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AN INEQUALITY FOR
VECTOR-VALUED MARTINGALES
AND ITS APPLICATIONS
Jushan
96-16
Bai
June 1996
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
AN INEQUALITY FOR
VECTOR-VALUED MARTINGALES
AND ITS APPLICA TIONS
Jushan
96-16
Bai
June 1996
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
:JUL18
1996
UBRAR!£5
96-16
An
Inequality for Vector- Valued Martingales and
its
Applications*
Jushan Bai
Massachusetts Institute of Technology
June, 1996
Abstract
In this paper,
we
derive a general Hajek-Renyi type inequality for vector- valued
martingales. Several well
general inequality.
We
We
known
inequalities are
shown
to be special cases of this
also derive a similar inequality for
dependent sequences.
then apply the inequality to the problem of strong consistency of least squares
estimators for multiple regressions.
AMS
1991 Subject Classifications. Primary 60E15, 60G42; secondary 60F15 62J05.
Key words and
phrases.
Hajek-Renyi inequality, martingales, mixingales, multiple
re-
gressions, strong consistency.
Running head: Vector- Valued Martingale
"Partial financial support
9414083.
is
Inequality.
acknowledged from the National Science Foundation under Grant SBR-
Introduction
1
mean and
For a sequence of independent random variables j:;,}^! with zero
ance, Hajek and Renyi (1955) proved the following inequality: for any
positive integers n
where
Ck (k
=
1.2,
N
and
...)
is
The Hajek-Renyi
erf.
numbers.
More
e
>
finite vari-
and
for
any
< N),
(n
a sequence of non-increasing and positive
inequality
is
numbers and Ezf
=
very useful in establishing strong laws of large
interestingly, this type of inequality plays
an important role in the
change-point problem for both parametric and nonparametric estimations
[see.
e.g.,
This inequality has been extended in the literature into various settings; see,
e.g.,
Bhattacharya (1987), Dumbgen (1991), and Bai (1994,1995)].
Chow
(1960) for martingales,
tingales,
and Sen (1971)
Birnbaum and Marshall (1961)
for vector- valued
Hajek-Renyi type inequality
this general inequality are
continuous time mar-
random sequences. More
and Serfiing (1990) extend the inequality to random
eral
for
fields. In this
recently, Christofides
note we derive a gen-
for vector-valued martingales. Several special cases of
then discussed.
One
of
them concerns
a sequence of quadratic
forms of martingales constructed using a decreasing sequence of positive-semidefinite
matrices (Corollary
1), as
opposed to a decreasing sequence of
This quadratic form inequality
is
scalars.
used to prove the strong consistency of least squares
estimators for multiple regressions. This latter problem has received considerable attention in the literature, e.g.,
Drygas (1976),
ity,
we
Anderson and Taylor (1976), Chen, Lai and Wei (1981),
Lai, Robbins,
and Wei (1978,1979) and Solo (1981). Using our inequal-
give an alternative and simpler proof of strong consistency.
Furthermore, we derive a similar quadratic form inequality
In particular,
we
for
dependent sequences.
allow the dependence sequence to be a mixingale, which includes linear
processes and strong mixing sequences as special cases.
This inequality
is
useful in
deriving the rate of convergence of change-point estimators for multiple regression models
with dependent disturbances.
The
2
Inequality
Let (Q,Jr P) be a probability space and Ti be an increasing sequence of
,
C T.
that Ti
Zi,Z2 ,... be
Let
cr-fields
such
a sequence of p-dimensional martingale differences relative
to {J-{] such that
E{Zn \Tn -,) =
=
Let 5^
Rv
on
E^=i
>
>
/^(z)
nonincreasing with respect to
k.
e
>
and for any
P (miK
Rp
h k (Sk )
>
=
1
(for
=
< - (E[h n (Sn
/i^,
>
and
1),
let
Z^
be a
N
(n
< A
)]
+ Yl
r
),
if
sequence
-
E[hi(Si)
we have
hiiSi-J]]
i=n+l
2
ck x
2
,
then
we
(1) specializes to
let
h(x)
.
