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working paper
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STOCHASTIC EQUICONTINUITY AND WEAK CONVERGENCE
OF UNBOUNDED SEQUENTIAL EMPIRICAL PROCESES
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
STOCHASTIC EQUICONTINUITY AND WEAK CONVERGENCE
OF UNBOUNDED SEQUENTIAL EMPIRICAL PROCESES
Jushan Bai
94-7
Jan.
1993
Abstract
This note explains
why
sequential empirical processes arise naturally in the context of
structural change and then provides an elementary proof for their stochastic equicontinuity.
and
in
An
application
is
considered for testing structural change in a linear regression
a single equation of a simultaneous equations system.
Key words and
phrases: Stochastic equicontinuity,
pirical process,
two-parameter Brownian bridge, structural change.
weak convergence, sequential em-
Stochastic Equicontinuity and
Weak
Convergence of Unbounded Sequential
Empirical Processes
Jushan Bai
Massachusetts Institute of Technology
(617) 253-6217
email:
jbai@athena.mit.edu
Revised: January 1994
APRS
1994
Introduction
1
A number
empirical processes for heterogeneous and dependent observations,
DeJong
first
[11],
and Hansen
A
[16].
review
is
given by Andrews
[3].
Andrews
[2],
In this paper
we
e.g.,
introduce a weighted sequential empirical process and then study
equicontinuity.
A
sequential empirical process involves partial
As such, an empirical process
cesses.
unbounded
of authors have recently studied stochastic equicontinuity for
case of a sequential empirical process.
in the usual sense
Andrews
[2]
sums
its
stochastic
of stochastic pro-
may be viewed
as a special
shows how empirical process the-
ory can be used in various applications in econometrics, particularly in establishing
asymptotic properties of econometric estimators and
test statistics.
paper that sequential empirical processes arise naturally
We
show
in this
in the context of structural
change. Therefore they will be useful for testing parameter constancy in econometric
models.
Sequential empirical processes and their related stochastic equicontinuity are
discussed by Bickel and Wichura
we
[9],
but with no particular motivation. In this paper,
The consideration
consider a weighted sequential empirical process.
version
is
motivated by structural change
next section.
The weights
in linear regressions, as
of a weighted
explained in the
are the regression variables or a set of instrument variables,
so that the process involves
unbounded summands
in contrast to the
one considered
by Bickel and Wichura. Also regression variables are generally stochastic and
correlated (e.g., autoregressive models); thus
we have a
in the literature.
from the existing
results.
empirical processes
Its
stochastic equicontinuity
In addition, the
demands highly
may
we
[10].
is
and
not discussed
not be directly derived
contemporaneous treatment of dependent
technical and abstract analysis. In this paper,
provide an elementary proof for stochastic equicontinuity.
extension of that of Billingsley
serially
stochastically weighted
dependent empirical process. This kind of sequential empirical process
much
first
The argument
is
we
a direct
Because of the concrete structure of the process,
are able to derive concrete sufficient conditions for
its
stochastic equicontinuity. In
addition, an elementary proof
is
instructive.
Application of the result to structural changes in linear regressions
The
cussed.
analysis
is
is
briefly dis-
applicable to changes in a single equation of a simultaneous
equations system. Tests based on a weighted sequential empirical process can detect
changes in regression parameters and changes in variance.
tests
can detect changes in error distribution functions that are not necessarily man-
ifested in the
form of changing variances.
In other words, the test
changes in higher moments of the data, whereas the
type of tests
2
Most importantly, these
may
not be able
CUSUM,
is
able to test
and Wald
fluctuation,
to.
Sequential empirical process
For a given sequence of random variables Z\, Zi,
of this sequence
is
V
is
z)-F(z)},
(k
=
1,2, :..,u)
t=i
the distribution function of Z,. Bickel and Wichura
and establish the stochastic equicontinuity of
The summand
process.
