INVARIANCE PRINCIPLES FOR U-STATISTICS AND VON MISES' FUNCTIONALS
IN THE NON-I.D. CASE
by
Pranab Kumar Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1404
June 1982
INVARIANCE PRINCIPLES FOR U-STATISTICS AND VON MISES' FUNCTIONALS
IN THE NON-I.D. CASE *
By PRANAB KUMAR SEN
University of North CaroZina, ChapeZ BiZZ
For independent but non necessarily identically distributed random
vectors, weak as well as strong invariance principles for V-statistics and
von Mises' differentiable statistical functions are established under
appropriate regularity conditions. Martingale characterizations of some
decompositions of these statistics playa fundamental role in this context.
1. Introduction. Let {X. ,i>l} be a sequence of independent random vectors (r.v.)
-~-------------1
with distribution functions (d.f.) {F.,i>l} , all belonging to a common class
1
-
of d.f.'s. Let g(xl, ••• ,x ) be a Borel-measurable function (kerneZ) of degree m
m
(~
1), and, without any loss of generality, we may assume that g(.) is a symmetric
function of its m arguments (vectors). For n
g(X. ,. • .,X. )
(1. 1)
1
1
1
~
m, a statistic of the form
(where C
={l<il< ••• <i < n} )
n,m
In-
m
is termed a U-statistic.A closely related one, termed the von Mises' differen-
tiabZe statisticaZ function
(1.2)
V = n
n
-m
n
L.
1
=1.
1
I
is the following
n
L l· =1 g(X. , ••• ,X. ).
•
1
1
1
m
When all the F. are the same, i.e., the X. are identically distributed (i.d.) r.v.,
1
1
central limit theorems for these statistics have been elaboratedly studied by von
AMS Subject Classification : 60F17, 62E20
Key Words & Phrases: functional central limit theorem, martingale(-difference),
Skorokhod-Strassen embedding of Wiener process, tightness.
*
Work partially supported by the National Heart, Lung and Blood Institute,
Contract NIH-NHLBI-7l-2243-L from the National Institutes of Health.
Mises (1947) and Hoeffding (1948); asymptotic normality results in the non-i.d.
case were also considered by Hoeffding (1948). Functional central limit theorems
for these statistics have been studied by Loynes (1970) and Miller and Sen (1972),
among others, and, based on the Skorokhod-Strassen embedding of Wiener processes,
Sen (1974) has obtained a strong invariance principle for these statistics ; all
these results are confined to the i.d. case where a special decomposition of U ,
n
due to Hoeffding (1961), and the basic reverse martingale property of U-statistics
playa vital role.
In the non-i.d. case, neither the reverse matingale property of U nor the
n
martingale structure underlying the Hoeffding (1961) decomposition may hold, and
the simple proofs of the functional central limit theorems in Miller and Sen
(1972) or Sen (1974) may not workout
well. Nothing particularly has been done,
so far, on such invariance principles in the non-i.d. case (though the case of
stationary dependent sequence has received considerable attention during the
past few years). The basic purpose of this study is to incorporate a sequential
decomposition of U
n
sition which enable
(viz., Sen (1960)) along with the Hoeffding (1961) decompous to use some martingale-difference representations for
these statistics, and this provide an easy approach to the study of the desired
invariance principles, under no extra regularity conditions.
Along with the preliminary notions, the basic results on U-statistics are
stated in Section 2 while their derivations are presented in Section 3. Section
4 deals with the parallel results for von Mises' functions. For simplicity of
presentations, throughout the paper, the specific case of m=2 has been condidered.
The case of m=l is trivial, while, for m > 3, a very similar but more lengthy
treatment holds.
2. Preliminary notions and basic results for U-statistics. For every
(2.1)
-2-
i,j~l.
let
Then, by (1.1) and (2.1),
n -1
(2.2)
e(n) = EUn = (2)
.. , for every n > 2.
