WEAK CONVERGENCE OF GENERALIZED U-STATISTICS
By
Pranab Kumar Sen
Department of Biostatistics
University of North Carolina, Chapel Hill, N. C.
Institute of Statistics Mimeo Series No. 837
AUGUST 1972
WEAK CONVERGENCE OF GENERALIZED U-STATISTICS
I
By PRANAB KUMAR SEN
University of North Carolina, Chapel Hill
Wichura (1969) has studied an invariance principle for partial sums of a
multi-dimensional array of independent random variables.
It is shown that a
similar invariance principle holds for a broad class of generalized U-Statistics
for which the different terms in the partial sums are not independent.
Weak
convergence of generalized U-statistics for random sample sizes is also studied.
The case of (generalized) von Mises' functionals is treated briefly.
AMS 1970 Subject Classification:
Key Words and Phrases:
DC[O,l] space, Generalized U-statistics, invariance principle, Gaussian
processes, relative compactness, random indices and weak convergence.
1)
Work supported by the Army Research Office, Durham.
TntToonction.
1.
Let {X .. , i>l},j=l, .•. ,c, be c(_>2) independent sequences of
~
J1.-
independent random vectors (irv), where X.. has a distribution function (df) F.(x),
J1.
J
XERP , the p(~l)-dimensional Euclidean space, for j=l, •.• ,c.
mj,j=l, ... ,c) be a Borel-measurable kernel of degree
assume (without any loss of generality) that g(
~
Let g(Xji,i=l, ... ,
= (ml, ••• ,m )' where we
c
) is symmetric in the m.(>l)
J -
arguments (vectors) of the jth set, for j=l, ••• ,c.
Let m =ml+ ••• +m , F=(Fl, ••• ,F ),
o
c
-
c
and consider the regular functional of F.
m.
6(F) = f'p~ fg(xll"",x cm )rrj~lrri=I dFj(X ji )
(1.1)
ROC
defined on F = {F: 16(F)1 < oo}.
For a set of samples of sizes
U-statistic for
(1. 2)
~=(nl, •••
,nc) with
nj~mj' l~j~c,
the generalized
6(F) is defined by
c n j -1
!!l
j - ( m. )
J
U(n)
_
.
I(*n )g(x.Ja , a-i.
,.··
,1..
,
Jl
Jm.
-
l~2.c) ,
J
* extends over all l<i.l<
where the summation I(n)
.•• <i. <n., l_<j_<c.
- J
Jmj- J
For various
properties of U(n), including its asymptotic normality, we may refer to Fraser
(1957) and Puri and Sen (1971), among others.
For the asymptotic normality, it is
assumed that 6(F) is stationary of order zero and essentially
(1.3)
where n=nl+ ••• +n '
c
lim
n
~
n./n=A.: O<A.<l, j=l, ... ,c,
J
J
J
Weak convergence of a stochastic process derived from the tail
sequence of one-sample U-statistics to a Wiener process has been studied by Loynes
(1970), while Miller and Sen (1972) consider a Donsker-type invariance principle
for one-sample U-statistics.
They show that a process derived from {U([kA ] , ••• ,
l
[kA ]), k2.n} converges weakly to a one-dimensional Wiener process.
c
The more
general and natural case of a c-dimensional time-parameter where we use the entire
set {U(k): ~<~2.~} (or the entire tail set {U(k): ~~~}) is considered here, and it
2
is shown that weak convergence to appropriate multi-dimensional Gaussian processes
hold under no extra regularity conditions; here
It may be noted that for
c~2,
~<~
means that
ai~bi
for all l<i<c.
the ordering of n is not defined, and as a result,
the treatment of Loynes (1970) or of Miller and Sen (1972), resting on the reverse
martingale property of U-statistics, does not work out.
Also, as is usually the
case with generalized U-statistics, U(n) in (1.2) involves a set of summands which
are not all stochastically independent.
Thus, Theorem 2 (or its Corollary 1) or
Wichura (1969) does not lead us to the desired result.
The task is accomplished
here by first extending Theorem 1 of Whichura (1969) to more general summands,
and then using a decomposition of U(n) which fits into this extension.
The preliminary notions and the basic theorems are considered in section 2.
The proofs of the theorems are presented in section 3.
The case of generalized
von Mises' (1947) functionals is treated in section 4.
In the last section, the
case of random sample sizes is also considered.
The results are useful in the
developing area of sequential inference based on generalized U-statistics.
For every d.: O<d.<m., l<j_<c, let
J
(2.1)
gd
- J- J
d (x· i ' i=l, ..• ,d j , l~j~c) =
J
1"· c
E{g(x,l,···,X'd ' X' d +l' •• ·'X, , l.:::J~c)},
J
J j
J j
Jm j
so that gOO .•• O
8(F) and g
mI·· .mc
( ) = g(
).
Let then
(2.2)
so that ~O .•• O(!)=O.