Z„, the sequential empirical process
defined as
Bn (k,z) = 4=Ei 7 (^ <
n
where F(z)
...,
in the process
sequential empirical process which
is
is
Bn for i.i.d.
bounded by
1.
not bounded and
Z\s. This
We
[9]
is
introduce
first
a two-parameter
shall consider a
may be
weighted
dependent. Weighted
sequential empirical processes arise naturally in the context of structural change. Consider the following linear regression model:
yt
where x
t
is
= x' P + e
t
(*
t
=
a vector of explanatory variables,
l,2,....,n)
/? is
(1)
an unknown vector, and e are
t
i.i.d.
disturbances with a continuous distribution function. Define the vector process:
n
-
<
-OC <
Z
<
In terms of parameter inference, observing this process
is
equivalent to observing the
Sn{z)
whole data
set,
YlxtI{Vt
Z),
OO.
with probability one. The vector process simply orders the original
data
is
according to the magnitude of the dependent variable. In this sense,
set
(may be a
a sufficient statistic for
"sufficient process"
Sn (z)
more appropriate). Now
is
suppose the true model obeys a two-regime regression:
To estimate
0\
The
belongs.
and
process
=
l,2,...,n 1 )
(2a)
et
(*
= m + l,...,n).
(2b)
x't
l
+e
yt
=
x't
2
+
Sn (z)
statistic (process) for 0\
(i
=
important to know to which regime a given observation
it is
02,
t
yt
does not convey this information.
and
2
It is
thus not a sufficient
However,
.
Sn (z) = f:x
1
t
<z)
I(y t
t=i
and
£
Sn2 (z)=
<z)
x t I(yt
t=m+i
are jointly sufficient for 0i and 02, because S„ only orders the
n^ observations
n
—
rii
However, when the regime-switching point n\
is
unknown, then
within themselves, and similarly S% orders the
selves.
first
last
observations within them(5*, S%)
longer a sufficient statistic. In this case, sufficient statistics are given by
(5*, S*) for
rii
=
all
is
no
pairs of
1,2, ..,n. Introduce
k
Sn (k,z) =
^x I(y
t
t
<
z),
t=i
which
is
a sequential empirical process up to normalization and centering. For k
Sn (k,z)
the process
when
rii is
unknown,
As an example
is
simply S*(z) and
Sn (n,z) - Sn (n
sufficient statistics are given
x
,z)
is
=
ri\,
simply S*(z). Thus,
by Sn (k,z), k
=
l,...,n.
of a dependent sequential empirical process, consider a time series
regression:
yt
Denote x t
But x
t
is
=
(l,y t -\,
=
fi
+
••, Vt—p, z[)'.
piy t -i
The
+
•• p y - P
P
t
+
z't 6
+e
t
sequential empirical process
serially correlated, yielding a
(3)
.
is
defined as before.
dependent sequential empirical process.
The above
model. To do
process
this,
Sn (k,z)
only describes the data,
we modify the process
it
does not incorporate the
to
k
Sn (k, 2, 7) = 51 x
t
I(y t
<
z
+
x't j).
t=\
A
linear structure
is
introduced in the above process. Also note, under model
Sn (k,z,/3) = J2 x tH £ <
(1),
z )-
t
t=l
However, the parameter
/? is
unknown, so Sn (k,
solve this problem one can replace
To
z, /?) is
not observable or computable.
by an estimator,
f3
$.
If
we put
it
=
yt
—
x't $,
then
k
Sn (k,zJ) = ^2x
<z)
I(e t
t
t=i
which can be considered an estimated sequential empirical process.
This process
embodies the model and data. The estimated parameter can be obtained from the
first
k observations or from the whole sample. In the former case, a sequence of estimators
is
needed, which
may
be obtained by recursive estimation. In our application, we use
a whole-sample estimator.
The
Sn (k,z, $),
Tests based on the weighted sequential empirical process
are
[8].
see Section 4.
parameter constancy
more powerful than those based on a non-weighted
To study the asymptotic property
Sn (k, z, /5), whose convergence
The weak convergence
we
test for
in turn
of the latter
is
of the test,
is
based on the process
process, as pointed out
by Bai
we need the weak convergence
depends on the weak convergence of Sn (k,
our focus.
By
of
z, 0).
normalizing and centering of
Sn
<z)- F(z)}
(4)
,
define the vector process
[ns]
Hn (s, z)
= (X'X)-^
£
x t {I(e t
t=i
where
of
Hn
X
=
and
(xi,X2, ...,x n )'. In the next section,
its
stochastic equicontinuity
weak convergence.
