). It is also possible to
replace the term M
I’L
2
by
M’L
q
for any q
> O, (but
then K will grow with q). Hence it
can be said that the main terms in theorem A are L, M’L 2/3 and
5/3
M’L
P
or
5/3-p
if p
p/2
M’log(L-1)
P
f p <
> 5/3.
Theorem A can also be used to give bounds in the case m > I. To do this, we
< [n/m], we set G
V
proceed by blocking. More specifically if for 0 <
jJ
where
ho,
k’
Ji
EJj
{j: im
’’
Jk
<
j < Inf((i+1)m, n)}, the fields G
h,,,
for j
.<
FI,
are l-dependent. Moreover, if
k, we have U
O<J k’<[n/m]
h’j,k
and it is possible
to apply theorem A to this U-statlstlc. (It is very useful at this point that we did
not assume that
h’j,k
is of the form h(X
i,
Xj)
and to allow the case
quantities involving the moments of the functions
terms of the moments of the
h,,.
However,
(h,k)
J
k!). The
can easily be expressed in
this does not seem to be the case of the
> I.
METHODS.
We shall use three basic techniques, viz. the method of R. Helmers and W. Van
Zwet [I], a method of V. Shergin [2] to deal with m-dependent r.v. and his result
about the convergence to normality of a sequence of m-dependent r.v. and an estimate
normalizing factor. This is why we do not state formally the result when m
3.
%4. T. RH.E
190
"’IEeitS
by the author of
as
where S
a sum of m-dependent r.v. [3] (which
based on Shergin’s technique).
We shall denote by K I,K2,... universal constants. No attempt has been made to
find small numerical values: their choice is made crudely, to check the consistency
of the construction.
I. We shall use Esseen smoothing inequality [4].
We suppose m
suPlP(U -1<.t )- (t)I-< KI{T
t
exp
g
Let
o
<-j <k<-n
Im
exp(itUo
I
T
-I
exp(itUo
Itl-11E
-T
(-t2/2)I
(3-I)
dt}.
We have
Yj
-I
exp(-t 2/ 2)
IE exp(it-Is)(exp(itA)-1)l
+Im exp(ito -I S) exp(-t2/2)l.
<.
The second term will be taken ca,e of by Shergin’s theorem. Considering that
le it-
21tl p, we have
IE exp(ito-Is)(exp(itA)-1)l
<
it
<. ItlIE(A
exp(ito
-I S))
21t PE IAI P
(3-2)
We shall evaluate these two terms directly. The above evaluation will De used
for t
<.
101og
L-I.
E exp(itUo
to bound
=>
For t
101og L
-I
.
We have, by expanding
we have
ep(-t2/2) <- L 5,
and so it is enough
Let I be an interval of [i, n],
appropriately). Let
AI=
-I,
o
-I
<nYj
k
<Ij(kI
j ,kI
2
(to be chosen
A- A
exp(itA2).
IE exp(ituo-l)I < IE exp(it(o -I S+A i))
ItIIE A2exp(ito-Is+A1)l 2ItlPEIA2 Ip
(3-3)
and we shall evaluate these three terms. In these evaluations, we shall several
times encounter the same difficulty. Say, for example we want to estimate
Elexp(it(
o
-I
-I (S-S’)
Let S’
S+AI))
A
fi"
depends on the F
Then S’ depends on the
-IF"
for ieI.
Moreover
for iI.
F. were independent, we would have
If we knew that the -i
IE
exp(it(o
-I
S+AI))
IE exp(Ito-Is’)l
for which good estimates are known (of the type exp
(-t2-2E(S’2))
for t not too
large). However, we must proceed in a different way. We shall use the technique
mentioned above to show that modulo a small perturbation one can (roughly speaking)
do as if S’ and
theorem
4-5.
-I (S-S’)
A
were independent and then use the estimate of
ASYMPTOTIC NORMALITY FOR M-DEPENDENT U-STATISTICS
In order to prove theorem A, one can suppose K
10
-3.
It then follows that if H
Moreover for
[log
-3,
so we can as.ume L <
I], we have HL <. 10 -2 and H .> 7.
each i,
Ef <. (Elfil3)
4.
L-I+
10
191
2/3 <
oeL 2/3 o2/1OO.
(3-4)
LEMMAS.
This section contains some of the technical tools we need.
LEMMA 4-I [4].