(2.3)
We assume that (i)
cr~ = m~ ~ 0
J
J
°
(F) > 0 for every 1~j~c,
j l " · jc ~
(where 0ab=l or 0 according as a=b or not), and (ii)
3
(2.4)
Later on, we shall see that (2.3) may be replaced by max l2j~c cr~>o, and the
J
neccessary modifications are trivial.
We know that for n.>m.,
J- J
l2j~c,
under (1.3),
(2.3) and (2.4),
(2.5)
Let now E =[O,l]c be the c-dimensional unit cube in RC, t=(tl, ••• ,t )
c
EE ' and let
c
c
~
[~E]
= ([nltl], ••• ,[nct c ]) where [s] denotes the largest integer
~s.
Then, for every n(>m), we define a process W(n)={W(t;n): tEE}
by letting
c
~
(2.6)
-~
~
W(t,n) =
~
~([~:];n)[U([nt])-e(F)],
~
~
[nt]>~,
= 0, otherwise,
where for k=(kl, •.• ,k ), k.>O, j=l, ••. ,c,
c
~
~(k,n)
(2.7)
so that
~
~(k,n)
J
= n
-~ \
c
~
\ c
J=
J J
J=
~ -1 -1
(L. lcr.A.)(L. lcr,A.k.)
J J J
is n-~ times a harmonic mean of kl, ..• ,k .
c
,
Consider now c indepen-
dent copies of a standard Brownian motion on [0,1] and denote these by
c
Finally, the space n [0,1] of all real functions
on E with no discontinuities of the second kind and the associated (extended)
c
Skorokhod Jl-topology are defined as in Neuhaus (1971).
Then, our first theorem
of the paper is the following.
THEOREM 2.1.
Under (1.3), (2.3) and (2.4),
W(~)
converges in law in the extended
Skorokhod Jl-topology on nC[O,l] to a Gaussian function W~{W(t): tEE }, where
~
c
4
~
( I..
W(t) _
(2.8)
~
a. /-. .) (
j=l J J
{
~
I..
-~ -1 -1
a. /-.. t.)
j=l J J
~
-~ -1__
a. /-.. t. -Wj (t . ) ], t>O}
[I..
J
j=l J J
J
J
0, with probability 1, if t.=O for some j:
J
We define a process W* (n) = {W* (t;n); tEE} as follows.
l~j~c.
Considering the
c
tail set {U(k): k>n}, let
~-~
*
(2.9)
W (t;n) = r
(2.10)
W
-1
e(F)],
(n)[U([n/t])
tEE ,
c
~
\ c 2
= (wl' ... ,w ) ... ; w. = a./-..-~ (I...
la. 1/-..) -~ , l<j~c;
c
J
J J
J= J
J
W(t) = (Wl(tl), •.• ,Wc (t c »
(2.11)
~
~
.. ,
tEE,
c
~
where the W.(t) are defined earlier, and let
J
W* = {W* (t): tEE}
(2.12)
c
W* (t)
=
w"'W(t), tEE.
c
Then, our second theorem of the paper is the following.
THEOREM 2.2.
Under (1.3), (2.3) and (2.4), as
* (~)
W
(2.13)
V
+
n~,
*
W , in the Skorokhod J -topo1ogy on Dc [0,1].
l
Theorems 2.1 and 2.2 provide multi-sample extensions of the weak convergence
results of Miller and Sen (1972) and Loynes (1970), respectively.
Related results
on von Mises' (1947) functionals are considered in section 4.
For simplicity of the proofs, we explicitly consider
the case of c=2; an essentially same but more laborious proof holds for general
c(~2).
First, we consider certain basic lemmas needed in the subsequent steps
of the proof.
Let B(j) be the a-field generated by {X.l, ••• ,X. } for
In
n
J
n~l, l~j<c; B(12)
nl n 2
5
denote the product a-field B(l) x B (Z) for
n
n
1
Z
n1~1, nZ~l.
Sii~
Further, let
=
S(X 11 ,···,X1i , XZ1' ••• 'XZi~)' for i,i~>l, be a sequence of random variables such
that
E(S ..
(3.1)
1J
IB~~:»=
S.~.
a.e. (almost everywhere)
1 J
1 J
for every i>i~>l and j~l, and for every j~~~l, n1~1,
(3.Z)
a. e. ,
where s(j) =
Finally, assume that for every
~k
E(S .. ) = 0,
(3.3)
1J
LEMMA 3.1.
a:.1J
= E(S:.) <
1J
i~l, j~l,
00.
Under (3.1), (3.Z) and (3.3), for every n >1, n >1,
2
1
(3.4)
Since Doob's (1953, p.317) inequality holds for non-negative sub-martingales,
the proof of the lemma follows precisely on the same line as in the proof of
theorem 1 of Wichura (1969), and hence, the details are omitted.
The extension
of (3.4) for a general c(>Z) is immediate; we need to replace 16 by 4
c
and a 2
n n
1 Z
by E[S2(X 11 ' ••• 'X 1 n , ••• ,X c l' ••• 'X cn )].