The weighting
can be a
we study the
set of
vector x t does not have to be the regression variables. Generally,
it
instrumental variables. Consider a single equation in a simultaneous
equations system:
yt
=
+e
z't
(5)
t
where z includes other endogenous variables so that
et
t
uses
limit
Zt in
Hn
place of x t in the definition of
then the process
,
Now
because the summands do not have zero mean.
of instruments that
is
correlated with z
consider the weak convergence of
t
will not
call this
t
t
.
t
is
a vector
Then we can
.
one
If
have a proper
suppose x
but independent of e
We may
(4).
correlated with z
is
still
process the instrumental-
variable weighted sequential empirical process. Tests based on a instrumental- variable
weighted process
will
have nontrivial local power only
if
X
in the sense that
plim(X'X/n) and p\\m(X'Z/n) have
uncorrected with
£t
,
where
Z=
a set of valid instruments
full
column rank and
X
is
(z[,...,z'n )'.
Stochastic Equicontinuity
3
To derive the
stochastic equicontinuity for weighted sequential empirical processes,
impose the following conditions
(A.l)
The random
(A. 2)
The disturbances
(A. 3)
The
is
is
(their implications are discussed below).
variables e t are
e t are
we
with a continuous distribution function
i.i.d.
independent of
F
contemporaneous and past regressors.
all
regressors {x t } form a triangular array (for simplicity the
dependence on n
suppressed) and satisfy;
plim- 22 x
n
where Q(s)
is
t
x't
=
Q(s)
€
for s
[0, 1],
t=i
a p x p nonrandom positive definite matrix for s
The convergence
is
necessarily uniform in
6,
because the
sum
is
and Q(0)
=
"monotonic" in
s.
>
0.
(A.4)
max n
-1 / 2
(A. 5) For every fixed Si, there exists a
that, for all s
>
" =
||a: t
"
l<t<n
||
ojl).
pv
'
random
variable
<
- Sl )Zn
si,
-
E
t=[nsi]
\\*t\\
(s
Zn (may depend on
s\)
such
with probability one. In addition, the
M<
and
There
exist
- J2
n
>
7
1,
a >
\
and
1
>C)< M/C 2(1+p)
K
<
.
00 such that for
E(x't x
t
y <K{v-u)
and
where
i
=
[nit],
j
=
<
all
<
u
<
v
and
1,
for
n
sumption (A. 3) allows
for
dynamic models,
is
tests, see, e.g., Ploberger,
and (A. 6) are unique
where
i
=
Y?t=i
for
is
=
2,
s
+
T =
and
\F{z)
—
when
The
If
E(x't x
2
t
x't x t )
)
2
<
M
<
{J2i=i[
regressors x t are
for all
Hn
left limits.
vector process
Hn
is
We
Hn
will
=
As-
(3).
g{t/n), for
some
Assumption (A. 4)
Assumptions
E
(
.
In (A. 5),
tail
set
Zn may
probability
t,
then (A. 6)
is
is
be taken to
also satisfied,
Zn = O p (l)
satisfied
uni-
=
with 7
2
x t x t) 2 ] 1/2 } 2 by the Cauchy-Schwarz
will
be
with metric p({r, y},
T
satisfied.
{5, z})
=
\s
—
that are right
equip D[T] with the Skorohod metric (Pollard
belongs to the Cartesian product space D[T] P equipped
,
implied by the
stochastic equicontinuity.
limiting process of
[18].
Let D[T] be the set of functions defined on
F(y)\.
space D[T] P
form x t
however, to choose
with the corresponding product Skorohod topology.
in the
,
model
autoregressive
bounded (A.4)-(A.6)
x 1Z be the parameter
continuous and have
[19]).
sj.
because £(£t-«
[0, 1]
a
our problem. They are the main assumptions for the
x t\\ provided the condition on the
inequality. Finally,
Let
e.g.,
used for obtaining normality.
[nsi] is fixed. It is generally impossible,
formly in both
and a
\\
u)
i<t<j
Kramer, and Kontrus
equicontinuity of the sequential empirical process
be max* k~ l
< K{y -
often maintained in recursive estimation for construct-
conventional for linear models and
(A. 5)
't
for trending regressors written in the
function g. This assumption
CUSUM
x x tV
[nv].