~l~i<j2? e 1)
--
Our first goal is to study weak invariance principles relating to the (partial)
~
_k
sequence {n \(U -6(k));2<k<n} and the tail-sequence {n (Uk
6(k)),k ~ n} ,
k
which would extend the results of Miller and Sen(1972) and Loynes(1970) to the
non-i.d. case. Also, we like to extend the results in Sen (1974) to the non-i.d.
case.
As in Hoeffding (1948), we let for every i( f j)
~
1,
e..
1)
(2.3)
1jJl ( 1)
. ) . (x.)
1 = Eg (x.1, X.)
) - 6..,
1)
1jJ2 ( 1)
.. ) (x.1,x.)
) = g (x.1,x.
))-
(2.4)
I'; (.
.) .
..
. = E1jJ (.
.) .
. (X. , ... , X. ).
C 1 ••• l
lc+l ••• 12;)c+l ••• J2
c l I .• ·1 C l c +l ••• 12 1 1
lC
C
1
1jJ (.
C 1
.) .
for all possible combinations of distinct i , j
r
and n( ~ m), the average over the possible
i
j
is denoted by
r' r '
E{U
(2.5)
6
n -
(n)
. (X. , •• OJ X. )
1 . · · l C J c + l ••• )2
r
1
l
1
C
and c=1,2. For a fixed c(=1,2)
I'; 's with 1 < i , j
c
-
r
< n and distinct
r--
I';c n • Then, from Hoeffding (1948), we obtain that
,
}2 = (n)-l ~2 (c) (n-2)
2
c=l 2 2-c I';c,n
n> 2.
The asymptotic normality of { Un - 6(n) } I { n -1 I';l,n }~
under some extra (mild)
regularity conditions, has been established by Hoeffding (1948). In particular,
the following conditions relate to the asymptotic normality result o Let
(2.6)
Assume that there exists a number A, such that for every
(2.7)
sup
i,j~l
J••• J g2(x l ,x 2)dF. (x)dF.(x) < A <
1
J
--
00
n~
2,
•
Further,
sup max
I/2+0
n l<i<n E 1jJl(i),n(X i )
(2.8)
<
00
and
(2.9)
lim { n
/- . ) (X. ) /2+0 I {~.n lE1jJl
- 2(. ) (X.)} 1+0/2} = 0,
~.1= IE 1jJl (1,n
1
1=
1,n 1
n~
where 0 (0 <
o~l
) is somepositive number; Hoeffding(1948) took
(2.9) insures that
nl';l
,n
goes to
00, as n
-3-
+
0 = 1.In particular,
00, though possibly in an arbitrary
=~l (x)
manner. In the i.d. case, ~l(i),n(x)
does not depend on (i,n) and ~l,n
~l is also independent of n. Hence, we may ,instead of (2.8)-(2.9), appeal to the
~l
classical Lindeberg condition, and this will only require that
> O. Note that
by (2.5), (2.6) and (2.9),
2
n E{U - 8(n)}
n
2
={4n ~l ,n HI
o(l)}
+
-2
}{
= 4{2::,n1= 1 E1/Jl(')
1,n (X,)
1
(2.10)
1 + 0(1) } •
The ingenuity of the Hoeffding (1948) approach lies in the (quadratic mean)
n
-
approximation of n[U - 8( )] by 2 L, 1 1/Jl(') (X,) and incorporating the central
n
n
1=
1,n 1
limit theorem for the triangular array {y , = ~l(') (X.), i=l, ••• ,n; n ~ I}.
n1
l.,n 1
For our desired invariance principles, we need to establish some maximal inequalities relating to { k[U k - 8(k)]-2
L~=lYki ;k~
2}
and
some invariance principles
k
relating to the triangular array { S k = L, lY ., k < n; n > 2}. Towards this,
n
1= nl.
we have the following.
