*This research was supported by the Air Force Office of
Research under Grant AFOS~-72-2386.
ON THE PATH A3S0lUTE CONTINUITY
OF SECOND ORDER PROCESSES*
by
Stamatis Cambanis
Department of Statistics
University of North Carolina at Chapel Hitl
Institute of Statistics Mimeo Series No. 935
July, 1974
~cientific
ON THE PATH Ar.~GLUrE CONTINUITY
OF SECOND ORDER PROCESSES*
bv
J
Stamatis Cambanis
University of TYorth Car>olina
SUMMARY
Necessary and sufficient conditions are given for almost all paths of a
Gaussian process to be absolutely continuous with derivative in
L
p
Also
sufficient conditions for almost sure path absolute continuity of second ordeT
processes are derived,
sli~1t1y
generalizing those previously known.
*This research was supported by the Air Force Office of Scientific
Research under Grant AFOSR-72-2386.
fV~f·
1970 subject classifications.
Key words and phrases:
Gaussian processes.
PriPlary 60G15. 60G17. 60G99.
Absolute continuity of paths, second order processes,
-2-
Suf:r:icient· condi tions for a stochastic proc,ess to "have
absolutely continuous paths with probability one were first derived for weakly
stationary processes in [6, pp. 536-7] and were later generalized to second
order processes in [7, pp. 186-7].
For a Gaussian process it is known that its
paths are absolutely continuous with probability zero or one [4], but necessary
and sufficient conditions for the two altenlutives were known only for stationary processes [2, 10], and thus also for processes with stationary increments,
and for harmonizable processes [2].
Theorem 2 gives several equivalent neces-
sary and sufficient conditions for the paths of a Gaussian process to be absolutely continuous with probability one.
These conditions, which are slightly
more general than those of [7], are shown in Theorem I to be sufficient for
almost sure path absolute continuity of a second order process.
The results of
Theorems I and 2 extend to the case where almost all paths have
n - I
uous derivatives with the
Throughout this paper
~
= {~(t,w)
bility space
, t
€
T}
(n,F,p)
contin-
(n - l)th derivative absolutely continuous.
T
= [a,b]
is a finite interval and
a real stochastic process of second order on the probawith correlation function
denotes the path of the process
~
R(t,s)
the absolute continuity of the paths of
of its correlation function
R on
w € n and
corresponding to
the random variable of the process corresponding to
~
TxT.
on
= E(~t~s)
t
€
T.
~(·,w)
~t
denotes
We will relate
T with the absolute continuity
Real valued absolutely continu0us
functions of two variables are defined and have properties similar to those
~eal
Thus
o~
valued absolutely continuous functions of one variable [9, Section 493].
R is absolutely continuous on
TxT
if and only if there is a
L(?bcsg"'_~3
-3-
r
integrable ,function
6
t
t
2 52
6
s R=
1
on
TxT
such that for all
J2 f 2 r(u,v)du dv , where
t
5
2 52
t 6s
s I l l
tl
1
R(tl,sZ) + R(tl,sl)'
By R(R)
~ake
t
+
t
R
l
2
= R(tZ'sZ) - R(tZ,sl) sp~:e
we denote the reproducing kernel Hilbert
of the nonnegative definite function
We will also
6
t , t ; 51' 52 e: T.,
use of the map
R.
T + L2 (Q) = L2 (Q,F,P)
defined by
T.
St ' which is a Hilbert space valued function defined on
So let us
introduce the following notation and properties that will be used in the
sequel.
Let H be a Hilbert space with norm
~ P <
11 11 • Lp[T,H], 1
0
denotes the spa.ce of (equivalence classes of) measurable functions
such that
fT Ilf(t) IIPet
Bochner integral
functions
g e:
f
every
E >
JTf(tJdt
T +
Lp[T,H] .
0
A
<
there is
f
T
II
f: T
0 > 0
<
E.
+ H
For every f e: L [T ,H] , 1 ~ P < co , the
P
is well defined. ~'Jl.p[T,H] is the set of all
+
f(t) = f(a)
rg(U)dU for some
a
H is called absolutely continuous if for
+
such that for all finite collections
{(ai,bi)}~=l of disjoint subintervals of T, I~=llbi - a~1
I~=lll f(bi) - f(a'i)
f: T
co.