Let X
X
m [
k
Let X
LEMMA 4-2 [53.
p
X
Xi Ir.
k
k
r-1
k
(4-1)
.
-
be a martingale dfference. Then for
xl
k
For a sequence
LEMMA 4-3 [2].
I,
ElXi Ir.
2, we have
1
r
r.v.. Then for r
be
k
lxl
k
of m-dependent r.v. of mean zero and
(Xi)n
2,
1
xl"
THEOREM 4-4. (V. Shergln [2]).
zero mean and e
ES2’
2
Let L
(m+)
"-
2 r/2
( x)
Let S be a sum of 1-dependent r.v.
0-3 [ Efi 3
&n
(fi)n
There exists a universal constant K
such that
t-1E exp(ItSe-I)
THEOREM 4-5. [3].
IE
exp(itSo
In particular, for o
IE
-I
K
)I
PROPOSITION 4-6.
$
(4-4)
(1+K21tl) Max (exp(-t2/80), (tK2L)
we have
It and ItlKmU
I/4 log L
-I
,
2
K2olt I)
(I
exp(itS)
(-t-R/2)dt K2L.
exp
Under the hypothesis of the above theorem one has for
-I
Max
(exp(-t2o2/80),
(toK2L)I/4 log L
R
With the above notation, for a
_2
F
{i
Elfil..
[I, hi: o
-I
).
(4-5)
let
400L2}."
2
Then
lrl
PROOF.
Let q
o
4
z
oo.
(4-6)
card F. We have
Z ml,l 2
F
z
I
mltll3)
$i n
ooq
IF mlr13) 2.
40oq
From Holder’s inequality, we have
3 2/3
I/3
F
So (4-7) gives
F
of
F
(4-7)
2
N.T. RHEE
192
iF
The result follows.
E[A] p
We -are going to bound at the sane tie
,
E[6[ p
(notice A
A
if
n3.).
LEMA 5-1.
where V
and
EIA2 Ip
{j, k:
PROOF.
K3o-P
<
j
(j ,)ev
EIYj
’
n, either j or k belongs to I}.
Since
A2
Yj, k
A"
A’
k
2
V
V
{j, k:
< j
{j, k:
2
<-
j
<
k < n, j e
<
k <
n, k
I}
I,
-
I},
k
it is enough by (4-I) to bound each of these sums. We bound the first one.
Th.e
I let Z
proof for the second is very similar. For k
k
Then
kI
kI
For
O, I, 2, let
[.
A
3+iI
The sequence
V
t
j,k"
Y3+i-1,3+i"
is a martingale difference, since if
Y3+i_1,3+
FI
Y
j.<k-2
j<3+i
Fj,
then
3t+ll-
z(z3t/i_
O.
So (4-2) implies
3+iI
Thus (4-I) implies
E
[
Yk -I ,k
kel
The sequence
H-k iV.<2k-IF,
then
for the sequence
(Z2k)2kei
Z2k
is H
(Z2k+1).
k
is a
p< 6
-I_
Y.
kel
p
,k
mrtingale difference. Indeed, if
measurable and
E(Z2k+21H_.k)
0. A similar result holds
So (4-I) and (4-2) give
p
1 kgl
[
zkl
’lzl p.
Let us fix k e I. Then, it is easily seen that both sequences
and
(Y2j+1,k)1<2j+1.<k-2
(Y2j ,k
12j<.k-2
are martingale differences. It then follows by (4-I) and
(4-2) that
<k-2
The result follows these estimates.
k
ASYMPTOTIC NORMALITY FOR M-DEPENDENT U-STATISTICS
6.
E(A exp it o -I S)
BOUND FOR
Let H- [log L -I
I].
Itl
For
.>
IOK2,
let
(1+K21tl) Max(exp(-t2/8000), (IOOK2tL) I/4(Iog
(t)
and for
193
Itl < IOK2,
LEMMA 6-I.
let (t)
L
-I
/1000)
(6-I)
I.
There exist universal constants
K
K4,
5
such that for
ItK4LI<
I,
we have
IE(A
exp(ito-Is))l
K5{t2V(t)ML 2/3+ t(t)NL1/3 MI(4OtL)H}.
<
We write
olE(a
exp(ito
-I
-I
E(Yj kexp(Ito S))I.