lim {(log
Theorem 2.1. (I) If n-t<X>
while, if
(II) If
)} = 0, then, for every s >0, under (2.7),
Ik(U
-~
P{kmax
~ n ~l,k
(2.12)
,n
I~}
+
> sen ~l,n)
k - 8(k))- 2S kk
lim { - 1 Ln
-I} _
n-t<X> n
k=l (k~l , k)
- 0, then, as n
P{kmax
~ n
(2.11)
n)/Cn~l
Ik(U
k - 8(k)) -2S kk
I
s nk
0, as n
+
lim {(n~
)-1}= 0, then, for every s > 0, as n
n-t<X>
1, n
(2.13)
while, if
p{ sup
k > n
~::
{n
sup
k > n
I (Uk
sen
- 8 (k)) -
Lk>n(k3~1,k)-1
~~~kl
,
(2.14)
p{
(III) If
Lk >2 (k2~ k,l f1
(U
}
-1
Skk
I
,then, for every
00
~l
o , then, as n
- 8 (k)) - 2k
k
<
-1
>
00
k
+
k
,
}
,n )2
s n2
S
+
00
0,
,
}
O.
+
> 0, as n
+
00
-
(2.15)
p{
sup
k > n
I{k(U k _
8 (k)) -2\k} I/ (k~l , k)
k
2
> s }
,
O.
+
}
00
,
00
2
>
+
+
,
O.
The proof of the theorem is considered in Section 3. The maximal inequalities
in (2.11) - (2.15) along with proper construction of stochastic processes lead us
to the desired results. In the i.d. case, { Skk; k > 1 }
-4-
-1
(or { k Skk; k > 1 } )
4It
form a martingale (or reverse martingale) sequence, and hence, the forward ( or
backward) invariance principle for U-statistics, considered by Miller and Sen
(1972) and Loynes (1970), follows directly by an appeal to the functional central
limit theorem for martingales (or reverse martingales). In the non-i.d. case,
neither the forward nor the reverse martingale property holds, and hence, a more
elaborate treatment is necessary. Note that in order to establish weak invariance
principles, one needs to study the convergence of finite dimensional distributions
(f.d.d.) and the tightness of appropriate stochastic processes constructed from
these statistics. By virtue of Theorem 2.1, it suffices to construct such processes
from the Skk or k
<••• <t
«
c-
-1
1), then, for any non-null
[nit.]), j=l, ••• ,c,
J
c(~ 1)
Skk' In this respect, if we consider arbitrary
Z
n
~c
A = (Al'" .,A )' on letting n
c
,
11\.S
J=
J njn j
I.J.
~c
,
j
=
and O<t l
[nt j ] ( or
-1
.
(or 1.J._ll\.n. S
), we may, by vIrtue
J- J J njn j
of the definitions of the Skk and Y , express Zn as (Znl+'''+Znn) (or Znll+'"
kj
+ Z
) where the Z . form a triangular array of row-wise independent r.v. 's.
nln l
nl
Thus, the Liapounoff-type condition in (2.8)-(2.9) may again be called on to
verify the asymptotic normality of Z • Thus, the convergence of f.d.d. 's may
n
be established under conditions similar to (2.8)-(2.9). However, these finite
dimensional distributions, though asymptotically multinormal, may not conform to
that of a Wiener process, particularly when
does not converge to a positive
,n
constant. Thus, in the non-i.d. case, for the invariance principles to be studied,
~l
the question of having a Wiener process in the picture remains of some good
interest. Secondly, even if, such a Wiener process does not come to the picture,
the question of weak convergence to some appropriate Gaussian function
..
needs to
be addressed. In both the cases, the basic task is to establish the tightness
property of the stochastic processes under consideration, and, then to examine
the covariance structure to decide whether the Gaussian function is of Wiener
structure or not. This will be studied here.
Let us introduce the following stochastic processes z~j)
-5-
=
{Z~j)(t);O<t~l},
j=1, ... ,4, all defined on the space DI 0, lJ, where
=
{O, 0~t<2/n.