Assumption (A. 2) allows
ing
£
E(-
i<t<j
r\
some p >
satisfies, for
n,
all
is
Zn
oo:
P(\Zn
(A. 6)
probability of
tail
The
finite
The weak convergence
of
Hn
dimensional convergence together with
latter condition also implies the
be continuous with probability one.
sample path of the
Theorem
Under assumptions
1
stochastically equicontinuous on (T,p).
8
>
(A. 5),
That
for any
is
>
e
0,
where
[8]
When
=
x
=
{{t1 ,t2 );t1
t
=
and Wichura
1
for all
[9].
When
sup \\Hn (r,y)-Hn (s,z)\\>T})
{r,y),r2
=
the equicontinuity of
t,
<
(s,z),p(tu t2 )
Hn
>
r\
Hn
is
there exists a
0,
<t
8} with
is
[8]
CT
implied by the result of Bickel
It
data that requires a different framework of proof.
a big obstacle for proving equicontinuity.
is
the statistical dependence
Dependence
m-dependent processes by Andrews
around the
for a
difficulty.
smoothed
And
there are
[2].
many
by Andrews
[3],
also see
[16] for
DeJong
[12] for
may be
[5]
and only work well
Levental's
for
The
[11] for
Andrews
It
[1]
unbounded
is
provided
bounded strong mixing
for a
differences (for each fixed z).
used.
extended to
unbounded absolutely
quence. For the proposed weighted sequential empirical process
of Levental [17]
is
unbounded mixingales. A review
Andrews and Pollard
unbounded martingale
it
successful results; examples are
strong mixing processes, Doukhan, Massart and Rio
Hansen
data could be
Recent development explores ways of getting
class of functions of near-epoch variables,
regular processes, and
in
Indeed, the powerful tool of symmetriza-
depends heavily on the independence assumption, although
tion
x T.
{(x t ,e t )} are independent, equicontinuity can also be proved
by extending the method of Bickel and Wichura.
are
the process
such that for large n,
P[
in
and (A. 6),
(A. 2),
(A.l),
(4),
the
se-
summands
seems that the method
conditions of Levental, however, are not primitive
bounded martingales, though
method may be extended
to
unbounded
it is
ones.
noted by Hansen
[16]
that
Hansen's own approach, as
pointed out by the author himself, does not work well for indicator types of functions.
We
shall offer
an elementary proof. Since our purpose
of processes as possible, our conditions are specific
structure of our process,
we
feel
is
and primitive. Given the concrete
an elementary argument
also interested in the limiting process.
not to cover as wide a range
is
more
instructive.
We
are
The proof
provided in the appendix. In the proof, we focus on the vector process
is
[«]
Yn (s,u) =
n-V^Xtim
<u)-u}
(6)
t=\
where U\, U2,
•••,
we
Effectively,
Un
are
uniform on
i.i.d.
[0,1]
replace e t by F(e t ) which
Yn
and
Corollary
Hn
Gaussian process
H
for
finite
z)'}
,
Hn
Hn (s,z) =
converges weakly to a
= Q(iy 1/2 Q(r A s)Q(l)-^[F(z Ay)-
F(z)F(y)}.
dimensional convergence to a normal distribution follows from
To
weakly to some process H.
Yn
t.
Q(l), a positive definite ma-
the process
martingale differences. This together with Theorem
value of
<
mean and covariance matrix
with zero
E{H(r, y)H{s,
Proof. The
-?-+
So that
[0,1].
for j
are equivalent in terms of stochastic equicontinuity.
Under assumptions (A.1)-(A.6),
1
independent of Xj
t
uniform on
is
(X'Xln)- x l 2 Yn {s,F{z)). By assumption, (X'X/n)
trix, so
U
with
for r
<
s
implies that
Hn
CLT
converges
verify the covariance matrix, consider the expected
=
and u
1
(7)
F{z)
<
v
=
F(y). Using double expectation and
martingale property, we obtain
[nT]
E{Yn (r,u)Y^s,v)}
which tends to Q(r)(u
We now
tests for
-
uv).