1
c
Consider now a two-sample U-statistic U
and denote by r 2 (n ,n )=Var(U
);
n n
1 Z
n n
1
Z
Z
1
note that by (Z.5), r 2 (n ,n ) is non-increasing in each of n and n •
Z
1
1 Z
LEMMA 3.Z.
For every N.>n.>m.(>l), j z 1,Z,
(3.5)
E[(n
-
J- J- J -
1
r- (n 1 ,n Z)\u k -UkN -UN q+U N N 1)2]
1- - 1 nZ_~ Z
q
Z 1
1 Z
~~N
~a~N
~ 16r- z (n 1 ,n Z) [r2(n1,nZ)-r2(n1,Nz)-r2(N1,nZ)+r2(N1,Nz)]
6
Proof.
For every
r~l,
we let
s~l,
(3.6)
(3.7)
Let, now C(j) be the a-field generated by the unordered collection {X. ' ••• ,X. }
n
J1
In
._
(12)
C(2) for
and by X.J n+l'X.J n+2' ••• ' J-1,2, and Cn n be the product a-field C(l)x
n
n
1 2
1
2
n1~1,n2>m2. It follows by some standard arguments that E(UkqICk~q~) = Uk~q~ a.e.
for every k<k~ and q~q~.
*
= s. ~.
E(s .. lc.~.)
1J 1 J
1 J
(3.8)
E(S(j)
(3.9)
a. e. ,
.
_1,
i~<
s (j~)
.> ,
IC*
.~) =
a. e., J_J
N1-n 1,J
~N1-n1
.~
~N1-n1
where
k~l,
Consequently, it follows by some routine steps that
C* - C(12)
1 k
1
(j)
kq - N -k N -q' ~~N1-n1' ~~N2-n2' and ~k
= ( Slj,···,Skj ) ~ , for
1
2
j~l.
Further, by (3.7), E(Sij) = 0 for all i~l, j~l. Finally, C*kq is t
in k and q.
Consequently, the same proof as in Lemma 3.1 holds, and (3.5) follows
by noting that by (3.7),
as r 2 (i,j) is
+ in
i and j.
We further note that {U
Q.E.D.
kN 2
' C
; nl<k~N1} has the reverse martingale prokN 2
perty, so that by reversing the index set, we obtain that
7
(3.11)
(3.12)
(3.13)
uniformly in
Nl~nl, N2~n2'
Hence, by Lemma 3.2, (3.11), (3.12), (3.13), the
c - inequality and the Chebycher inequality, we obtain that for every E>O, there
r
max
I>r(n ,n )K }< E,
P{n __
<k<N 1 n _maxI
<~
<N 2 Uk q -8(F)
~
l 2 E
2
l
(3.14)
and hence,
I>r (n)K
} < E.
>n IUk -8 (F)
~
~
E
sUP
(3.15)
P{k
We now consider a typical decomposition for U(n), defined in (1.2), when c=2;
the case of c>2 follows trivially on the same line.
and for
Let us write
~>~,
(3.16)
*
kl
k2
d l +d 2 k l k 2
Uk(~) = Ld =OL =0 (-1)
(d )(d )U d d (n); Q<k<m,
d2
1
1
2
1 2 ~
~-~-~
(3.17)
Ud d (n)
1 2 ~
l<i<d., 1_<j_<2) ,
--J
*
UOO(~)
= 8(F)
8
* extends over all l<ajl< ••. <ajd,<mj' j=l,Z.
where the summation L(n)
-
Then, by
J
(l.Z), (3.16), (3.17) and a few routine steps, we obtain that
(3.18)
* (~)
where each Uk
(O<k<m) is a generalized U-statistic.
"V-"V-"V
A little readjustments
lead us to
(3.19)
where
(3.Z0)
n l -1
Ul(n l )
(3.Zl)
UZ(n Z)
and the summation \(*
L
n.
()
ml
=
(
nZ
)
mZ
-1
L(*n
l
)[g O(X l , , ... ,X l ,
m I ll
1
L(*n ) [gam Z(X Z1,
Z
, ••• ,XZ'
- 1
1
1
) extends over all l<i <.•• <i
J
)-e(F)],
ml
<n"
m.- J
J
)-e(F)]
mz
-
j=l,Z.
We first consider the proof of Theorem Z.Z which is relatively simpler.
write [by(3.l9)]
(3.ZZ)
and note that
(3. Z3)
*
U3(~)
is a generalized U-statistic for which
*
[r * (~)]2 = E[U3(~)]2
r2(~)
- E[U (n )]2 - E[U (n )]2
l l
z z
We
9
Thus, from (2.5) and (3.23), we note that under (1.3),
(3.24)
*
and hence, using (3.15) for {U3(~)'
~~~} and (3.24), we conclude that for every
E>O and n>O, there exists an nO [=nO(E,n)], such that for n>n '
O
(3.25)
---
Therefore, under (1.3), (2.3) and (2.4), as
n~,
(3.26)
Let us now define
-1 ~.
