I
I
I·
I
I
I
I
I
I
I.
I
I
I
I
I
I
I
..
I
CONVERGENCE OF SEQUENCES OF REGULAR FUNCTIONALS OF
EMPIRICAL DISTRIBUTIONS TO PROCESSES OF BROWNIAN MOTION
By
Pranab Kumar Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 726
January 1971
~
I
I
Ie
I
I
I
I
·1
I
CONVERGENCE OF SEQUENCES OF REGULAR FUNCTIONALS OF EMPIRICAL
DISTRIBUTIONS TO PROCESSES OF BROWNIAN MOTION 1
BY PRANAB KUMAR SEN
University of North Carolina, Chapel Hill
For partial cumulative sums of independent and identically distributed
random variables (iidrv) having a positive (finite) variance, weak convergence
to Brownian motion processes has been established by Donsker (1951, 1952).
result is extended here to von Mises' (1947) differentiable statistical functions
and Hoeffding's (1948) U-statistics.
~.
Let w
=
Few applications are sketched.
{x ,X , ... } be a sequence of iidrv with each Xi having
l
2
I.
I
I
I
I
I
I
I
(1.1)
6(F)
defined over F E ~
= {F:
~
lWork supported by the Army Research Center, Durham.
I
The'
a distribution function (d.f.) F(x), XERP , the p(~l)-dimensional Euclidean space.
Let g(Xl, ••• ,X ), symmetric in its arguments, be a Borel-measurable kernel of
m
degree
m(~l),
and consider the regular functional
= f---f
~m
g(xl,···,x )dF(xl)---dF(x ),
m
m
6(F)<oo}.· For a sample (Xl, ••• ,X ), consider the
n
empirical d.f.
(1.2)
F (x,w) = n -1
n
Ln.
1=
1 c(x-X.), XERp ,
1
I
I
••I
I
I
I
I
I
••I
I
I
I
I
I
I
••I
2
where for u = (ul, •.. ,u ), c(u) = 1 if
p
ui~O, l~i~,
and c(u) = 0, otherwise.
Then, the corresponding regular functional of F (x,w) is defined by
n
8 (F ,w)
(1. 3)
n
JoooJ
Rpm
g(xl, ..• ,x )dF (xl,W)ooodF (x ,W).
m n
n m
Under certain regularity conditions, von Mises (1947) derived the asymptotic
!.:
normality of n 2 [8(F ,w)-8(F)]; a comparatively simpler proof was subsequently
n
devised by Hoeffding (1948), who primarily considered the unbiased estimators
(U-statistics)
U (w) = n -em]
n
(1.4)
I'
g (X . , ••• , X. ),
Lp
1
n,m
1
1
{n ooo (n-m+1)}-1)
(where P
= {l<i fooofi <n} and n-[m]
n,m
- 1
m-
n~m,
m
'
and showed that under
!.:
essentially the same conditions, n 2 [U (w)-8(F)] is asymptotically normal and
n
!.:2
P
n [8(F ,w)-U (w)]+O, as n+oo.
n
n
In particular when m=l, 8(F ,w) = U (w) = n
n
we assume that E{[g(X)-8(F)]2} = 0
n
2
-lIn.
1=
1 g(X.) = gn(W)' say.
1
If
is positive (finite), and let
~
Y (t,w) = {[nt]g[nt](W) + (nt-[nt])g(X[nt]+l)-nt 8(F)}/{0n },
n
(1. 5)
where [s] denotes the integral part of s(~O) and t E l = {t:
O~t~l}, then by
the Donsker theorem [cf. Billingsley (1968, p. 68)], as n+ro
Y (w)
(1. 6)
n
[Y (t,w),t E I]
n
l'
+
W
where W ,t>O, is a standard Brownian motion, and
t
-
distribution.
[W ,t E I],
t
fJ
+
stands for convergence in
I
I
.e
I
I
I
I
I
3
In the present paper, the above result is extended to general 8(F ,w) and
n
U (w), and some possible applications are briefly sketched.
paper, we assume that m>2.
~.
(2.2)
Our basic assumptions are (I,IIa) or (I,IIb), where (I) 8(F) is stationary of
order
Ie
I
I
I
(2.3)
I
I
° i.e.,
~1
~m(F)<oo,
(F»O, (IIa)
and (lIb)
2
max 1<,1 < ••• <'1 <m{g (X.1 , ... ,X.1 )}
- 1-
[Note that (lIb)
~
Y (t,w)
n
(IIa).]