1
= -E
\
(
\Tx x'A
t
From (X'X/n)-' / 2
1
-^
-
(u
Q(l)
_1
,
uv)
we
introduce a Brownian bridge type process which
parameter constancy
arrive at (7).
is
Xk =
Let
in linear regressions.
(8)
closely related to
(xi, ...,Xk)',
X
=
(xi,x 2 ,...,x n )', and
A k = [x'xy^ix'Mix'x)The matrix A[ns converges
]
Q(s)
= sQ
matrix.
for
some
to A(s)
=
1 '2
Q(l)~ 1 ^ 2 Q(s)Q(l)~ 1 / 2
positive definite matrix Q, A(s)
=
si,
(9)
.
.
In the special case that
where /
is
a p x p identity
the assumptions of Corollary
Corollary 2 Under
Vn (s,z) = Hn (s,z) - A
V
converges weakly to a Gaussian process
E{V(r,y)V(s,z)'}
=
{A(r
As)-
[ns]
the process
A(r)A(s)}{F(y A
by Corollary
(10) follows easily
As noted
s.
F(y)F(z)}.
Hn
The
(10)
and the conver-
limiting process of
from
= H(s,z)-A(s)H(l,z).
(7).
when Q(s) = sQ
earlier,
covariance matrix of
is
-
1,
V(s,z)
Now
z)
from that of
follows
]
is,
defined as
and covariance matrix
zero
gence of A[ ns to a deterministic matrix A(s) uniformly in
Vn
Vn
Hn {l,z)
mean
with
Vn
Proof. The stochastic equicontinuity of
1,
V becomes
said to be a two-parameter
(r
for
Q >
some
A(s) becomes si and the
0,
As — rs){F(z Ay) — F(z)F(y)}I. A
Brownian bridge on
[0, l]
2
if it is
process B(s, u)
a zero-mean Gaussian
process with covariance function
EB(r, u)B(s,
We
see that V(s,z) has the
same
v)
=
(r
A
s
—
rs)(u
A
v
—
uv).
distribution as B'(s,F(z)), where
B*
is
a vector of
p independent Brownian bridges.
An
4
Application in Structural Change
Consider the structural change model
Ho
'
=
ft\
02 with Hi unknown.
example, Andrews
process.
by
it
=
We
yt
—
[4].
Define the p x
Tn (-, z) =
(X'X)- 1 ' 2
1
The
There
Here we construct a
OLS
estimate model (1) by
x't /3.
(2).
is
objective
t
I(i t
to test the null hypothesis
a rich literature on the problem, for
test using
or other
vector process
£x
is
an estimated sequential empirical
methods and compute the residuals
Tn
,
<z)- A k (X'X)-^ JT x
2
t
I(i t
<z)
(11)
and the
test statistic
M
k
where
\\y\\<x>
=
max{|t/i|,
...,
n 2 different values, so the
M
maximum
n
z
maximum
the
\y p \},
||Tn (-,z)||oo
norm. The process
value always exists.
£x
(X'X)- 1/2
- A k (X'X)- 1/2
t
so that I(£ t
<
Therefore,
.
z)
<
can be replaced by I(£ t
Tn
is
actual computation of
£x
t
=
for all k
z)
—
F(z) without changing the value of
centered (only approximately centered because
of e ( ). Recognizing this,
Tn
takes at most
t=i
t=i
Tn
The
Tn
straightforward. If x t contains a constant regressor (we do assume this), then
is
n
k
- max sup
n
we
see that
Tn
uses estimated residuals while
Vn
Using the result of
Bai
this paper,
a root-n consistent estimator of
/?,
is
Vn
the same as
F
is
not the
d.f.
defined in Corollary 2 except
uses true disturbances.
[7]
shows that
if
the residuals are obtained from
then
Tn ^-,z)=>B*(s,F(z))
[
(
where B*
= (Bi,B 2 ,.-.,Bp
bridges defined on
[0, l]
2
.
is
)
A
n
a vector of p independent two-parameter Brownian
similar result
is
obtained by Bai
weighted) sequential empirical processes based on
[6]
ARMA residuals.
for
By
bounded (nonthe continuous
mapping theorem,
M
n
—
max
||2?*(5,u)||oo-
0<s,u<l
The
test
M
[8].