(3.27
(3.28)
*
W.(t,n)
J
j
w.(n) = r
J -
=
cr. n:u.([n./t]),
J J J
J
{
-1
0,
O<t~l,
t=O, for j=1,2;
-~.
(n)cr.n. , J=1,2.
- J J
Then, from (2.9), (3.26), (3.27) and (3.28), we obtain that
2
(3.29)
sup IW* (t;n)- L\' w.(n)W.(t.,n.)
*
E2
tE
.
-1
J
J J J
J-
I ~ ° as
n~.
Now, by (1.3), (2.5), (2.10) and (3.28), we obtain that
(3.30)
lim
n~
w.(n) = w. for j=1,2.
J -
J
* j)
Also, Wj(n
*J J J O<t<l},
{W.(t.n.),
j=1,2, are stochastically independent, and by
-the results of Loynes (1970), as n~,
(3.31)
W~(n.)
J
J
eW
j
= {W.(t):
J
O<t~l},
j=1,2
in the Skorokhod J -topo1ogy on D[O,l], where WI and W are independent copies
2
1
of a standard Brownian motion on [0,1].
Finally, (3.31) implies that
10
sUPO<t<l IWj(t;n j )
I,
j=1,2, are measurable and
(3.32)
Consequently, (2.13) follows from (3.29), (3.30), (3.31) and (3.32).
Q.E.D.
To prove (2.8) [i.e., Theorem 2.1], we note that by using (3.23) and some
standard inequalities among the {~dld2 (!): ~~~~~},
(3.33)
where C(F) <
(whenever (2.4) holds) and it does not depend on (n ,n ).
l 2
show first that under (1.3), (2.3) and (2.4), as n~,
(3.34)
00
~
----
We shall
o.
......
For this, we partition the set A(n) = {k: m<k<n} into three disjoint subsets:
----
.......
(3.35)
min
<
~}
Al(~) = {kEA(n) : 1~j~2 kj_Eln ,
(3.36)
A2(~)
(3.37)
A3(~) = {~EA(~):
1
min k. < [n4/5]},
{kEA(n): E n~ <
1
1<j~2
J -
min
1~j~2
k
j
> [n 4 / 5 ]},
where £1(>0) is arbitrarily small and will be chosen later on.
(3.38)
so that by (1.3), 0<0<1.
Then, by (2.7) and (3.38),
(3.39)
Consequently, by (3.35) and (3.39),
Let then
11
(3.40)
max
* I
~£Al(~)
¢(~,n) IU3(~)
-1
{max
~ 0 £1 ~£Al(~)
I U3(~)
*
I}
-1
*
{U3(~)'
where by (3.15) and (3.33), we obtain for
{suPI *
~ 0 £1 ~~~ U3(~)
~~
}
I} ,
SU P \U3(~)
*
that k>m
--
I = 0p(l),
so that (3.40) can be made arbitrarily small (in probability) by letting £1 (>0)
arbitrarily small.
*
~
Again, for ~£A2(~)' r ([£ln~], [£In ])
1
~ -1
= O([£ln]
),
by (3.23) and (3.33).
max (n)o/'''(k
< u~-ln3/l0.
On teat
her h an,
d by (3 • 39) 'k£A
h
_,n ) _< n-~~-1[n4/5]
u
- 2-
Thus, for
(3.41)
~
0-l{ O(l/(£ln1/5 )) }
+
0 as
~.
On the other hand, by (3.15), (3.23) and (3.33),
(3.42)
Consequently, by (3.41) and (3.42), we have for n+oo,
(3.43)
Finally, for
max
I*
~£A2(~)¢(~'~) U3(~)
I = 0p(l).
~£A3(~)' ¢(k,n) ~ n-~ max (kl ,k 2) ~ n~, while r*([n 4 / 5 ],[n4 / 5 ])
O(n 4 / 5 ), so that
(3.44 )
max
*
~£A3 (~) ¢(~,n).r ([n
4/5
], [n
4/5
])
= O(n-3/lO).
On the other hand, by (3.15), (3.23) and (3.33), as n+oo,
(3.45)
~:: (nJr* ([ n 4/ 5 ] , [n 4/ 5 ] ) ] -1 ~
u; (j~) I
12
<
sup
-k . >n 4/5 ,
op (1).
J-
Thus, from (3.44) and (3.45), we have for n-+oo,
(3.46)
o (1),
p
and, as a result, (3.34) holds.
On D[O,l], we now consider independent processes W.(n.)={W.(t,n.): O~t~l},
J J
J
J
j=1,2, where for O<t~l,
1
{
([n.t]/a.n~)u.([n.t]), m.<[n.t]<n.,
J
J J J
J
J- J - J
(3.47)
W.(t,n.)
J
Also, let for every
J
0, t < (m.-l)/n.; j=1,2.