=
-
1
TIl
Let then, for every
~ -1
{m[n~l (F)]}
~*(F)<oo.
m
n~l,
{([nt]+l) (nt-[nt])8(F[nt]+1'W)
(2.4)
+ [nt]([nt]+1-nt)8(F[nt],w)-nt8(F)}, t € I.
Also, we define
Y~(t,w)
as in (2.4), with 8(F ,w) being replaced by Uk(w) for
k
m'::'k'::'n, and 8(F ,W) by 0 for k<m.
k
(2.5)
I
Ie
For every h(O.::.h.::.m), let
(2.1)
I
I
I
Throughout the
n
(2.6)
Y (w)
n
Let then
= [Yn (t,w),t
p (Y n (W),Y*(W»
n
€ I], Y*(w)
n
=
= [Y*(t,W),
n
t € I];
sup t€ IIYn (t,W)-Y*(t,W)I.
n
I
I
II
I
I
I
4
Then, we have the following theorem:
Theorem 2.1.
.I
n~,
fl
yil(W) -+ W
n
'
(2.7)
while under (I) and (lIb), as
(2.8)
I
I
II
I
I
I
I
I
I
Under (I) and (IIa), as
y
n
(w)
~
n~,
W, and p(Y «(I'),yic(W)) ::. O.
n
n
The proof of the theorem is postponed to section 4.
v n, hew)
(3.1)
fOOhOf
gh(x1,···,xh)rr~
P
1d [F (x.,w)-F(x.)],
J=
R
n
J
J
so that
s
(3.2)
Now, writingdF (x.,w)
n
J
= dF(x.)+d[F
J
n
(x.,w)-F(x.)],
J
J
n
(w),
say.
i~j~m,
we obtain from (1.1),
(1.3), (3.1) and some simplifications that
reeF ,w)-e(F)-mV l(w)]
n
n,
(3.3)
Let then
(3.4)
an
be the a-field generated by {X.,i<n}, and let
1
V*n, 2(w)
-
f
g2(x,x)dF(x).
I
I
••I
I
I
I
-I
5
{V* 2(w)+n[8(F)-8*2(F»), ~ } forms a martingale sequence.
n,
n
Lemma 3.1.
Proof.
By (1.2), (3.1) and (3.4),
V~,2(W)
=
V~_1,2(W) + 2I~:i f~2pf
g2(xl,x2)d[c(xl-Xi)-F(xl»)d[c(x2-Xn)-F(x2»)
(3.5)
Also, for all i<n-l,
(3.6)
I
I.
I
I
I
I
I
I
I
I·
I
(3.7)
(8
x x
l 2
where 8
x x
l 2
)dF(x )-dF(x )dF(x ),
l
l
2
is 1 or 0 according as x =x or not.
l 2
Hence,
(3.8)
and the lemma follows.
Now, for all r.>l, j=l, .•• ,~(_>l), \~ 1 r.=2s, s_>l,
J-
(3.9)
lJ=
J
= 0, if at least one of rl, •••
,r~
= 1,
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
6
and hence t the maximum
~
for which (3.9) is different from zero is St where
r 1= ... =r s=2.
Lemma 3.2.
Under (2.3)t for every E>Ot
lim
(3.10)
Proof.
n~
p{w:
0.
For every E>Ot there exists an nO(E)t such that
Hence t for
n~nO(E),
by definition in (3.4)t
p{w:
(3.12)
< p{w:
Now, under (2.3), by (3.1) and (3.9)t
(3.13)
< Ck
-
1
2
s*(F) '
where Cl«oo) does not depend on k.
Hence, by theorem 1 of Chow (1960)t (i.e. t the semi-martingale extension of
the Hajek-Renyi inequality), Lemma 3.l t (3.12) and (3.l3)t it follows that for
I
I
7
,e
I
I
I
I
I
I
'.I
I
I
(3.14)
<
Lemma 3.3.
(3.15)
Proof.
,.
I
2
n]/n£ )
+
0 as
n~.
Q.E.D.