It is
interesting to realize that the limiting process T„ does not
n is
asymptotically distribution
free.
estimated parameters. The underlying reason
The estimation
terms.
This
is
effects are
that the process
canceled out in the
first
in contrast with the classical goodness-of-fit test
does not go away, see Durbin
As
is
Critical values are tabulated in Bai
[13]
for the limiting distribution
and
Tn
depend on the
consists of
two
term and the second term.
where the estimation
effect
[14].
under
tion in a simultaneous equations system.
local alternatives,
we consider a
The model under the
10
single equa-
null hypothesis
is
given
by
Suppose the alternative hypothesis postulates that
(5).
= z' pt + e
yt
where
f3 t
=
(3[l
+
t
A^g(t/n)] with g a vector-valued function defined on
Riemann-Stieltjes integrable. Suppose x t
assume
plicity,
in (A. 3)
sQxz, a p x r matrix.
Mn
where p(u)
if
if
g
column rank.
M
n will
hypotheses.
test
is
[8]
0<5,U<1
and G(s)
0.
Also assume plim^- J2™{ x t z
=
2
+ p(u)Q^ Q xz G(s)\\ 00
J* g(v)dv
-
Note that G(s)
sftg{v)dv.
a constant vector, implying no change in
local
power
=
't
shows that
-±+ max \\B*(s,u)
have non-trivial
test to
that
Then Bai
and
[0, 1]
a vector of instrumental variables. For sim-
is
= sQ xx for some Q xx >
Q(s)
= /(F -1 ^))
and only
(12)
t
non-constant
for "all"
g's,
t
=
In order for the
.
must have a
Q xz
Q xz G(s) e
Otherwise, there exists a non-zero G(s) such that
full
so
have the same limiting distribution under both the null and alternative
In
summary, when
valid instruments are available,
changes in a simultaneous equations system and
M
n
can be used to
M
n possesses non-trivial local
power. Regressors themselves constitute valid instruments when they are independent
of error disturbances.
The same
test
can also detect changes in the following type:
Vt
with e* having a continuous
t
>
ni.
Bai
[8]
argues that even
variance, as long as
and Wald
tests
may
F^
fail
G,
F
d.f.
if
it is
=
for
t
x't P
<
+
rii
e't
and
a*
having a continuous
the two distributions have the same
detectable by
Mn
,
G
for
mean and same
whereas the fluctuation,
to diagnose this kind of shift.
11
d.f.
CUSUM
A
Appendix
Lemma
A.l Assume
such that for
all Si
<
the conditions of
and
s2
u\
—
Si
<
a
>
-
Ul ) {s 2
Ul )
,
a
s,, u,
+r
7l
,u 1 )||
=
<
<
oo,
1,2)
2 'Y
+ n-^-^K(u 2 -
Sl )
K
Ul )(s 2
-
7, since \u 2
si).
—
<
Ui\
and
1
Moreover, when
1.
0,
lemma
the
This inequality
Proof. Write
,ui)
and j
1 (z
there exists a
I
U2
_ Ul
rn -(-y-D/2(Q -i)<
and
=
is
r\ t
t- 2 1*-V](u 3
I(ui
+ Yn (si,ui)
< U <
t
for the
u2)
is
t
gale differences,
where Tt
is
-
u2
+
=
Q
Sl )
(14)
.
198).
= Yn (s 2 ,u 2 - Yn (si,u 2 )
n~ x l 2
Y^i<t<i^tT}t
)
with
i
=
[ns\]
a sequence of (nonstationary) vector martin-
the u-field generated by
inequality of Rosenthal (Hall and
-
([10], p.
and Y*
ui
moment. Then Y*
Note that {x t T} ^t)
[ns 2 ].
(13)
+ Yn (s uUl )\\^
a
Ul ) (s 2
-
analogous to (22.15) of Billingsley
=
_ Si
S2
implies
< K[\ +
2
<
( Sl
E\\Yn (s 2 ,u 2 )-Yn (s 1 ,u 2 )-Yn (s 2 ,u 1 )
Yn (s
Then
1 hold.
one can assume that a
loss of generality,
Tn -(7-i)/2M)<
for r
<
where
,
)
< K(u 2 -
I-52
u2
- Yn (s u u 2 - Yn (s 2
E\\Yn (s 2 ,u 2 )
Without the
<
Theorem
Heyde
[15] p.