J
J
E>~'
Wj(~'~)
--
(3.48)
-
=
~([nt];n){a.n~j/[n.t.]}
J
J J
~~
2
= n
= (
-~
2
\
~
\
(L. a.A.)( L.
j=l J J
~
-1
~
a.A./[n.t.])
(a.n./[n.t.])
j=l J J
J J
J J
J J
2
~
2
~
-1 ~
L
ajA.)( L a.A./[n.t.]) (A.a./[n.t.])
j=l
J
j=l J J
J J
J J
J J
[1+0(1)].
Then, for every n.t.>l, 1<j<2, w.(t,n) is positive and is bounded above from
J JJ ~ ~
2
~
(L'=la.A.)(n./nA.)
J
J J
J
J
(3.49)
~
= (
~
~
L a.Aj)[l+O(l)],
j=l J
j=1,2.
Also, for every:>9,
lim
~
~
~
-~ -1 -1
-~ -1
.
w.(t,n) = ( L. a.A.)( L. a.A. t.) (a.A. t. ), J=1,2.
n-+oo J ~ ~
j=l J J j=l J J J
J J J
Finally, it follows from Miller and Sen (1972) that Wl(n ) and W2 (n 2) are
l
stochastically independent, each converging in law to a standard Brownian motion.
Consequently, for every £>0 and n>O, there exist a 6>0 and an nO' such that for
(3.50)
p{0:~~6IW.(t;n.)1
> £} <~,
_ _
J
J
(3.51)
o:~~llwj(t;n.)
_ _
J I = 0 p (1), j=1,2.
j=1,2,
13
Thus, (2.8) follows from (2.6), (2.7), (3.22), (3.34), (3.47), (3.48), (3.49),
(3.50) and (3.51).
Q.E.D.
We define the empirical distri-
butions by
1
t:l.
F.(x;n.) = n~ lJ c(x-X .. ), XERP , n.>l, j=l, ••• ,c,
J
J
J i=l
J 1.
J-
(4.1)
where c(u)=l or
° according as all the p components of u are non-negative or at
least one negative; we let F(.,n) = (Fl(.,nl), •.• F (.,n).
-
-
c
Then, the von Mises
c
(1947) functional corresponding to (1.1) is
8 (F (. , n»
(4.2)
I···I
RpmO
Here, we define a process Wen)
c
g(xll,···,x
-
m.
).IT 1..U l J dF.(xj.;n.).
cmc J- I
J
1. J
{W(t;n); tEE} as in (2.6), where we replace
-
c
Similarly, on replacing {U(k); ~~~} by
-* (n)
{8(F(.,k); k>n} in (2.9), we define W
---
{W-* (t;n); tEE}.
_ c
Finally, we strengthen
(2.4) to
*
(4.3)
1; (F)
max
max
{ 2
}
·
1
E g (Xl
' ••• ,X
) <
1_<J_<c
<a
.1<·
.•
<a.
<m.
all
cacm
- J - Jm.- J
c
J
THEOREM 4.1.
Under (1.3), (2.3) and (4.3), as
00.
n~,
-*
~
Wen) and W (n) converge in law
in the extended Skorokhod Jl-topology on DC[O,l] to Wand
w*,
respectively, which
are defined in (2.8) and (2.12).
Proof.
(4.4)
(4.5)
Again, we consider the case of c=2, and for
=
I··d·f
RP
0
~<~~~,
2
define
d.
gd(xll,···,x ld ,x 2l ,···,x 2d ).lll1..llJl d[F.(x .. ,n.)-F.(x .. )],
_
1
2 J
J J 1. J
J J 1.
8(F(.,n»
14
where
(4.6)
(4.7)
Now, proceeding as in the proof of Lemma 2.6 of Miller and Sen (1972), we
have for every k>m"
-J
E[V,(k)-U,(k)]2 < C(F)k- 3 , for j=1,2,
(4.8)
J
J
where, under (4.3), C(F) <
-
00,
--
and U,(n,), j=1,2, are defined in (3.20) and (3.21).
Since [by (3.39)], ",o/(k,n) <n -~o-l k"
-
-
J
J
J
'_
J-1,2,
by (4.8),
p{ ~ka~ 1JJ(k,n) IU,(k.)-V.(k.)
~-
JJ
JJ
I
> £}
(4.9)
so that for every £>0, the right hand side of (4.9) converges to 0 as
n~.
Similarly, on noting that r2(~), defined by (2.5) is bounded below by
min 1<'< (cr.2/ n.), we obtain that as
_J_c
(4.10)
J
J
n~,
under (1.3), for every £>0, j=1,2,
15
Consequently, to prove the theorem, all we need to show is that under (1.3),
(2.3) and (4.3), for every d>l,
~-~
(4.11)
-----
(4.12)
where both £(>0) and n(>O) are arbitrarily small, and n is chosen adequately
large.
Note that
*
*
= Ull(~)
Vll(~)
for all
~~.