J
=
x
2
,
we obtain the Kolmogorov
is
obtained. For p
Then hk(x)
is
>
1, let
let
hk(x)
hk{x)
=
=
CfcX
+
=
sup A ^ (A'/lA) _1/
'
Ck
2
|A'x|,
k.
where
A
For this
Sen (1971).
consider another special case of the theorem, which
M
and nonincreasing
positive-semidefinite. Let
hk(x)
we
c,tmax{0,x}, then
convex and nonincreasing with respect to
(1) yields the inequality of
If
the inequality of Hajek and Renyi (1955)
Further,
of positive-semidefinite matrices (p x p)
(2)
nonnegative and
given in the following theorem:
(i
the strong consistency of least squares estimators. Let
is
is
Let c k be a sequence of nonincreasing and positive numbers.
positive-definite.
We now
/i^
l
e)
independent random variables).
sequence of
oo
n and
\
Chow's inequality
is
is
Thus
.
an increasing sequence of a -fields {Tic}-
to
positive integers
(the one dimensional case),
type inequality.
choose h(x)
<
with Ehi(Si)
- -
p
/? p
x £
general inequality
martingale differences relative
Then for any
When
for all
Let hk(x) be a sequence of convex, nonnegative, and nonincreasing (with
1
R p -valued
(1)
>
•••
The
respect to k) functions defined on
of
S,-.
Let /it(^) be a sequence of real-valued convex functions defined
Z,-
such that h\[x)
Theorem
=
and E{ZiZ{)
a.s
= x'Mk x.
k
(k
is
=
useful in establishing
1,2,...)
be a sequence
in the sense that
Mk — Mk+\
convex, nonnegative, and nonincreasing with respect to
Clearly, hk(x)
is
Corollary
Let
matrices,
1
and
P
let
(
M
Theorem
be given in
Zk
max
S'k
MS
k
N
> /) <
k
\
£ HM&i
I
ltr(Mn J2^) +
t
= n+l
tr(-) is the trace function.
Proof of Corollary
particular h k
.
We
1:
only need to compute the right hand side of (1) for this
Now E[h n (Sn )] = E{S'n Mn Sn = tr{Mn E[Sn S'n }) = tr{Mn £?=1
-
hk(Sk-x)}
=
- S'^MkSk-^}
E[S'k MkSk
= E(Z'k MkZk)-2E[S'k _ MkZ k
1
= tr(MkE[ZkZk }) =
MZ
have used the fact that E[S'k _ x
k
k]
and
£,-)
)
E[hk(Sk )
We
have
then
1,
t=l
where
We
of p x p positive-semidefinite and nonincreasing
be a sequence
k
k.
}
tr(MkXk).
= E[S k -\Mk E{Zk \Tk -
=
l ))
0.
This proves the
x)
l l
corollary.
Remark
1.
Because h k (x)
positive-definite,
we
=
ck
sup A ^ (A'y4A) _1,/2 |A'2;|
see that Corollary
1
,
1
event {sup n < k < N h k (Sk)
1:
>
are disjoint,
The proof
t).
It
Define
is
is
2
,
where
A
is
is
The
due to the nonincreasingness of
in (2) is
due to the
this latter feature that
makes the
similar to that of
Ak =
=
N
N
h n (sn )dp
fc=n
+
f
<
t
Chow
for n
(1960).
<
r
Let
A
f h k (Sk )dP
JA *
h n+1 (sn+1 )dp
+ jrf
be the
< k;h k {Sk) >
A. Thus
= eJ2P(A k )<^2
k =n
< Eh n (sn - /
{h T (ST )
Ak G Tk and U^=n Ak
eP(A)
)
l
suitable for the least squares problem mentioned in the Introduction.