Z
<
S))
<-j <k<n
We shall evaluate each of the terms of the rlght-hand’side
LEMMA 6-2
<
PROOF.
First Case
If
k
<
IOOK21tlL
IE(Yj, kexp(ito-ls)
j+2, we have for
k+1
k13/2)2/3(o -3 i-J Elfll3) I/3 EIYj
241tl(t)(EIyj
First step:
We prove that one of the following cases occurs.
< j satisfying the following conditions:
Seoond Case
>
There exists s
f
I< <-j
and S
i’
follows from (3-4) that
ESa 49.2/100
(or even
>
s
E
+
J
El i=s’l +1 fil 2 <
98,2/100.
a
o2/10
12< 400 Lo 2
s
Elfll
2
(6-4)
and
2
2
ESI
<
400 Lo
ES
l
i"’ +1
(6-5)
2Efjf j+
It
Let s
Let F- {lI:
Ef ’
3.10-3. 2
/
2
I El2
2(2H+3)400L
t:F
16,10
3.2
<
12.
2/ 100
J
>.ES
48o2/100
2 +
2
2
We prove that the first case occurs
It then follows that
l I
El 1<i<S’
(6-3)
]s, J-2[ contains 2H indices which do not belong to
and L < 10 -3 from (4-3) and (4-6), we get
J
2
< s’ <
We have o
S
ESZ 24.2/100.
-2
for k
]k+2, s[ for which
S
2
ES21 ES
since HL $ 10
<.
o2/10,
a
be the largest integer such that
For s’
(6-2)
k satisfying the followlng oonditions:
there are at least 2H indices
$I=
and
]s, j-2[ for which Elf
I,
E
I
El i=S’
Indeed, let
Z fll 2 >. o2/I0
< J, E
there are at least 2H indices
F
kl(40tL) H.
Their exist s
for s < s’
if
< I:
I j,
-El i-S’+1
12o2/100
-2Ef
S’
fs ’+I
2o2/100 => o 2/ 10
2o2
N.T. RHEE
194
ES
Similarly, the second case occurs for
Second Step.
We suppose that the first case occurs, the second being similar.
< H, indices s < p() < j-2 such that p() F and
p(+1) < p()-2 for
.<
S
Let
o
o
-I
<. H-I),
-I
-I
Z.
o
-I
Z2 +I-
Let us define
o
.
i>p(1
let
(resp.
fp()
Z0
S
z-
f.
j-1 =<i.<k +I
<-H (resp.
Z2
.
<. H-I. Let
<
Zo=
and for
.
<
can pick for
49o2/100.
-I
We
f-z o
f
p(+1)<i<p()
Y
I. It is easy to check
exp(itZ)
that
So
Yj,kexp(it
-I
Yj,kexp(ItZ 0)
exp(It S I)
(6-6)
2H-I
" Yq
Yj kexp(itZo
-I
(itS
q-1
+I
2H
Yj kexp(itZ 0)
..IYql-<
Note that for each q,
2
and
has an expectation bounded by
2HEIYj
since
Y
q=l
q
EIZ2q
o
-I
exp(itS ,+1
qeXp(itS2H).
EIYql,.
<
ItlEIZ q I-
The last term of (6-6)
H
k
q
< o-I (Ef
Elfp(2q)
q=1
=[lIEl2ql
(2q))
<
I/2
EIYj,kI(4OtL)
.< 20L. For
<
H
(6-7)
< 2H-I,
is measurable for the o-fleld generated by the
k+2, hence independent of
So, if
Yj,k"
YO exp(itZ0)-1,
F
since
for
j-2 and
E(Yj,k
we get
E
E(Yj
k
exp(itZ
E(Yj ,k
Since
S+
is independent of
O) qIlI[ YqeXp(itS+1
))
N Y exp(itS
q
+I
q-0
we get
Yj,kq II0 Yq,
lEvi ElY j kVOI 2qs,n EIX2ql.2[’/2]+llE exp(ttS+l)
[/2]
< 2ElY
(40tL)
IE exp (itS+l)
j ,kxO
Since
S+
is of the form
follows from step
that
o’
L,-
S1+1
2
o
2
s
fl
> I/I0
ES+
o ,-3
-I
s’
E
i=1
where s
< s’ <
So we have
Io -1 fil 3
s
o2o
-1
L.
j by construction,
it
ASYMPTOTIC NORMALITY FOR M-DEPENDENT U-STATISTICS
I02K21tlL <.