Also, by extending the proof of
Lemma 2.6 of Miller and Sen (1972), we have for every d>l[under (4.3)],
~-~
(4.13)
Consequently, if d >2, d >2, the proof for (4.11) and (4.12) follow
2
1
for k>m.
~--
trivially by using (4.13), the Chebychev and the Bonferroni inequalities.
we need to consider the case where min <o<2 dJo=l, but d +d >3.
l 2
l 2J_
Thus,
We consider
explicitly the case of d=(1,2); the case of (2,1) follows similarly, while for
any other d: 1=min(d ,d ) <
l 2
max(dl,d2)(~2),
a similar but more laborious proof
holds.
By direct evaluation from (3.16) and (4.4), we have
(4.14)
for every
(4.15)
~~,
where for
m2~2,
**
1
kl
k
U12 (k) = - k
1 2 i =1
I
1
Consequently, to prove (4.11) and (4.12) for d=(1,2), it suffices to show that
as
n~,
(4.16)
max I *
I
m<k<n U12 (~) =
-----
16
(4.17)
note that
Since
*
1jJ(~,n) /k
U1Z(~)
*
{U1Z(~)}
Z = O(n
-~
), V~~~ and r
-1-1
(~)kZ -+
0 as
n~,
for every k>n.
is a generalized U-statistic, the proof of (4.16) and (4.17) for
follows directly from (3.15) and the fact that as in (3.33),
E[U~Z(~)]2 ~ C(!)k~lk;Z
for every
~>~.
Now, we define the sequence of a-fields {C kq , k~l' ~Z} as in section 3
[following (3.7)].
Then it follows from (4.15) that for every k'>k and
**
E[Ulz(k,q)
(4.18)
Ie
It q'
q'~q,
** , ,q , ) a.e.
] = U1Z(k
Consequently, following the same method of approach as in the proof of Lemma 3.Z,
we obtain from Lemma 3.1 that for every N>n>m,
......---......
(4.19)
**
~ l6E[UlZ(~)]
2
~ l6C(!)n
-1
Thus, the proof of (4.16) and (4.17) for
and the Chebychev inequality.
-Z
l nZ •
**
{UlZ(~)}
follows immediately from (4.19)
Q.E.D.
We now consider the case where in (1.3)
we allow (Al, ••. ,A )'=A to be a stochastic vector with positive elements.
c
-
(1)
precisely, let {N = (N
-n
n
, •.• ,N
(c),
n
) , n>l
-
}
More
be a sequence of vectors with positive
integer valued random variables, such that
(5.1)
n
-IN P"\
-n -+ ~
where A , j=l, •.• ,c are positive random variables defined on the same probability
j
space as of the original' {X .. ,
J1
i~l},
j=l, ••. ,c.
17
We define W(N ) as in (2.6) when N > 1; otherwise, we let W(N ) = 0.
-n
-n - -n
Similarly, we define W*(N ) as in (2.9) when N > 1; otherwise, we let
-n
-n - W*(N ) = 0.
-n
Finally, we define W(N ) and W*(N ) as in section 4, with n being
-n
-n
-
replaced by N.
-n
Our basic problem is to study the weak convergence of W(N ),
-n
W*(N ), W(N ) and W*(N ), when
-n
-n
-n
THEOREM 5.1.
n~
and (5.1) holds.
Under (2.3), (2.4) and (5.1),
~t~h~e~e~x~t~en~d~e~d~S~k~o~r~o~k~h~o~d~Jl-topology
W(~n)
and
W*(~n)
converge in law in
c
on D [0,1] !£ W and W*, respectively, defined
in (2.8) and (2.12), while under (2.3), (4.3) and (5.1), ~(N ) and W*(N ) weakly
--
-n
--
-n
converge to Wand W* respectively.
Proof.
We shall only consider the case of W(N ) as the other cases follow
-n
similarly.
where
°
=
For k(>l), let {t = (t (1) , ... ,t (c)) "
- -
q
t6j)<tij)< .•.
q
Q _< q _< !c} be the set of points
q--
<t~~)~i, l<~~C. The~,
by Theorem 2.1, we have for every
J
~~
and !q' ~<~~, as n~, under (1.3), (2.3) and (2.4),
(5.2)
Also, by Lemma 3.2, (3.11) and (3.12), it follows that for every positive £ and
n,
there exists a 0>0, such that under (1.3), as
n~,
(5.3)
where Ixl denotes maxI'
~J~c
-
Ix.l·
J
In fact, (5.3) readily extends to the set of
k-\n)lu([Nt ])-U([nt ])1: IN-nl<on, O<~k}.
-
--q
--q
- -
--- -
Thus, by (5.2), (5.3) and theorem 2
of Mogyorodi (1967), we conclude that under (5.1), (2.3) and (2.4), as
(5.4)
O<~k] ,
--...
-
n~,
18
for every k>l and arbitrary {! : O<~k}C: [O,l]c.
---
q
- - -
Hence, the convergence of
finite-dimensional laws is established.