H of the form
function
co
It can be shown [1, appendice]
< 0
t
implies
1
1at a function
H is absolutely continuous if and only if f e: W1 ,1[T,H] . Also
f e: Wl,p[T,H] , 1 ~ P < co , if and only i f it is absolutely continuous, diff~i.·­
df
entiable a.e. [Leb] on T, and dt e: Lp[T,H]
Yihen H is the real line we
+
have the familiar results for real functions and. we will use the notation
and
WI •p [T]
for the corresponding spaces .
In Theorem 1 we give sufficient conditions for the paths of a seconct
order process to be absolutely continuous with probability one.
In Thcor(;,;'l 2
-4-
we will show that these conditions are also necessary for Gaussian processes.
THEOREM 1.
~
If
= {~(t,w)
, t
€
T
= [a,b]}
is a real separable sto-
chastic process of second order on the probability space
lation function
(i)
R(t,s)
= E(~t~s)
with corre-
, the following are equivalent.
With probability one the paths of
~
there is a measurable second order process
t h(n~)
(1)
(Q,F,P)
dt <
n
are absolutely continuous and
= {n(t,w)
, t
€
T}
such that
ex>
a
and
,(t.w) " '(a.w) +
(2)
I:
n(u,w) du
for all
t < T
with probability one.
(ii)
(iii)
with
(3)
(4)
R(r)
TI1e flap
T
+
L (n)
2
defined by
t
+
St
is absolutely continu0us.
There is a measurable nonnegative definite function
separable and
I: {r(u.u)}~
du
<
~
r(u,v) du dv .
r
on T x 7
-5-
(iv)
(R
R(t,·)
is absolutely continuous on
T for every fixed
t ET ,
is absolutely continuous on TxT)
and there is a measurable subset
2
of T with Lebesgue measure zero such that a R(t,s)/atas exists for all
t , sET - To
To
and satisfies
<
(5)
00
•
In general, (i) is stronger than the absolute continuity of the paths of
~
with probability one
(for an exceptional case see Theorem 2), and (iii)
and (iv) are stronger than the absolute continuity of
R on
TxT
(the
stationary case is an exception here).
Condition (iv) may in some cases be
easier to verify than condition (iii).
Trivial examples show that the
presence of the zero Lebesgue measure set
(iv) implies that for each
t E
T -
To
in (iv) is necessary.
To ' aR(t,o)/at
Also
is absolutely continuous
on T.
t~en
~
is harmonizable or weakly stationary with (two- and one-dimen-
sional respectively) spectral distribution
valent to
JJ:ool~~ldF(A'~) <
00
and
J~oo
F , then condition (iv) is equi-
2
A dF(A) <
00
respectively, and the
fact that (iv) implies (i) is known [6, pp. 536-7; 7, pp. 186-7].
tionary case (iv) is also equivalent to the absolute continuity of
function of two variables on TxT.
If
~
In the
R as a
is strictly stationary, it is
clear that (1) of (i) may be dropped, but it seems that we still need to
require
n to be of second order.
The relationship between the almost sure path absolute continuity of
and the absol\tte continuity of the map
t
+ ~t
has also been
con5id~red
~
in
-6-
(11] where, under certain conditions (not applicable here), it is shown that
the former implies the latter.
get)
PROOF.
(i)
= nt
l'le will show that
.
impZies (ii).
Define the function
g
g : T
L [T , LZ (~1)].
l
€
+
L (O)
2
by
In order to show that
g
is measurable, since T is compact, it suffices to show that (a) for every
h
€
L2 (O) ,E[hg(t)]
subset C of
t
€
L2 (O)
(5, p. 92].
T
measurability of
that
is Lebesgue measurable, and (b) there is a countable
such that
€
€
C , the closure of
E(hg (t)] = E (hn ) and the product
t
n is second order and measurable it follows
n.
Since
L2 (Q)
of the linear space generated by
T} , is separable [3, Theorem 1].
C of H(n)
C , for all
(a) follows froJ!1
H(n) , the closure in
{n t ' t
get)
satisfies (b).
Thus
Then any countable dense subset
g is measurable and
g
€
Ll[T , L2 (O)]
follows from (1).