I02K21tlL <. I,
It follows from (4-5) that for
Moreover,
since K
2
.>
and
195
IE exp(itSE+1)
we have
we have (20tL) [/2] < 2
(t).
-[E/2].
Finally,
EIYj,kOI
-<
t(EIYj,kI3/2)2/3(EIZoI3) I/3.
-<
tEIYj,kZ01
In the same manner, we have
elYj,kexp(itZo)exp(itS1)
.<
EIYj,kYOI(t).
EIZo 13 from
The lemma follows from these estimates and the estimate of
LEMMA 6-3.
If k
>
cIE(Yj
-<
IOOK21tlL
j+2, we have for
40t2(t)(EIYj k13/2)2/3(0
(4-I).
I,
exp(itc-Is))l
k
-3
_<
El f 13)I/3
Elf iI 3 )I/3"(0-3 7.
7.
EIYj,kI(40tL) H.
PROOF.
One of the following cases occur:
First Step.
first case, second case:
identical to the first and second case of lemma 6-2
<
respectively. Third case, there exists j+2
conditions: for j+1< s’<
<
s
s
<
2
k-2 satisfying the following
s"< k-l, we have
s1< s2<
fi 12
7.
E
S
<i <-S"
there are at least 2H indices
e ]j+2,
>.
o2/10
st[
(6-8)
and
Elfi 12 400c2L
for which
(6-9)
e ]s 2, k-2[ with the same property.
and 2H indices
The proof uses the same method as in lemma 6-2. We omit it.
Second Step.
We shall treat only the first case.
the third uses the same idea.
that
Eft(f)
400o2L
Z’
and for
Let
<
o
2
H
q-
[i-j l1 fi’
Z
>-
j+3). Since
E(Yj, k exp(itZ’)W)
Z"
& H-I.
0 and
YO--
c
and Z
ll-kll fi
.<
For
generated by the E
E(Yj,kI_.G
<-
Let
O-
Z’
+
Z",
2H-1, the r.v.
for
O, we have
I-JI
(exp(itZ’)-1)(exp(itZ")-I),
we have
kexp(itZ 0) n YqeXp(ItS+
q=l
E(Yj kTO
H
q-1
q
>.
exp(ltZ")W
exp(itS +I )).
))
is
2. (Here" we use the
E(Yj, k exp(itZ")W)
O. It follows that if
E(Yj,kW )
E E(Yj
such
as in lena 6-2. Then (6-6) holds.
and
qeXp(ltS+l ).
measurable for the o-field G
fact that k
The second case is identical and
H, we pick indices s <. p() < j-2
and p(+I) & p()-2 for I
< 2H, define
W
.<
For
0. Similar
N.T. RHEE
196
The rest of proof is entirely similar to lemma 6-2, except that we estimate
EIYj,kYOI
<
t2EIYj,kZ’Z"I .< t2(EIYj,klB/2)2/3(EIZ,Z,,I3) I/3
t2(EIYj,klB/e)2/3(EIZ, 13)I/S(EIZ,,13) I/3.
-<
Follows from lemmas 6-2 and
PROOF OF LEMMA 6-I
If, in the estimate of
EIY j,kY0 I,
..
Itl(t)Np1/2Q1/q MI (40tL )H
I/2 2/q
Q
K5(t2(t)M p
n
-q
Elfi lq.
i=I
7.
we use Holder’s inequality
p/(p-1), we get in lemma (6-I) the bound
with exponents p and q
where Q
by the use of Holder’s
K4-- I00K3).
inequality (with
REMARK:
6-3
HOW TO CHOOSE
Le
10
-2 >
8 >
3.103HL 2
which will be chosen later. The next lemma shows how
to pick a subinterval I of {I,
that interval {I
n}
which roughly speaking will play the role
inS]} would play the i.i.d, case. Let
a {I
F
zlfil
n}
L2}.
>-400
{I
There exists an interval I
LEMMA 7-I.
2
n} which has the following
properties.
I is reunion of ten subintervals
<- 10,
I
each
.
n
Elf iI 3
iI
If A
a-P
(j,k)A
PROOF.
24008
i=I
<
{j, k; j
Let s(1)
fi )2
a
28a
2.
Elf iI 3
(7-2)
k, j or k belongs to I}, we have
p <
5000eM p
EIYj ,k
I.