Proof of Theorem
Then the Ak
A~
whereas the nonincreasingness of h k defined
nonincreasingness of the matrix sequence Mkresult of Corollary
c k {x'
extends the inequality of Sen (1971).
nonincreasingness of Sen's h k (with respect to k)
the scalar sequence c k
=
h k (s k )dp
e}.
< Eh n (Sn +
I
)
(Sn+1 )
[h n+1
-
h n+1 (Sn ))dP
JCl-An
N
r
The
fixed k, the sequence
{.F,-}
/
Jn-(A
n UA n +
because h k (-)
£,
=
h k (Si)
Si
/i„+i(Sn ).
1,2,...)
is
4*
A
'
key observation
is
that for each
a real-valued submartingale relative to
a martingale [Doob (1953), p. 295].
is
[h n+ i(Sn+1
/
(3)
=
(i
convex and
is
>
h(Sk )dP
/
n+2'7
1 )
from h n (Sn )
last inequality follows
r
h n+1 {Sn+1 )dP+Yl
Thus
)-h n+1 (Sn )]dP>0
J An
and more generally
/ U---uA
JA
n
k
(4)
It
[h k+1
(Sk+1 )-h k+1 (Sk )]dP>0
(k>n).
follows that
Eh n (Sn + E{h n+1 (Sn+1 -
eP(A) <
+
jQ-{A„uA n+ i)
[h n+2
-
h n+2 {Sn+ i)]dP
N
h n+2 (sn+2 )
/
Jn-(A n uA n +iUA n+ 2)
The theorem
{Sn+2 )
r
-
h n+1 (Sn )}
)
)
r
+ J2
h k (sk )dp.
/
n+3^^*
follows by repeating the above
argument and making use
.
of (4).
Next, we generalize the inequality to dependent sequences. Let {Jr,}^_ 00 be a
quence of increasing
Heyde
(Hall and
and
{%l)
(i)
:
m
>
1980, p.
That
21).
0} such that
t/>
m
is,
as
1
that {Z,, F,} forms an
2
Z.
-mixingale sequence
there exist nonnegative constants
m—
>
oo and for
all
i
>
and
m
>
{6,
0,
:
i
>
1}
and
E\\E(Z \^. m )f < bUt,
x
B||Z,- B(Z,|^ +m )|| 2
(ii)
where
m
Assume
cr-fields.
se-
J
J
||i||
(iii)
=
Em=i
<6
2
(Y%=\ X 1Y^ f° r x £
rn
1+8 tp
m <
oo, for
2
t
to satisfy conditions
+1
,
R p We
-
some .6 >
Throughout the remainder of
assumed
^
assume
0.
this paper,
(i)-(iii).
in addition that
any L 2 mixingale {Z ,Ti}
The sequence
t
6,
is
called
is
implicitly
mixing norms and
^>,-
is
called
mixing
Mixingales include martingale differences, linear processes,
coefficient.
and strong mixing processes
as special cases.
Andrews
special cases of mixingales; see
<
with J2JL-oo d]
<
field{?7j; j
Theorem
°° an<^
Then
n).
M
2 Let
{Z,, Ti]
increasing matrices, and
P
(5)
Zk
let
m^ M S
S'k
(
k
2
>
k
— a2
e
=
ij>f
^jVn-j
Tn
—
o-
E|i|>» d\-
p x p non-random, positive-semidefinite and non-
<
)
Let
.
and
(for all i)
an L 2 mixingale. Then, for every
be
= E^-oo
Z„
let
zero and finite variance a 2
a mixingale with b 2
is
be a sequence of
k
other dependent structures are also
For example,
(1988).
mean
i-i-d.-
Vj
Many
£
tr(MB
f
)
£
6
2
+
£
2
6
>
e
0,
tr(JU-)
•
J
uAere
C=
4(^
2
+ 2E~
2+2
Proof of Theorem
V
We
2:
E
=
Zfc
j
1
2
+ 2E°°=1 J- 2-26 <
)(l
oo.