To establish the relative compactness (or tightness) of W(N ), we define
-n
the uniform topology
(5.5)
p(x,y) = SUPtEE
-
Ix(!)-y(£)I,
c
and the modulus of continuity (where 0>0)
(5.6)
Now, using our Lemma 3.2, (3.11) and (3.12) in place of Theorem 1 of Wi chura
(1969) and thereby extending his (2a) to our statistics, we obtain by the same
technique as in his proof of Theorem 3 (on pp. 686-687) that as
n~,
under (1.3),
(2.3) and (2.4), for every E>O,
(5.7)
Consider now a sequence of generalized U-statistics
(5.8)
=
0, if t.<k(j) In. for some
J
n
J
l~j~c,
**
where g*(xll, ••. ,xcm ) = g(xll, •.. ,xcm )-8(E), the summation L(~!) extends over
(")
c
c
all k J +l<a.l< ••. <a. <[n.t.], l<j<c, and
n
- J
Jm.J
J J
- -
19
= 00 but lim
n-~ k(j) =
n-7<Xl
n
(5.9)
Then, on replacing
U([~!])-e(f)
in (2.6) by
U'([~!]), !E~c'
we define a parallel
process W'(n) = {W'(t;n): tEE }.
-
LEMMA 5.2.
- -
c
Under (1.3), (2.3) and (2.4), as n-7<Xl,
p(W(~),W'(~»
(5.10)
Proof.
-
+
0, almost surely.
As before, we consider only the case of c=2.
_-_-_-_n _
For O<t<m<k , k-k-n--_
>m-t,
_
we define
where the summation f*(!c) extends over all l<a'l< •.. <a,n <k(j)<a..
<... <
_
- J
JN - n
J1
.+
t
l
j
a .. <k., 1<j<2. Thus Un(k,k ) may be interpreted as a generalizea U-statistic
J1 - J
~ - -n
I1J
basea on four samples of sizes (k(l) ,kl-k(l) ,k(2) ,k -k(2». Consequently, as
n
n
n
2 n
in our Lemma 3.2, (3.11), (3.12), (3.13), (3.14) and (3.15), it can be shown
that for every E>O, there exist a positive K «00) and an n (E), such that for
E
0
n>n
(E), under (1.3) and (2.4),
-0
(5.12)
Also, by (1.2), (5.8) and (5.11), we obtain that for
[~!] ~ ~n'
2
m k -k(j) k(j) k
j
j
n
n
j
1
II1{()(
)( Nn. )(m.)- }Un([~!],~n)'
.
t.
m.-t.
N
t=O J=
J
J J
J
J
m
(5.13)
U([~!])-e(f)
!
where .we let k.=[n.t.], j=1,2.
J
J J
Consequently, by (5.12) and (5.13), as n-7<Xl,
20
s~p km~~<n{n-~[min(kl,k2)]IU(~)-U'(~)I}
(5.14)
-Il----
Thus, by (2.6), (2.7), (3.39), (5.5) and (5.14), as
(5.15)
sup
IW(t;n) - W'(t;n)
- n -1 k <t<l
I+
n~,
0, almost surely.
-n----
On the other hand, for
t£E~n)
= Ec-{t:
that by (2.6), (2.7) and (3.39), as
sup
(5.16)
tEE
(n)
c
=
n-l~~!~l}, W'(t;~)=O
[by (5.8)], so
n~,
IW(t·n)-W'(t·n) I
-'-
sup
(n)
tEE
- c
-'-
IW(t.n)!
-'-
--= 0(1)·0(1) = 0(1), almost surely, by (5.9) and (5.12).
Hence, (5.10) follows from (5.15) and (5.16).
Q.E.D.
We recall that the random vector A in (5.1) and the X.. ,
J1
j~l,
i~l,
are
all defined on a common probability space, say, (Q,A,p).
LEMMA 5.3.
If A£A, then for every £>0 and n>O, there exists a 0(>0), sufficiently
small, such that
21
lim sup
n P{W (W'(g)) > E IA} <
(5.17)
8
n.
By (5.7) and Lemma 5.2, for every E>O,
Proof.
Also, by definition in (5.8), W'(n) depends only on the set {X .. , k(j)<i<n.,
-
J1
n
-
J
Hence, using (5.9), Renyi's (1958) idea of mixing sequence of
j=l, ... ,c}.
sets and proceeding as in Lemma 3 of Blum, Hanson and Rosenblatt (1963), it
follows that (5.18) implies (5.17).
Q.E.D.
We return now to the proof of Theorem 5.1.
By virtue of (5.1), for every
8'>0,
(5.19)
Also,
~
is a vector of positive random variables, so that for every n>O, there
exists an a (n), such that
o
(5.20)
Then, for every E>O,
(5.21)
P{p(W(N ),W'(N ))>E}
-n
-n
< p{ln-lN -A.I>8'} + p{lm~n
-
-n -
~J~c
-1
A..<a (n)} +
J-
0
P{p(W(N ),W'(N ))>E, In Nn-~1~8',
-n
-n--
min
l~j~c
A..>a (n)}·
J
0
22
Thus, if we let
O<o'<~
a (n), then by (5.19), (5.20) and Lemma 5.2, we conclude
o
that (5.21) can be made arbitrarily small by choosing n(>O) small and letting
n~.