.
contains at least 2H elements which
F, and is such that E(
does not belong to
(7-I)
which
Ii0
F, and such that for
extremities does not belong to
<
Ii,
(7-3)
We construct by introduction a sequence s(i) in the
following way:
s(i+1)
<
Inf{s: s(1)
s < n, s-l, s
at least 2H elements, E(
Z
s(i)<<s
F, is{i), s[
\
F contains
f)2 >. 28o2}.
The construction goes until we reach an integer s(h) such that eltheP s(h)
n or no
s e ]s(h),n[ satisfies the required conditions. In the second case we set
s(h+1
n.
We
index s
.
show now that 58(h-I)
.> s(i)
such that E(
>. I. For
s(i)<.t<_s
f)2
each
28o
easily if A
]s’(i), s(i+1)[ F, then if B
card B. -< 2H
card A.. For each i, we have
2.
<
h, let s’(1) be the largest
The definition of s(i+1) implies
]s’(i), s(i+1)[
\
Ai,
we have
197
ASYMPTOTIC NORMALITY FOR M-DEPENDENT U-STATISTICS
Ef
f)2 .< E(
[
E(
2
s’Ci)
Ef
2
s’Ci)+1
2
.
[
A.
f)
Ef2
2
cB.
Ef
It follows that
2
<-2he
h
2
(E
i-2
f2s(i)-1
h
Ef2s(i)
[
i=I
(Ef2s’(i)
Ef
2
s’(i)=1
ieF
F it follows easily
Since s(i)-1, s(i)
o
h
But
card B.
i=I
-
2 <
2
2h8o
[
4
iF
1600hL2o 2
Ef
h
2
[
card B
i=I
card F, from (4-6) we got
2Hh
800L
+
400L2card
F
.
2
io
o2/20,
so we get finally
o
2
602/20
ho2(20
Since H
>. 7, We get 7o/10 <. 3ho 2. So h
5e(h-1)
->
1600L
+
1600HL 2)
2
10-28
7/30. Since
I, we have h
20, so
1.
<.
For
<. h-l,
let
0-3
a
eJ
where A
.<
{(j,k):
j
<
k
,
]s(i), s(i+l)[. Let
Ji
<-
:lr.l
o-p
,:,.
(J ,k)eA
lj.l
p
n, j or k belongs to J }"
It is easy to see that
[
a. S L
[
and
b
S
Si &h-1
.<i-<h-1
2Mp..
.< <. h-1 for which
It follows that there are at least 19(h-I)/20 indices
-I
-I
a <. 40(h-I) L and b S 80(h-I)
Since (h-l) >. 19, it is possible to find ten
Mp.
I+10 with thls property. The lemma follows by
consecutive indices i+I, i+2
10
letting
8.
I Ji+
for
<- 10 and I
exp(it(o
S+A
}E
BOUND FOR
I))I
and
U
=1
IE
I
A2exp(it(o
We shall bound the above quantities when I is chosen as in the preceding
Let
paragraph.
(1+K21tl)Max(exp(-t2e2/80), (2400tK2L)I/4 log(L-tel/2/2400))
(t,e)
>. -I K 2
for t
that
[Sl, s 2]
S’
o
and
Indeed, if
I
otherwise. We first show that if
contains one of the intervals
-I
s is
(t,)
fi’
we have
Elexp(itS’)
2
(s’, s"), we have
IE
(2
Sl,
s
I are such
2
E < 9) then, if
< (t,) whenever
24001tlK2L
<
I.
N.T. RHEE
198
E(S’)
2
E(
s. <i<s’
2 Ef
Moreover o -3
fi.)
fs
s’-1
2
:
E(
ieI
fi)
Efs,,f s "+I
2
2
(
,"<
28o
: <s fi)
2
2
1600o2L 2
2
8o
2
Elfi 13 .< 24008L, so the result by (4-5).
If
ItlK6 L <- I, then
EIA 2 exp(it(o -I S At)) < K5MI((t,O) (8OiL)H).
[
s <.i <s"
LEMMA 8-I.
PROOF.
<
Let us fix j
k. Then, among the intervals
J1’ J2’ J3 which do not
< H in
J1 \ F such that
possible to find three consecutive intervals
j or k.
for
<
<
<- H-I
<-H-I.