)
can write (Hall and Heyde 1981, p20),
Zik,
with Zj k
= E{Z k \Jrk ^
J )
- E(Zk \Jrk -i-i)-
j= — oo
/2
Ml S k =
Thus
Ef=1
1/2
Z,
£f=1
fc
Note that
for
E£_oo
P
(6)
(
each
aJ
/2
max
||Mfc
*
\n<k<N"
=
E
n
E°l_oc Af
1/2
fc
£?=1
1/2
= ££_«,
Zji
Mfc
where Sjk
5,fc,
=
m«
IIM^S.fcU
<
— P
V
max
z_^ „<fc<7v
i
< N]
^
= E{Zlt Z'-
i
)
>
<
e
/
oo
-oo
j
,
y
is
1/2
||Mfc
"
*
S,fc||
m
J
>
-
a martingale. Let a ;
2
\
P
U
- -
t \
.
i
>
K'%11
N
n
*r(Mn
°7
£
j = -oo
/
2
R p we
e)
<
1
- -
<T =E
Ej,-
>
—
for all j
such
Then
1
where
5fc||
m
{5"^,^-^;
j,
1.
\j = -oo
i £
=
Z„. Thus
P
that
M
£E ,-)+ E
i
j=l
>
\
:=n+l
(positive-semidefinite),
Substitute Zj, for x and take expectation to obtain
a,e
MM.-E,-.-)
/
and the second inequality follows from Corollary
have x'xl — xx'
>
where /
.E||Zj,||
2
is
1.
apxp
I — Ej,
>
0.
For any vector
identity matrix.
Furthermore, by
Z
the mixingale property and the definition of
Thus
for all j.
C >
- £* >
4&?^j£|2
;I
easy to show that
it is
2
<
46
2
B>A
for
hand
follows that the right
It
£]|Zj,-||
< tr(CB)
Using the fact that tr(CA)
0.
we have tr{MiL JX ) < 4b 2 tpf^tr(Me).
0,
,
2
V> J
i
and
side of (5)
is
bounded by
I
a
tA
7
2
oo
\
t
n
/
A'
£ *M M
sVS, )[tr{Mn )Y,bU
1
=1
t=n+l
"
It
remains to choose appropriate a^s such that £j a~
Vj
=
j~ l ~ s
>
for j
2E
=iJ-
where
1),
Then £j
0.
c
J
>
(j
2
-2
a,
=
>
as given in
By assumption
1.
<
semidefinite E,, then from tr(MtEji)
P
(7)
which
is
where
i, is
{e ,Jr
}
z
We
t
( max S'k
\n<fc<W
e,
=
k
>
2
e
<
)
/
bounded. Let
is
=
i/
(V»o
2+2
+2E,"i
J
and
1
i/j/(l+2X3fc=i "*) an ^ «-j
Ej ajVjj| =
see that
if
tr(MeT,i)tp?j,
£
(tr{Mn
%
y
£
.
£,,
<
^
2
=
Qj
)(l
+
positive-
we have
(6),
£,-)
+ f)
=1
some
for
S,-^»jl-|,
and
tr(M,-S.-)
1=n+ l
This situation arises
1.
.
)
/
in the following setup: Z,
nonrandom constants and
is
Hj=-oo
e j<i so
* nat
= Xix'iE^A <
EiZjiZ'ji)
setup
k
vector of
1
=
Let a ;
.
-,
e, is
random
=
x,£,-,
variable such that
forms an L 2 mixingale with mixing norms {b 2 } and mixing coefficients {V'™}-
can write
£* =
MS
analogous to Corollary
a p x
(iii),
we
Inspecting the proof
2.
(iii).
ij}
D
')<oo-
Remark
6
2 ,/,2
2
%i
=
XT^-co
(xix'^bfyfa.
^i«
We
w ith Z =
;1
i,£j,.
can choose E,
related to regression models, where x, are regressors
and
=
£,
It
follows that
(46?)x,-x;.
This
are disturbances.