Hence, to establish the relative compactness of {W(N )}, it suffices to
-n
show that
for every E>O.
> E} + 0 as 0~0,
lim sup p{ws(W'(N »
(5.22)
n
u
-n
We note that by (1.2), (2.6), (2.7), (5.8), (5.13) and (5.18),
for every E>O and n>O, there exists a 0>0 and an n (E,n), such that for
o
n -> n 0 (E,n),
(5.23)
Hence, using the inequality that
(5.24)
it suffices to show that
(5.25)
lim sup . {
n P wo(W'([n~]»
(5.26)
p{p(W'(N ),W'([nA]»
-n
-
}
> E +0 as
> E}+O as
0~0,
n~.
Consider then a multiple array of events
(5.27)
{~:
-
a (n)+h.o' < A. < a (n) + (h.+l)o', l<j<c},
0
J
J -
J
0
- -
and let
(5.28)
(a (n) +
o
(h.~~)o',
J
l<j<c) , h>O.
- -
---
Then, for every E>O, 0>0, by (5.20), (5.24) and (5.27),
~_>~,
---
23
(S.29)
A.<a (n) }
+ 'p{w~(W'([nA]))>£, A.>a (n), l<j_<c}
min
< P {1<.<
-
_J_c
J-
0
U
J-
-
0
~
~ n/3
+
L
p{A(h)}p{W8,(W'([n~]))>£IA(h)}
h=O
~
<n/3 +
L
h=O
P{A(!:) }p{p (W' ([n~]), W' ([ne(h)])) > ¥
1A(h)}
00
+
L
h=O
p{A(h)}p{W8 (W'([n e (n)])) > ¥IA(h)},
where we let p(BIA(n))=O when P{A(n)}=O.
Now, in the same say as (S.17) follows
from (S.18), it follows from (S.23) that for every £>0 and n>O, there exists a
8>0 and an no' such that for A£A
and n>n
,
-0
(S.30)
Since when A(h) holds, IA-a(h) 1<8', we obtain from (S.30) that for every £>0
and n>O, there exists a
n>n
(£,n),
-0
8'(0<8'<~a
•
o
(n)), and an n (£,n), such that for all
0
(S.3l)
Consequently, by Lemma S.2 and (S.3l), it follows that for n
(S.32)
which proves (S.2S).
P{W8(W'([n~]))
Finally,
> £} < n,
~
no(£,n),
24
(5.33)
p{p(W'(N ),W'([nA]»
-n
-
> E}
00
+
L P{A(~)}P{P(W'(~n),W'([n~]»>E,ln-l~n-~1 ~
o'IA(h)},
h=O
--
< P{ln-lN -AI> Of}
-
-n -
+ n/3 +
max
00
< n, by (5.19) and (5.30).
Hence the proof of the theorem is complete.
The theorem is of great value in studying the asymptotic distribution of
generalized U-statistics, useful in the context of sequential procedures based
on these statistics.
As an illustration of the uses of Theorems 2.1, 2.2, 4.1 and 5.1, we consider a simple case where p=l, c=2 and
00
(5.34)
and we assume that both F
=
l
p{X l · < X2 ·},
1 J
and F2 are continuous everywhere.
is the Wilcoxon rank statistic
I,
(5.35)
c(u,v) =
{ 0,
u>v
Then
e(F('~)
U(~)
25
so that Wichura's (1969) results do not hold.
If the two distributions F
l
and
F are mutually overlapping, then (2.3) holds, while (2.4) holds for all F , F ,
l
2
2
as c(u,v) is bounded.
results further simplify.
Theorem 5.1 for the Wilcoxon statistic is useful
for the problem of sequential testing and estimating e(F ,F ).
l 2
26
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[2]
DOOB, J. L. (1953).
[3]
FRASER, D. A. S. (1957).
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[4]
HOEFFDING, W. (1948). A class of statistics with asymptotically normal
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[5]
LOYNES, R. M. (1970). An invariance principle for reverse martingales.
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[6]
MILLER, R. G. JR., AND SEN, P. K. (1972). Weak convergence of U-statistics
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-41.
[7]
MISES, R. von. (1947). On the asymptotic distribution of differentiable
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MOGYORODI, J. (1967). Limit distributions for sequences of random
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[9]
NEUHAUS, G. (1971). On weak convergence of stochastic processes with
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Nonparametric Methods in Statistics.
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[10]
PURl, M. L., AND SEN, P. K. (1971).
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RENYI, A. (1958). On mixing sequence of sets.
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[12]
WI CHURA , M. J. (1969). Inequalities with applications to the weak convergence of random processes with multi-dimensional time parameters.
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