Now define the second order process
X(t,w)
=
ft
n(u,w) du
for all
a
t
E T
with probability one, and the function
integral
Yet)
Y : T
+
L2 (O)
= Jt
by the Bochner
g(u) du. By using the basic properties of the Bochner
a
integral, the measurability of the processes n and X, and FUbini's theoTe~
justified by (1) we have for all fixed
ELY 2 (t)]
= Jt
a
E(g(u)Y(t))du
t
€
= Jt
T ,
ft
a a
E(nunv)du dv .
-7Hence for all fixed
= ~a
Yet)
= ~a
e T , E[X
t
Jta
g(u)du
=0
- Y(t)]2
t
.
L2 (f2)
in
and thus by (2),
~t
= ~a
X =
t
+
g e LI(T , L2 (f2)] , i t
follows that the map t + ~t belongs to WI,l(T , L (f2)] and thus it is ab2
solutely continuous [1, p. 145].
+
+
(ii) impZies (i).
~t = ~a
Jt g(u)du
a
+
is absolutely continuous,
such that
L2 (Q).
the equality being in
for all
Since
g
t E T
is measurable, there is a measurable
N of T with Lebesgue measure zero such that
~
Now define the process
= {~Ct,w)
~ = {gCt)
t
Then
~
L2(~)
, and hence so is
it.
t + ~t
Since the function
there is age LI[T , L2 (f2)]
subset
Since
Let
by
t
E
T}
is separable as a subset of
, the closure of the linear space generated by
= E[g(t)g(s)]
and
K(t,s)
= E(~t~s)
, t,s
E
T.
Since
E
T, Q(t,o)
is measurable,it is weakly measurable arid t1lUS for all 'fixed t
is measurable.
Since for fixed
it follows that
K(t,o)
t e T imply that
~
t
E
T ,K(t,o)
is also measurable.
Theorem I of [3] shows that
sepa~able.
,teN
{~t'
H(~)
is
• t e T - N
0
is of second order,
QCt,s)
, t e T}
geT - N)
H(~)
and
Q(t,o)
g
agree on T - N,
An inspection of the proof of
separable and
K(t,o)
measurable for all
has a measurable modification,denoted by
natively one can easily show the product measurability of
T).
Alter-
K from the
-8-
separability of R(K)
of
(which is isomorphic to
B(l;).) and the measurability
K(t,o) , and then apply Theorem I of [3] directly.
E(n~) = E(l;~) = E[g2(t)] and g E L1 [T , L2 (n)]
Now for all t
implies (1).
€
T - N,
This in turn
implies
and thus
p(no)
n(o,w)
= o.
Ll[T]
€
with probability one, i.e. for all
X = {X(t,w) , t E T}
Now define the process
X(t,w)
__ {toca,w)
Clearly the paths of
+
fat n (u,(ll)du
t
w
n - no with
€
by
T , WEn - no
€
X are absolutely continuous with probability one.
Also
it is easily seen, as in the previous part of the proof, that for all fixed
t ET ,
E(~t
- Xt )
2
=0
and thus
P{w En:
~(t,w)
= XCt,w)} = l .
a countable dense subset of T which is a separating set for
P{w En:
~(t,w)
= X(t,w)
, t
€
S}
=1
with probability one it follows that
and since
P{w En:
~,
If
S
we have
X has continuous paths
~(t,w)
= X(t,w)
r(u,v)
= E(nuv
n )
, t E T}
=1
and thus (i) is satisfied.
(i) impZies (iii).
This is obvious, with
note that the measurability of the second order process
is measurable and
R(r)
is separable [3, Theorem 1].
, when we
n implies that
r
is
-9-
(iii) impZies (ii).
From (4) we have that for all
= ft
that
= r(u,v)
<feu) , f(v»H
H = R(r)
instance take
and
and
f(t)
II and a function
{f(t)
€
T ,
T}
t
an,l. ret n,l.,t)
1=
LI[T, L2 (n)]
<t"
~t
and since
€
T} : h
€
T , <h,f(t»~I~
and we have for all
_ t"
~a'
t"
_ t" >
~s
~a L (n)
2
r
is measurable,
H is separable.
t,s
= <Jta
It follows that there is an isomorphism
E;
g: T
A ft f(u)du
a
t
€
H; for
H is the limit of
= limn
=
h
€
f
n
is weakly
Then (3) implies that
T
f(u)du , JS f(v)dv .>
a
H
A
between the closure in H of the
linear space generated by {Jt f(u)du , t € T} and the closure in
a
the linear space generated by {E;t - E;a ' t € T} , such that
Define
H such
-+
is complete in
Every h
€
f: T
N
r. nl
measurable and also measurable since
f
€
{f(t) , t
r·
= limn
t
= r(t,o)
a sequence of linear combinations from
N
n
hn = 1= l a n,l. f(t n,l.) . Thus for all
€
JS r(u,v)du dv .
a
a
There exists a separable Hilbert space
t,s
t
-
E;
L2 (n)
-+
= Jt
= A Jt
a
f(u)du
for all
t
L" (~n
~
of
T
€
a
by g(t)
g(u)du
= Af (t). 111en g
[8, p. 83].
a
T , and (ii) is satisfied.