(2.16)
Z(1) (t)
n
(2.17)
~
(1)
Z(2) (t) =
{sl ,nisI, [nt]} Zn (t),
n
(2.18)
Z(3) (t) = n~2{U (t)
n
n
(2.19)
z(4)(t)
n
e([nt]))/2{n s l ,n
. [nt] (U [nt]
~
e(n(t))}/2S{,n
{Sl,n/Sl,n(t)}
~
(3)
Zn (t)
p
, t2:. 2/n ;
and 0, otherwise;
2/n~t~l,
net) = min{ k:n/k~ t},
O~t~l
;
O<t<l,
Our goal is to study weak conveegence of these processes to appropriate Gaussian
functions. Corresponding to the d.f. {F. ,i> I} and the kernel g(x,y), we define
1
(2.20)
-1 n
F( ) = n L. IF.
(2.21)
glen) (x) = J g(x,y)dF(n) (y) ,
(2.22)
Sl (F(n)) = J
n
1=
-
, for n -> 1,
1
g~(n)(X)dF(n)(X)
- (JJ g(x,y)dF(n) (x)dF(n) (y) )2
and
2
~n = n
(2.23)
-1 n
Li=l(
-
-
J gl(n)(x)d[F i (X)-F(n) (x)]
)
2
Then, following Hoeffding (1948) and Sen(1969) , we have
(2.24)
sl , n = sl(F(n)) -
~
2
+
n
o (n -1 );
~2
n ~ sl(F(n))·
As such, if F(n) converges to some F , sl (F) is a continuous functional of F in
some neighbourhood of F and ~2
n
T
~l
(-F) _
A
D
2 >
a,
as n
-+
to some positive limit
00
converges to some ~2 < sl (F), then, sl , n
. There may be other situations where
( .)
,n
S*
may converge
S* without requiring that F(n) converges to a limit F .
Theorem 2.2. If (2.7)-(2.9) hold and sl
ZJ
n
Sl
-+
,n
-+
s* > 0, then, for each j(=1, ••• ,4),
converges in law to Z = {Z(t);O<t< I} , a standard Wiener process on [0,1].
--
The result holds if in (2.16)-(2.19), 8(k) is replaced by 8(F(k)) =JJ g(x,y)dF(k) (x).
dF(k)(y) , for all possible k.
In the other cases where sl
may not converge to a positive limit, though the
,n
f.d.d. 's of the Z(j) may converge to some multinormal distributions, the covariance
n
function may not conform to that of a Brownian motion. Nevertheless, we may like to
study the tightness property of these processes o In the sequel, it will be
-6-
assumed~
that
n~l
(kin),= n
~l I~l k ' k < n. Note that h (kin)
,n is t in n. Also, let hn
,
n
> kin, 'd k ~ n. We may also need the following
= 0(1),
max{ h (kin) : k < n}
(2.25)
n
-
(2.26)
Ih (q/n) - h (kin)
n
n
for some
0:
I -<Klq/n
- k/nlO:
> llr, for every n,k and q, such that q/n> kin
depend on
8
~
8 >0, where K may
but not on n.
then~ under (2.7)-(2.9), the tightness of z(l)
and
Theorem 2.3 •. If Um
n
- ~l ,n >o~
z(3) are in order~ whiZe~ if in addition~ (2.25) and (2.26) hoZd~ then z(2) and
n
n
Z(4) are aZso tight.
n
When lim
~l,n
may not be positive, we may still be able to establish the tightness
property of the z(j), provided we impose more stringent conditions on the kernels.
n
For every n
(2.27)
~
q >
Z
=
n;q,k
l~
n
2, we define
-~{ k
L:.1= 1 [1jJl (1
. ), q (X.1 )
Theorem 2.4. If for some r >
2~
for every n,q and
k~
we have
2
Ir / {EZ n;q~
}r/2 < K <
k
k
( .)
then~ the tightness property of the Z J hoZds under the hypothesis of Theorem
n
(2.28)
Elz
n;q~
2.3~
without the condition that Zim ~l
> O.
- - ,n
Finally, we consider the strong invariance principle for V-statistics. In Sen
(1974), the basic tool of martingales and reverse martingales were incorporated
in the derivation of the main result. In the non-i.d. case, these martingale or
reverse martingale property may not hold and hence that method may not be applicableo
By virtue of (2.15), the desired strong invariance principle would follow if we
can establish the Skorokhod-Strassen embedding for the sequence {Skk;k ~ n~ 2}.