If m>3 and (2.3) holds, then for every £>0,
lim
n~
0.
p{w:
By (2.3), (3.1) and (3.9), for all
k~l,
Hence, by the Bonferroni and the Markov inequality,
(3.17)
I
I
I
I
(8C1~*(F)[log
~n
2
-1
~ Lk=l {C2~*(F)/n£}k
which converges to
°as
n~.
2
~ C (log n)~*(F)/n£ ,
2
Q.E.D.
From (3.2), (3.3), (3.10) and (3.15), we have for every £'>0,
(3.18)
I
I
.I
I
I
I
I
I
•I
I
I
I
I
I
I
.I
8
n~m,
Now, by (1.4), for all
U (w)
n
n
-[mho
Lp
n,m
f···f
Rpm
g(xl,···,x )rr~ 1 d[c(x.-X. )]
m J==J
J]
(3.19)
where Un, lew)
vn, lew) is given by (3.2), and for
2~h~m,
(3.20)
Lemma 3.4.
{n[h]un,h(W),CB } forms a martingale sequence for ever
n
l~h~m.
The proof is similar to Lemma 3.1 and may be found in Hoeffding (1961) .
Lemma 3.5.
1i
~m(F)<oo,
lim
(3.21)
n-+oo
then for every
h(2~~m)
and E>O,
o.
p{w:
By virtue of Lemma 3.4 and the Chow-Hajek-Renyi inequality, the proof follows
2
-h
exactly as in Lemma 3.2 after noting that E{Uk,h(W)} == O(k ), 2<h<m.
and Lemma 3.5, we obtain that for every E'>O,
(3.22)
lim
n-+oo
o.
p{w:
Now from (3.1) and (3.20), for
n~m,
By (3.19)
I
I
••I
I
I
I
I
I
••I
I
I
I
I
I
I
••I
9
v*
n,2
(w)-n[2]U
n,2
=
(w)
(3.23)
Z* (w),
say.
n
Then, by the same technique as in Lemmas 3.1 and 3.2, we have the follo,Ttng.
Lemma 3.6.
{Z*(W)+n[8(F)-8*2(F)],CB} forms a martingale sequence, and hence,
n
n
under (2.3), for every E>O,
(3.24)
limn+oo p{w:
maxl<k<n k
-I,
I
Z~(w) >En~ }
O.
Finally, from (3.18), (3.21), Lemma 3.3 and 3.6, we obtain the following.
Lemma 3.7.
Under (2.3), for every E>O,
lim
p{w:
n+oo
(3.25 )
O.
For tEl, define
(4.1)
a
Y (t,w)
n
where Skew) is defined by (3.2).
Then, by the Donsker theorem [cf. Billingsley
(1968, p. 68)], as n+oo
(4.2)
=
a
[Y (w,t),tEl]
n
cp
+
W = [W ,tEl].
t
Now, by (2.5), (3.2), (3.18) and (4.1), under (2.3),
I
I
.e
I
I
I
I
I
I
10
p(Y (w),yo(w»
n
n
= SUP t £ rlYn (t,w)-yo(t,w)
I
n
(4.3)
Hence, by (4.2) and (4.3), under (2.3),
Y (w)
(4.4)
n
P
~
Y° (w)
n
'f)
~
W.
:P
~
W.
Similarly, by (3.19) and (4.1), as n+oo,
(4.5)
P Y° (w)
Y*(w) -+
Further, by definition, Uk(w)
=0
n
n
for k<m, and hence,
Ie
I
I
I
I
I
I
I
•e
I
(4.6)
where by the Chebyshev and the Bonferroni inequalities, for every £>0,
(4.7)
<
2
[(m-l)~*(F)]/[n£ ~l
(F)]-+O as n-+oo, [by (2.3)]
and hence, by Lemma 3.7 and (4.6), p(Y ,Y*)
n
n
P
~
0 as n-+oo.
Q.E.D •
-I
I
.e
I
I
I
I
I
I
·eI
I
I
I
I
I
I
•e
I
11
(I)
Asymptotic normality for random sample sizes.