-<
<.
q(),
and indices
-I
<-H,
"
i<p(1)
and for
J3
\ F’such that
q(+I)
o
fi
-I
i>q(1)
A
fi
o
<. H-I,
o
Z2_I
-I
-I
.
fq())
(fp()
o
f
-I
[
f
q(+1)<i<q(t)
p()<i<p(+1)
i"
Let
S
,o -I S
g
I)
[
Y
and
Zi
exp(itZ)
I.
We have
Yj,kexp(ito
-I S)-=
Yj,k exp(itZl) exp(itS2)
2H-I
Yj
k
7. exp(itZ1
=2
[[
q=2
YqeXp(ItSz+1
2H
Yj ,kexp (itz )q.2YqeXp(ItS2H)
So, by using the same type of majoratlons as in section 6, and since
lexp(it S) <- (t,8) for 2 & <- 2H by the preceding remarks, we get
E
-I
Yj,kexp(ito S)I
4EIYj,kl(t,8) + EIYj,kI(8OtL)
and the lemma follows by summation (with K
6
2400K
H
K4).
2
A comparable but simpler proof yields the following.
LEMMA 8-2.
If
ItlK6L
<. I,
then
Elexp(it(o -I S+AI))
9.
K5((t,8)
+
(80tEN)).
PROOF OF THEOREM A.
Since we can suppose K
(6-I) shows that for
6
>_
2400K2,
0, I, 2,
->
p()+2
q(+I) < q()-2 for
[et
Z2
<
< H in
it is
contain either
Let
ZI
and for
<
So we can pick indices p(),
Ii0,
11,
straightforward computation from
199
ASYMPTOTIC NORMALITY FOR M-DEPENDENT U-STATISTICS
I
e
Itli(t)dt < K?.
12
(9-I)
<
,ItlK6L
Let us denote J(T) the integral in the right hand Ride (3-I). Let
T
Inf ((80000-1og L)
O
(e-4OK6L) -I)
I/2
It follows from (3-2), theorem 4-4, lemmas 5-I and 6-I, and 9-I that
(since K
> 40)
6
J(T
For T O
<-
t
-<
(e
<.
O)
-40K6
L
K8(L
-I
(log
10
-2
-2
log
-I
IL I0).
24002L 5/6)
and it is indeed possible to.assume
(24000)
ince we can take K
NL I/3 +M
(if such t exists) we let
Max (8000t
The first term is
ML 2/3
L-I)p/2Mp
2
e
< 10 -2 for otherwise
and theorem A will be automatically satisfied.
3HL.
Hence choose I as in section 6 and use the estimates (3-3)
Notice that for this choice of e, straightforward
8-2.
and lemmas 5-I, 8-I and
Moreover
e >- 4.10
<-
computation gives (t,O)
K8L4.
It then follows that
J((exp(-40)K6L)-1)
J(T O) <.
K9(L
MIL3
M
P
L5/6-P).
If we put all these estimates together we get theorem A, with the bound as stated,
but where the quantities with a "dash" are replaced by corresponding quantities
without a dash.
However, since for
.<
r < p we have
Elgj,k
lemma 4-I shows that the "undahed" quantities are bounded by a universal constant
This concludes the proof of theorem A.
In order to get the extensions of theorem A mentioned as remarks after the
5/6
statement of this theorem one replaces e -40 by e-(8q+2) in the choice of T O and L
times the "da.hed" ones.
by L
Y
in the choice of
.
REFERENCES
I.
HELMERS, R. and Van Zwet, W.R. The Berry-Esseen bound for U-statlstlcs,
preprint.
2.
SHERGIN, V.V. On the convergent rate in the central limit theorem for
m-dependent random variables, Theor. Prob. appl. XXIV (1979) 782-796.
3.
RHEE, W. On the charaterlstlc function of a sum of m-dependent random
variables, to appear.
FELLER, W. An introduction to probability theor[ and its applications,
vol. 2 (1971) Wiley, New York.
4.
5.
CHATTERJI, S.D. An
LP-convergence
theorm, Ann. Math. Statist. 40 (1969)
1068-1070.
6.
7.
CALLAERT, H. and JANSSEN, P. The Berry-Esseen theorem for U-statlstlcs,
Ann. Statist. 6 (1978) 417-421.
SERFLING, R.J. Approximation theorems of mathemetical statistics (1981)
Wiley, New York.