However, for a double sequence of independent (row-wise) r.v., we may not have
such a powerful result to utilize in the current context.
and (2.21), we conceive of a sequence{1jJk(X ) ;k
k
2:.l}
assume that there exists a positive c > 1, such that
(2.29)
=
L:~1= 1
E[
~l (.1), n (X.1 )
-7-
Keeping in mind (2.6)
of random variables and
*2
n
and that sn = Ik=l
as n
lim
(2.30)
7
00 • More specifically, we assume that
1';* > 0
n700
Typically (2.29) and (2.30) hold when F(n)
F as n
7
!g(x,y)dF(y) - !!g(x,y)dFk(x)dF(y), k ~ 1, where the
00 , so that
7
1JJ
lJJ * (X)
k
k* (X ) need not be i. i. d.
k
(as in the multisample situation where the F belong to a finite class {Gl, ••• ,G }
m
i
of d.f. 's.).
Let Z
{Z(t):O
2
t < oo}
be a random process on [0,00 ), where
Z(t) = Z(k) for k < t < k+l and
(2.31)
Z (k)
=
0, k < 2,
1
= k(U k - 8(F(k)))/2(1';*)~ , k ~ 2.
Finally, let W =
{W(t):O
2
t < 00 } be a standard Wiener process on [0,00 ).
2.2~
Theorem 2.5. Under (2.29) and the hypothesis of Theorem
k:
Z(t) = Wet) + oCt 2
(2.32)
)
almost
surely~
as t
00 •
7
The result can be extended to the case where s * may not go to 00
n
k:
of n 2
,
at the rate
but at a slower rate. In that case, we need stronger order of convergence
in (2.29). We will not elaborate it
further.
3. Proofs of the theorems. We start with the following (sequential) decomposition
of U
n
(3.1)
[ c.£. Sen(l960)]. For m=2, by (1,1), (2.1), (2.2)
i-I
n
n
{i-l
}
n
(2)[ Un - 8(n)] = I i =2 Ij=l [g(Xi,X j ) - 8 ij ] = I i =2 I.J= 1
Further, we write, for every i
(3.2)
and (2.3),
~
1JJ ( .. ) (X. , X . )
2 1J
1
J
2,
*
~i-l
i-I
i-l{
Ui = '"'j=l lJJ l (i)j (Xi) + Ij=l lJJ l (j)i (X j ) + Ij=l
1JJ 2 (ij)
(Xi,X j )- lJJ l (i)j (Xi)
- lJJ l (j)i (X j )}
*
*
**
= Uil + Ui2 + U ' say.
i
Let ~k be the sigma-field generated by (Xl, ••• ,X ), for k
k
(3.3)
and E (U *
il
&>.1- 1)
=0 ,
~
\J i
1 • Then, note that
> 2 ;
-
however, this martingale (difference) property does not hold for the U* • Also,
i2
note that for every n > 2,
-8-
(3.4)
(n-l)
-1 n
* *
n n
L.1. =2{Uo1. l +U·1. 2} = L 1. =1 '/'1(0)
(Xo) = Lk=lYn k = Snn ,
'Y
1 . , n - 1.
where the Y k and S
are defined after (2.10). Hence,for every n ~ 2,
n
nn
(3.5)
n[Un - S(n)] - 2Snn = 2(n-l)
(3.6)
E[IRn
-1
**
n
Li =2 Ui = Rn , say,
and, conventionally, we let R = O. Note that by (3.3) and (3.5),
l
I IcB n _l ]
.:.IE(Rn
I Q)n-l) I = ~=i
IRn_ll , \j n.:. 2.