For every index variable
r(>l), let N be an integer valued (non-negative) random variable and n is a
r
r
real (positive) number, such that
(5.1)
lim
~
n =00 and lim
(N In )=1, in probability.
r
~
r r
This implies that for every 8>0,
lim
(5.2)
r-+oo
1
P{ln- N -11>8} = 0.
r r
Now, by the tightness property [cf. Billingsley (1968, p. 54)] of W ,
t
t~O,
for every £>0 and n>O, there exists a 8>0, such that
(5.3)
Hence, by theorem 2.1, (5.2) and (5.3), we have as r-+oo,
N:[8(F
N
£
,w)-8(F)]
n:[8(F
r
N
,w)-8(F)]
r
._~ n~[8(F
r
n
,w ) - 8(F)] ,
r
and the asymptotic normality readily follows.
A similar result for UN (w) has been obtained by Sproule (1969) by an
r
indirect method involving some elaborate analysis.
2•1 , (5 •2) and (5. 3) •
Our proof follows from theorem
I
I
,e
I
I
I
I
I
I
'e
I
I
I
I
I
I
I
,e
I
12
(II)
Limiting distributions of SUPtsI Yn(t,w)
~~~.
By virtue of theorem 2.1
and well-known results on Brownian motion processes [viz., Billingsley (1968,
p. 79)], we obtain that for all A>O,
lim p{w:
SUPIIY (t,W)I>A} = lim p{w:
tS
n
n~
n~
(5.6)
00
L
= 1 -
k=-oo
1
where ~(x) = (2rr)-"2
1
2
-"2t
e
dt,
lim p{su p Y*(t,W»A}
n~
tsI n
lim p{sup Y (t,W»A}
n~
tsI n
(5.5)
sUPIIY*(t,w)I>A}
ts
n
(-1)k[~((2k+l)A) - ~((2k-l)A)],
2
X
exp(-~t )dt, -oo<x<oo.
J
_00
(III)
An application to rank statistics.
(5.7)
1,
Consider the kernel
° or -1 according as x l +x 2 is >,
l
and assume that the X. have a continuous d.f. F(x), xSR .
1
or < 0,
Then
00
(5.8)
8(F) =
J
[1-2F(-x)]dF(x)
(=0 when F is symmetric about 0).
_00
The corresponding 8(F ,w) is given by
n
(5.9)
2 W , say,
n8(F ,w)
n
where R . = Rank of Ix.
nl
1
I
n
among Ixll, •.• , Ix
n
I,
sgn u=l, Q or -1 according as u
is >, = or < 0, and W is the classical Wilcoxon signed rank statistic.
n
here
Also,
I
I
Ie
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
13
00
~1 (F) =
(5.10)
which equals to
(5.11)
1/ 3
f
2
2
[1-2F(-x)] dF(x)-8 (F),
when F is symmetric about O.
Thus, if we let
1
Yn(t,w) = {(nt-[nt])W[nt]+l + ([nt]+1-nt)W[nt]-~nt8(F)}/[n~1 (F)]~.
O<t<l, we have from theorem 2.1 that if
Y (w)
(5.12)
and thus the sequence
n
{Wk-~k8(F); k~l}
~l
~
+
(F»O,
W,
though not linear in the basic random
variables is attracted by the Brownian motion process.
REFERENCES
Convergence of Probability Measures.
John Wiley &
[1]
BILLINGSLEY, P. (1968).
Sons, New York.
[2]
CHOW, Y. S. (1960). A martingale inequality and the law of large numbers.
Proc. Amer. Math. Soc. 11, 107-111.
[3]
DONSKER, M. (1951). An invariance principle for certain probability limit
theorems. Mem. Amer. Math. Soc. ~, 1-12.
[4]
DONSKER, M. (1952). Justification and extension of Doob's heuristic
approach to the Kolmogoro1l-Smirnov theorems. Ann. Math. Statist. 23,
277-281.
[5]
HOEFFDING, W. (1948). A class of statistics with asymptotically normal
distribution. Ann. Math. Statist. 19, 293-325.
[6]
HOEFFDING, W. (1961). The strong law of large numbers for U-statistics.
Inst. Statist. Univ. N. Carolina Mimeo Rep. No. 302.
[7]
MISES, R. V. (1947). On the asymptotic distribution of differentiable
statistical functions. Ann. Math. Statist. 18, 309-348.
[8]
SPROULE, R. N. (1969). A sequential fixed-width confidence interval for the
mean of aU-statistic. Doctoral Dissertation, Univ. N. Carolina, Chapel Hill.