...,x t ,x t+ \] ...,Ut -i :
U
t
23), there exists a constant
.
By
the
M < 00
only depending on 7 and p such that
e\\y:\F
=EI
(
U£
xtnt]'
<ME[-J2
n
\
Note that x
t
addition, Erft
is
£
x hVh
)
)
^{(zJxO^t-i}) +Mn~T
i<t<j
j
measurable with respect to Tt-\ and
<u — u\
2
and
En^ <
u2
— U\.
12
These
£
E{(x't x
t
r^}.
(15)
i<t<j
r\t
is
independent of Tt-\.
results together
In
with assumption
The
(A. 6) provide bounds for the two terms on the right of (15).
term
first
bounded
is
by
M(u 2 and the second term
yE
is
bounded by
I-
T
Mn-^- x \u 2 - ux)-
A. 2 Under
\\Yn (s,u)
(x't x t
< MK(u 2 -
)
)
<MKn~
E{x'x t V
MK as K, the lemma follows from (u
Renaming
Lemma
£
Ui
-
<
J;(«i,ui)||
where the term
we have for
(A. 5),
O p (l)
\\Yn (s 2 ,u 2 )
uniform
is
rn
-
<
s\
>
in s (s
<
s
( Sl ,
2
—
s2
'
( 1
- x)
ui)
7
{u 2
<
-u
(u 2
—
a
-
Ul y(s 2
Sl
)(s 2
l
Ui)
)
-s
a
,
and u\ < u < u 2
l ).
for
>
7
a.
,
Ul )||
+ O p (l)n 1/2 [(u 2 - u x ) +
Si),
does not depend on u and u\ and
-
(s 2
Sl )]
satisfies
P(\0 P (1)\ > C) <
Proof. First notice that
Otherwise write x t
and x^(i)
=
=
(0, ..0,
M/C 2(1+p)
of the
all
VC>0,
,
=
(0, ..0,i t ,-,0, ...,0)'
—x
Yn
can be written as a linear
tt
,
0,
...,
1
0)' if
<
x (1
t
xtI(U
<
u)
<
a,
6,
for all
Yn (s,u) -Tn (5
1
1
A new
<
\\x t
[ns]
/
+n 1/2 (-E^)("2-u) +
n
is
-Yn (s
)
1
/2
and
\\
and xj{i).
||x t
It is
(0ll
— x
ll
*ll-
1
-
t
>
0,
easy to show
1
,ui)
[nsj]
^
x {I(U
t
t
1
( Sl , Ul
)-Yn (s,u)<nV
2
/
["«]
(-J2xt)(u-Ui) + n 1/2
13
1
-
<u )-u 2 }
2
M
£
So
6,
the vector functions
and
Yn
>
thus enough to
piece of notation, for vectors a and
in u. It
< Yn (s 2 ,u 2
,ui)
||x*(i)||
components. Since x
and x t u are nondecreasing
\lx t i
most 2p processes with each process having
satisfied for xf{i)
assume that the x are nonnegative.
mean
In this way,
0.
or -1) of at
assumptions (A. 5) and (A. 6) are
b to
0.
components of x t can be assumed to be nonnegative.
nonnegative weighting vectors. In addition,
<
>
Y%=i x t(i)~ Ef=i x 7(i) where xf(i)
combination (with coefficients
take a
for some p
x / {7(f/t
<
u)
-
Ul
}
The lemma
follows from the boundedness of the indicator function
Proof of Theorem
We
1.
shall evaluate directly the
u 5 (Yn ) =
sup{\\Yn (s',u')-Yn (s",u")\\; \s'-s"\
We
show that
shall
any
for
>
e
and
>
rj
0,
<
modulus
\u'-u"\
8,
<
there exist a 8
and (A. 5).
of continuity. Define
8,s',s",u',u"
>
€
[0, 1]}.
and an integer n
,
such that
P{u s (Yn ) >
Since
[0, l]
<
e)
n > n
77,
2
has only about 8~ squares with side length
2
every point
(si, Ui)
£
[0, l]
2
every
,
e
>
and
>
77
0,
.