Therefore, by the Birnbaum-Marshall (1961) inequality, for every sequence {a }
k
of positive numbers, t > 0 and nl' n
(3.7)
max
P{n <k<n
1-- 2
I
I
a k Rk
>t
}
< t
-2
: n
2
2
> n
~
l
2 ,
n2
2
-2
2 2
2
Lk=n (b k - k (k-1) bk+1)ER k '
1
where
(3.8)
b k = max { ak,k
-I
(k-1)a k+1 ,· •• ,(n 2-l)
-I}
(k-l)a
Note that by (3.20, (3.3) and (2.7),for every n
(3.9)
~
n2
2,
2
-2 n
** 2
-2 n
ERn = 4(n-l) Lk =2 E{(Uk )} .5.8A (n-l) Li =2(i-l) = 4An/(n-1) .5. 8A.
Thus, if we let n l =2, n 2=n, ak=1,2~k~n (so that bk=l, 2.5.k.5.n) and t= E(n~l n)
,
then, (2.11) follows from (3.7) and (3.9) when (n~l )-llog n + 0 as n + 00.
;,,;2
and a
-1
for k > n. Note that (n~l,n) Lk >n k
-3
2
,
,n
and the proof follows on parallel lines.
-~
= ~l,k
k
For the proof of (2.13), we take n = n, n
1
2
For (2.12), we take t= En
;,,;
+
00,
2
t = E(n
-1
-1;";
ER k = O(n~l,n Lk>n k
~l,n)
-3
2
and a
1
k
) = O((n~l,n)
= k- ,
-1
),
and hence, (3.7) insures(2.13) under the assumed condition. The proofs of (2.14)
and (2.15) are very similar, and hence, omitted. This completes the proof of
Theirem 2.1.
In the proof of Theorem 2.2, we may note first that by (2.1), (2.2), (2.7)
and (2.20),
(3.10)
18(k) - 8(F(k))I .5.
(k_l)-lA~
, for every k.:. 2,
so that max{ kI8(k)- 8(F(k))I//n :2<k<n}
converge to 0 as n
+
00
•
and
Hence, we may use
max{
n~
18(k)- 8(F(k))I
:k~ n}
both
8(k) and 8(F(k)) interchangeably
Since the convergence of the f.d.d. 's follow along the line of Hoeffding (1948,
Theorem 8.1), we shall only prove the tightness property of the z~j), j=1, ••• ,4.
For this purpose, we define for each k .:. 2,
-9-
(3.11)
k
.
-1 i
k
W = L. 2(1-1) L. 1 lji1(i)j(\) = L =2
k1
i
1=
J=
lji1 (i-I) (\)
,
(3.12)
k
.
-1 i
k
W = L =2 (1-1) L =1 lji1(j)i (X ) = L. 2
k2
i
j
j
1=
iiJ1 (i-1)i ,
say.
Then, by (3.1), (3.2), (3.11) and (3.12), for every q
(q-1)
(3.13)
-1 q
Li=2Wij + (k-1)
By (3.4) and (3.13), for every q
n -~ I (S qq - Skk)
I 2.
~
2
L.
~
2, we have
-1 k
Li=2Wij , for j =1,2.
2 and n,
-~!
{W.
qJ - WkJ'
I
I
I 1 q
k W" },
+~q_1L1'=2
W. ' - - 1k
1J
- 1L.1= 2 1J
= n-(1+0/2)E!L q
k+1
iiJ.
1(1-1)
(X.)
1
12+0
(_k)1+0/2 -(1+0/2) {( _k)-l Lq E I;J;
(X ) 12+0 }
q
n
q
k+1
~1(i-1)
i
C [(q_k)/n]1+0/2 , where c +
2 0
<
~
0,
2 0
(3.15) n-(1+0/2)E! W _ W 1 +
q1
k1
c 2+0
k
J =In
Now, under (2.8), for some 0 >
<
k
-1 q
*
-1 k
*
L i =2 Uij - (k-1)
Li =2 Uij
= (W qj - Wkj ) - (q-1)
(3.14)
~
say
and C are finite positive constants.
Similarly,
n -(1+0/2)E! W - W 12+0
q2
k2
(3.16)
1
2+0
i 1+0/2 E liiJ.