€
It follows that
LI [T,
E;t
1. (n)]
2
= E;a
+
and
Jt g(u)du ,
a
-10-
(iv) impZies (iii).
t,s
€
T - T
and
0
2
R(r)
H(~)
t
€
is the closure in
and similarly for
R(r)
and
H(~)
,
=s
€
This is shown as follows.
~
for
0
L (Q)
2
= ret,s)
, t,s
= ~t
~t
by
R(R)
and
and
\'Jhere
H(~) c H(~)
T and
€
{l;t
of
~
' t € T - T~
of the linear space generated by
Now
Since
, the mean square derivative
T - T
,
t
€
T}
are isomorphic, and so are
H(~)
and thus the separability of
If
= a2R(t,s)/atas
ret,s)
elsewhere, everything in (iii) is obvious
E(~t~s)
.
H(~)
(iii) impZies (iv).
THEOREM 2.
t
111 en
To
by
Define the process
o
=o ,
r
is separable.
exists for
exists on T - T" .
~t
=0
ret,s)
except perhaps that
a R(t,s)/atas
If we define
R(R)
implies that of
R(r).
This will be shown in the proof of Theorem 2.
= {~(t,w)
, t
€
T
Gaussian process on the probability space
= [a,b]}
(Q,F,P)
is a real separable
with correlation function
R , each of the conditions (ii), (iii) and (iv) of Theorem 1 is necessary and
sufficient for the paths of
~
to be absolutely continuous with probability
one.
For harmonizable and stationary Gaussian processes the equivalence of
almost sure path absolute continuity to (iv) was shown in [2, 10].
PROOF.
It sufficies to show that if the paths of
~
are absolutely con-
tinuous with probability one then (i) of Theorem 1 is satisfied; in fact it
will also be shown that (iv) is also satisfied proving thus that (iii) implies
(iv) .
Since
~
has with probability one continuous paths it is product
measurable [12, p. 122].
If
Td(w)
is the set of points in T where the
-11-
path
~
~(·,w)
implies
is differentiable, the almost sure path absolute continuity of
=0
Leb{T - Td(w)}
T where the paths of
~
a.s.
Also, if
Td
is the set of points in
are differentiable with probability one, it is shown
Leb{Td (w)6Td } = 0 a.s.. It follows that
Leb{T - Td } = 0 and thus \'Ie will take To = T - T
For every t e Td =
d
= T - To the paths of ~ are differentiable with probability one, hence ~
in [2, Tneorem 3(ii)] that
is mean square differentiable at
(since almost sure convergence of a
t
sequence of Gaussian random variable implies convergence in the mean square)
2
and thus
a R(t,s)/atas
p (n~ = 0
exists for all
be such that for all
ous and define
~
= {~(t,w)
w
€
t,s
n - no'
, t e T}
o e F with
Now let
Td .
€
~ (0 ,w)
Q
is absolutely continu-
by
,
I;(t,w) =
Then
I;
t
€
o
t
=b
{
t e [a,b) , w e no
for
~(o,w).
define
n
n+co
and
w
is product measurable and for all
= ~1(t,W)
of
lim sup n[~~t + ~ , w) - ~(t,w)]
n
Also for all
= {n(t,w)
, t
€
n (t,w)
It is clear that
n
t
T}
€
€ Q -
~I(o,W)
t e Td(w) - {b} ,where
[a,b) , w
Td - {b} , /;(t,w)
•
€
Q - Q
o
W € Q .
no ' we have
=
~(t,w)
denotes the path derivative
= ~1(t,W)
a.s . .
Now
hy
__ {O/; (t ,w)
is product measurable and also Gaussian, since for all
-
~
t
)
a.s. and
is Gaussian.