Inequality (7) will be used in proving strong consistency of least squares estimators in
regression models with mixingale disturbances.
Corollary 2 For
sup, b
2
<
b
<
oo.
Z,
in
Theorem
Then, for some
2,
'UsM'i?*
Let M = k~
Then
2
pESn+i
some
k
2
'
& /1
2
M < oo.
=
0(l/nj.
.
that the
mixing norms
b
2
are such that
M < oo,
1
Proof:
assume
I.
Thus the
-')-
M
for all
tr(Mn ) £?=!
right
hand
6
2
<
N>n>
2
p6 /n and
side of (5)
is
1.
££ B+1
&
2
*r(M) <
bounded by M/(e 2 n)
for
The
corollary
is
Note that since the probability
useful in change-point problems.
bound does not depend on N. the
corollary holds
when we
replace
A
r
by
oo.
This yields
the strong law of large numbers for mixingales as well as certain rate of convergence for
the strong law.
Application
3
The
application concerns the strong consistency of least squares estimator of the param-
eter Bj in the following multiple regression:
yt
(8)
where the vector
is
+
XflA
=
•
•
+
•
(x tl ...,x tp )'
,
x ip BP
>
(i
the regression coefficient, and {e,}
sequence of
(j/i, ...,
cr-fields
y n )' and
and hence for
is
x,-
=
all
{Ti}. Denote by
Un =
(e\,
..
.,£„)'.
n > m. The
+
£i
=
xtf
+
£{
(z
known
1) consists of
=
1,2,...)
constants, B
=
(/5 l5 ...,
Bp )'
an L 2 mixingale with respect to an increasing
is
Xn
=
(xi, ...,x n )' the design
Assume (X'n Xn )
positive-definite for
is
least squares estimator b n
,
Y„
=
some n =
m
matrix and
based on the
first
let
n observations,
given by
bn
Thus strong consistency
=
(X'n
is
{X'n
(9)
When
Xn )-
l
X'n Yn
= 6+
{X'n
Xn )-'X'n Un
.
equivalent to
X n )-
l
X'n Un -»
with probability
1.
the regressors are non-random, Lai, Robbins, and Wei (1977, 1979) establish the
strong consistency under the minimal condition
(x'n
(io)
together with the assumption that
an increasing sequence of
(10)
is
significantly
£, is
7
cr-fields {.T ,}
xn y l
o
a martingale-difference sequence with respect to
such that sup, E{e 2 |^i-i)
<
weaker than the conventional assumption that
to a positive definite matrix.
Under the
latter
°°
(a.s.).
Condition
(.X'n
Xn jn)
converges
assumption, the condition number of
Xn
(X'
(the ratio of the largest and smallest eigenvalues) will be uniformly bounded,
)
and the strong consistency
and
(1979). For
be
Condition
trivial to prove.
when the
sufficient condition
et al.
will
are
e,-
i.i.d.
with zero
(9) is
mean and
dynamic models, strong consistency
is
known
as a necessary
finite variance; see Lai
also studied
many
by
authors
Anderson and Taylor, 1979, Chri Christopeit and Helmes, 1980, Lai and Wei, 1982).
(cf.
Using our inequality, we give an alternative and simpler proof of strong consistency.
We do
impose a stronger condition than
correlation structures.
minimum
the
<j)(x)
Y^
(f>
—
X'n Xn
a >
log(x)[loglog(x)]
1+l5
>
(S
This condition
is
members
and
weakest possible
(j-fields {J~i}-
=
We
of $.
<f>(X max (n))
x
for
some x
=
x5
<f>(x)
,
=
is
rjini-i,
2
<
Theorem
Then
where
77,
Ee 2 Note
.
(a.s.), as is
are
i.i.d.
00,
3 Let
We
1+6
and
= 0(Amin (n)).
is
close to be
<
b
2
<
oo.
that sup,
assumed
normal.
in
Let
When
Ee 2 <
£,-
is
a martingale-difference
oo can be significantly weaker
the previous literature. For example,
=
J- x
(T-fieldj^; j
<
i)
<
.