.1 2+0 ) (L~
i - (2+0) /2 (1+0)) 1+0
1(1-1)1
l=k+l
< n- (1+0/2) (L~
-
_ -(1+0/2) 1 q
- n
E Li =k+1 lji1(i-1)i
1=k+1
where, under (2.8), E ,-ljil(i-l)i 12+0
.-1-0/2
= 0(1
), so that by some standard steps,
the right hand side of (3.16) can be bounded by
1+0/2
C[(q-k)/n], for every k
(3.17)
2. q 2. n; C
<
00
•
Also, note that
(3.18)
(q-1)
-1 q
Li =2 Wij - (k-l)
-1 k
Li =2 Wij
= [(q-k)/(q-l)]{(q_k)-lL~ k lW" - (k_l)-lL~ 2 W.. }, for j=1,2.
1= + 1J
1=
1J
Thus, under (2.8), a moment-bound similar to (3.15) and (3.17) applies to (3.18)
as well. Using (3.4) and (3.13) through (3.18), we conclude that under (2.8),
for every n > q > k >
(3.19)
Eln-~
(S
qq
- S
2, we have
kk
)
1
2 0
+
<
K[(q_k)/nJ l + 0 / 2 , 0 >
0,
where K is a positive constant, independent of n. By virtue of Theorem 12.3 of
Billingsley (1968) and our Theorem 2.1, (3.19) insures the tightness of Z(l) when
n
-10-
~l
,n
+
~* > O. The tightness of Z(2) follows from (2.17) and the tightness of
n
Z~l) where we use the fact that for every t
E
(0,1],
~l,n/~l,[nt]
1, as n
+
+
00.
For the process Z(3) in (2.18), we again use Theorem 2.1 and note that
n
(3.20)
~
n (k
-1
Skk -q
-1
~
Sqq) = n [(k
-1
-q
-1
)Skk - q
-1
(Sqq-Skk)]'
As such, proceeding as in (3.14) through (3.19), we obtain that for every
n < k < q < 00, under (2.8), for some 0 > 0,
<
where K«
K[(n/k_n/q)]1+O/2,
00) does not depend on n. Thus, Theorem 12.3 of Billingsley (1968) may
again be recalled to verify the tightness of Z(3) • The tightness of
n
from (2.19), the tightness of
Z~3) and the fact that ~l,k
+
~*
Z(4) follows
n
> 0, as k
+
00 •
This completes the proof of Theorem 2.2.
To prove Theorem 2.3, we note that Theorem 2.1, (3.19) and the assumption that
Ijm ~l,n > 0 insure the tightness of Z~l), while (3.21) and the other conditions
insure the tightness of Z(3)
• Further, (2.25)-(2.26)
and the tightness of Z(l)
n
n
( or Z(3) ) insure the tightness of Z(2) ( or Z(4) ). Hence, the details are
n
n
n
omitted.
For Theorem 2.4, we may note that under (2.28), for some r > 2,
(3.22)
for every 0< s
~
t
~
1 ,
where K « 00) does not depend on n. A similar moment-
bound holds for the Z(3) also. Hence, Theorem 12.3 of Billingsley (1968) and (3.22)
n
insure the tightness of Z(l) and Z(3) • The case of Z(2) and Z(4) follows by
n
n
n
using (2.25)-(2.26) along with the tightness of
n
Z~l) and Z~3) •
Finally, we consider the proof of Theorem 2.5. Note that by (2.15) and
Theorem 2.2, for every positive E
,
as n + 00,
O.
(3.23)
Let us define now
(3.24)
k
= l:.1= 1
1jJ.* (X.) , k _> 1.