Also
-12for all
WEn - no
= r;(c,w) = ~t(·,w)
n(o,w)
and since
is absolutely continuous,
~(·,w)
that with probability one
b
p. 391],
I
k
2
E 2 (n )du <
that for all
u
n(o,w)
~ ,
u
a
a.e. [Leb] on
E
Ll[T]
~'(o,w)
T
E L1[T].
It follows
and hence by a result in [13,
i.e. (1) is satisfied.
(5) follows if we note
d - {b} ,
E T
= d2R(t,s)
at()s
/
t=s=u .
We also have
~(t,w) = ~(a,w)
+
It
n(u,w)du , t
€
[a,b] ,
a
WEn - no
Now an application of Fubinits theorem justified hy (1) gives
R(t,s)
=
= R(t,a)
t2
S2
t l
51
I I
+
I: E(~tnu)du
t,s
E
T
E(nunv)du dv
and since by (1) the functions inside the integrals are Lebesgue integrable,
it follows that
R(t,o)
t
R is absolutely continuous on
E
T and that
is absolutely continuous on
T for every fixed
TxT.
Thus (i) and (iv)
-1.: -
t:
tl1f.t if t>.e !'rocess
is Teal l:.32Sur".:'].e 2nd. C2ussian C'.r.:::
wit2: protal:ility ene if
ex>
r'~/?.. c' t
{..., (t ~ t)
J
only i f
&'"ld
1::; l' <
<
ex>
~
"re also need to introduce
(13, p. .391]
integer and
1 ::; P <
co
:I
of all functions
continuously cifferentiable on
are
~aant
ant::
:f
of all
functions on
THEORH1 3.
If
( n..• -l)
= {t:(t.w)
R and. if
j
a positive
~
u
~
:l.:re
(n
l)-tiF~cs
(t!ie derivatives
r,~,·q
"1 ,;:'
.,., ,-" ....-
E:
• t
E:
1
= [~.b~}
~ave t~e s~2ce,
is a real separable sto-
C;1Bstic process of secone order on the probability space
relation function
-.~
.~
'/1!lic:~
it·
....
T;T
!'11
1):'"
atsolutely continuous with cerivative in
I
t:
T -+
n
(n - 1) -times continuously <H£ferentia:.:le red
,. (n-I)
T with
f
fr;'
"'r
sp8.ce
is the reBl line we
in the strong sense).
d.enote(:I, LY
t~"e
n
(n,r-.p)
is a positive integer, t:le
~rith
::0 11owing
cor-
are
equivalent.
satisfies (i) of
(ii)
(iii)
Tee ;.:ap
T
-+
L"Cn)
,.
",2(n-l)r(t
"')/".0<.. . n-l'"oS :1-1
o
.'..,",
t -+ t: t
exists on
Trieore~
belongs to
TxT
1.
n,l r'r'
'." • L ? (0)1
11
and satisfies
_u
(~11)
-'
of
TheoreB 1.
(iv)
TheoreI:! 1.
a7.(n-l):.. . (t,s)/atn -. l as Tl - 1
exists on
TxT
and satisfies (iv) of
-14-
Again when
~
is harmonizab1e or stationary. (iv) is equivalent to
ff~= IAlnl~lndF(A,~) <
THEOREM 4.
~
If
00
and
J~oo
= {~(t,w)
2n
A dF(A) <
, t
T
E
n
1
is a positive integer and
(i)
(ii)
~(o,w) E
The map
W
n.p
[T]
~
P
<
= [a,b]}
(Q,F,p)
sian process on the probability space
00
,
respectively.
00
is a real separable Gaus-
with correlation function
the following are equivalent.
with probability one.
T -+ L2 (Q)
defined by
t -+ i;t
belongs to
Hn,pf'I', L2 (Q)].
(iii) and (iv) are as in Theorer.! 3 with 1/2 replaced by p/2 in (3) and
(5).
n
-15-
REFERENCES
[1]
Br~zis,
Op~rateurs
H. (1973).
Maximaux
~Ponotones
Contractions dans les Espaces de f;ilbert.
[2]
Cambanis, S. (1973).
Cambanis, S. (1974).
or~.eT
[4]
J. ftfultivariate Anal., 3,420-434.
TIle measurability of a stochastic process of second
Proa. ArneI'. Hath. Soa., in print.
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Department of Statistics
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J.