8, it suffices
to
show that
for
there exist a 8 £ (0,1) and an
integer no such that
<28 2
P(sup\\Yn (s,u)-Yn (s 1 ,u 1 )\\ >5e)
n > n
r),
(16)
.
(6)
where
(8)
=
{(s, u); Si
For a given 8
>
<
<
s
and
>
77
+
Si
0,
8,
ux
< u <
choose
C
U\
large
+
8}
2
f] [0, l]
.
O p {\)
enough so the
in
Lemma
A.
satisfies
P(|O p (l)| > C) <
By Lemma A.2
-yn (-si,^i)||
<
=
e/(n^ 2 C) and
m=
X(iJ) =
max
3
||y„(.si
l<i,]<m
(S)
en
(17)
(see also (22.18) of Billingsley [10], p. 199),
sup||rn (5,ii)
where
8\
[n^ 2 C8/e]
Yn (si
+
1.
+ie n ,u 1 +
+ie n ,ui +
when |O p (l)| < C,
-
jc n )
3^(si,ui)
:
||
+
2e
Write
je n )
-Yn (si,Ui).
Then
P(sup
||yB ( a ,«)-yB ( ai ,u 1 )||
>
5e)
< P(|Op (l)| > C)+P( max
l<i,j<m
(5)
Now
for fixed
i
and k
(i
>
k) write Z(j)
=
X(i,j)
(e/C)n-W( a -V < el{Cn" 2 =
)
14
—
en
\\X(iJ)\\
X(k,j). Notice that
<
jen
,
j
>
1,
>
e).
(18)
< n -i/2
which follows from n -(T-i)/2(«-i)
E\\Z(j)
-
2
Z(/)|| ^
=
where, from (14) with r
C =
e
Thus by Theorem
P( max
where K\
let
^%
-
l)e n }
a
1
,
2(Q ' 1)
-
<
/
for small
<
(14),
m
<
j
Ki =
2
a
(19)
e.
we have
^[(i - k)t n }°8°
<
k)e n }°(me n )°
K\K. The
(20)
last inequality follows
from
<max\\X(i,j)-X(kJ)\\=max\\ZU)l
is
then (20) implies
k)t n
12.2 of Billingsley once again
sum
5, of
random
>
\V(i)\
e)
variables
£;,
a
]
6
[let
l<k<i<m.
a
,
£h
=
V(h)
— V(h —
1),
so that
we obtain
in Billingsley's notation],
S^me,)"^ < ^*>
<
~
6
'
(_
a generic constant and
K$ = 2°K[K2- Note that max,
\V(i)\
=max, maxj
\\X(i, j]
(18)
P(sup
(19), the
- Yn {s
\\Yn (s,u)
By Lemma
1
,u 1
>
)\\
be)
< 8\ +
second term on the right hand side above
^1
hand
2(C/e)
>e)< ^^[(i -
V{k)\
l<t<m
By
<
]
||X(t,y)||,
P( max
Thus by
2(a - l)
(Clt)
- max \\X(k, j]
— maxj
V(i)
the partial
where K[
-
[(j
and
:
Thus by Theorem
is
a
}
(13)
Because
for large n.
P(\V(i)
V(i)
k)t n
By
7.
e/C,
>e)<
\\Z(j)\\
1
we
-
e [{i
<a <
1
12.2 of Billingsley ([10], p. 94),
max||X(z,;)||
if
KC
a generic constant and
is
(me n ) < 28
+
[1
<
because
2
S °
C =
A. 2, one can choose
side (21)
becomes K(e,
choose 8 such that K(e,
r))8
a
T])8
<
a
r),
<
,
8
2
is
K'zCt z 2
-Ar~L 6
bounded by
™\A C8f^\
'
(M/t))
2 ( 1+p >
where K(e,
(16) follows.
D
15
77) is
(21)
8~^ t+p' to assure
(17)
a constant and a
The proof
of the
and the
= 2 ^~
theorem
is
1
>
p
>
0.
left
By
completed.
References
[1]
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