00
Then
is
e,
is
a
equivalent
which does not hold.
{ei,J-i} be
an L 2 mixingale with sup,
the least squares estimator b n
Proof.
[\og(x)]
series
an L 2 mixingale with respect to an increasing sequence of
Further assume that sup, b 2
E(e 2 \J-,-i) < oo
T]
and the
assume, for some </>£$,
sequence of martingale differences. The condition that sup, E(e 2 \!F,-i)
to sup,
such that each
for strong consistency.
sequence, b 2 can be taken as
let £,
<f>(x)
<f>(x)
A^^n)
and
weaker than (2.24) of Lai and Wei (1982) but slightly stronger than
Next, we assume {e,}
sup,-
>
x
Xn
For dynamic models, Lai and Wei show that condition (11)
their (1.17).
than
set of functions
For example
0.
0) are
be the
in the interval
A mjn (n) -> oo
(11)
$
Let
.
and non-decreasing
converges for every
r^rr
—
Let A max (n) denote the largest eigenvalue of X'n
eigenvalue of
positive
is
but we allow the disturbances to have broad
(9),
is
-0\\
2
-
\\{X'k
2
<
b
2
strongly consistent for
observe that
\\b k
b
X
k
Y'X'k Uk
2
\\
<
/?.
00.
Assume
(11) holds.
<
(12)
X
\\(X'k
)-^f\\{X'k X k )-^X k Uk
k
2
\\
= (Xninm-'iU'MX'M-'XiUk)
=
Let Zi
H!=i x £
i'
t
Then
x,e,.
=
i? p -valued
a
is
P(
—
||6^
2
-/3||
\\b k
S'k
>e 2 )
2
sup
<
8\\
MS
k
£
where E,
=
1
with c*
S[M,S, >
sup
(b
2
We
A min (A;).
Thus by Remark
(12).
tr(Mn
2
=
\
x
Let
}.
see that
Sk =
M
k is
2,
2
e
)
J2^)+b
£
2
*r(M,-£,))
i=n+l
1=1
/
Note that
x,x^.
tr(Mn
(14)
by
k
< P(
<
{T
mixingale with respect to
= ^(XlXk)'
X'kPk, and let Mfc
non-increasing, and
(13)
Z,
£
=
Si)
c^
1
ir((X:Xn )-
1
^X
=
n)
pc^
1
=
P /A mn (n).
-
\A,^\)/\A,\
t=i
Furthermore,
ir(M,£
where
A =
x
equivalently,
:
=
)
{X[X{) and
1
l^l
^ <
= c-'x^XlX^x, =
x'iMiXi
we have
A max (n),
-l
(\A
<K|/ln|
<
1/p
)
t
\
Because \A n
the determinant of A,.
is
|j4,-|
c
<£(A m ax(n))
<
\
A max (n) p
= 0(A mjn (n))
by
,
or
(11).
This implies that
.-.,,
1
1
,w,
,
r
(l^,|-|^-l|
c- (|A,-|-|A_ 1 |)/[A,|<l,
,
,
^
\Ai\<f>(\Ai\Vr)
for
some L <
Thus
oo.
00
£
(15)
Because J2
test.
V
£
(r(M-E,)=
'
l/( n </ ( nl/ p ))
)
Combining
<
(13)- (15),
oc
K
iS:
£
"
- ,i)/l 4
^'(KI-IA-iD/IAISI
r (l 4
'
'
'
l
'
'
1
(|4
I
—
I
4
,
"
+ ,M.wi-y"»)-
the above series converges by the integral comparison
i
and
letting
N—
oo,
we have
'
p(
The
right
hand
III
«!!>-,
A<r C /_^!_
side above converges to zero as n
10
i-tff
—
>
oo.
V-
(Mi| -|A;-i|)
MIT LIBRARIES
3 9080 00988 2652
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12
Date Due
Lib-26-67