1
1
Note that the Sk* involve independent (but, possibly non-i.d.) summands, and, by
-11-
(2.29) , (3.24) and the definition of the Skk ' we have
1
1
{sup
->2->2
*
P k2. n k t:l,k IS kk - Sk
(3.25)
> s }
1
~ Ik>n P{(kt: l k)-~ISkk - Sk*
,
< I
-
I
k> n
(kt: l , k)
(kt:l,k)
k>n
-1
-1
E( Skk - Sk* )2
I
Sk
side of (3.25) converges to 0 as n
-+ co
. Thus,
from (3.23) and (3.25), we obtain
-+ co
1
(3.26)
-1 [O(k-l(log k)c) ]
,
k>n t: l , k
*
t: l , k converge to a positive limit t: • Hence, the right hand
where c > 1 and the
that as n
> s }
(nt:
l ,n
I
)->2
n[U
n - e(F(n))] -2S:
I
o,
-+
almost surely.
On the other hand, for the sequence {S~ ; k.:::.. l} , we directly appeal to the
Skorokhod-Strassen embedding of Wiener process [ c.f. Strassen (1967)] and
conclude that under (2.8),
(3.27)
~
S * /ft, *
n
Wen) + o(n 2
almost surely, as n
)
-+
co.
Thus, (2.31), (3.26) and (3.27) imply (2.32). Q.E.D.
4. Invariance principles for the von Mises' functional. Recall that by (1.1) and (1.2);
(4.1 )
In addition to (2.7), we now assume that
(4.2)
sup
i>l
!
2
g (x,x)dF. (x)
1
.
B <
<
co
Let us now write
(4.3)
(4.4)
,. i -> 1
e~ = Eg(X. ,X.) and t:~ = Var{g(X.1 ,X.)}
1
111
eon
Then, under
1
= n-lL~ e~ and e*(n)
1=1 1
(4 2),
0
=
-;:;-OS - S
n
(n)' n.:::..
2
•
by the Kolmogorov law of large numbers, as n
(4.5)
-+
-+ co
0 , almost surely •
Further, as in (3.21), for some 6 >0,
(4.6)
E!k-lSkkI2+6
=
O(k- 1- 6 / 2), for every k.:::.. 2.
Hence, by (2.15), (4.6) and the Borel-Cantelli Lemma, under (2.8),
(4.7)
U
n
Sen)
-+
0 almost surely, as n
-12-
-+
co
•
..
Therefore, by (4.1), (4.5) and (4.7), we obtain that as
(4.8)
n(U
n
- V) n
eon +
e,In )
*
= n(Un - e(n))
-1
{n(Vn - Un) - ~
+
[(n-l)e(n)
+
*
-:ct)
en ] and noting that n(Vn - e(n))
e(n) }, we conclude that as n
+
* ) - n( Un - e(n))
n( Vn - e(n)
(4.9)
,
0, almost surely.
-+
Consequently, on letting e(n) = n
n-+ oo
0
-+
-+
00
,
almost surely.
Thus, if in (2.18),(2.19) and (2.31), we replace the
k(U
- e(k)) by the
k
I·
and ZO '.
k(V k - e *(k) ) an d denote t h
e resu
tlng processes by zo(3).
n ' Zo(4)
n
respectively, then the invariance principles in Theorems 2.2, 2.3, 2.4 and
2.5 hold for these processes as well. Similarly, under (4.2),
(4.10)
max { n -~
Ik-1 ~i=lg(Xi,Xi)
k
* I: k
- e(k)
2
}
n
P
0, as n
-+
-+
00,
while, by the Theorems in Section 2,
p
1
(4.11)
max{ n-~ IU
- e(k) I : k < n }
-+
o,
k
Consequently, under (4.2) and (2.7)-(2.9), as n
(4.12)
1
max{ n-~
I
*
k(V k - e(k)) - k(U k - e(k))
as
-+
I
n -+
00
00
:k < n}
P
-+
O.
* , and denote
Vk- e(k)
the resulting processes by Zo(l) and ZO(2), respectively, then, under (4.2) and
Thus, if in (2.16)-(2.17), we replace the
n
Uk - e(k) by
n
the hypotheses of Theorems 2.2, 2.3 and 2.4, the invariance principle holds for
ZO(l) and Zo(2) also. This leads us to the following.
n
n
Theorem 4.l.Under the additional assumption (4.2)J the invariance principles in
Section 2 hold for the von Mises' functionals as well.
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