1. This research was supported by the Air Force Office of Scientific Research
under Grant AFOSR-68-1415.
ON SOME CONTINUITY Arm
DIFFERENTIABILITY PROPERTIES
OF PATHS OF GAUSSIAN PROCESSES1
Stamatis
C~mbanis
Department of Statistios
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 828
June, 1972
ON SOME CONTINUITY AND DIFFERENTIABILITY
PROPERTIES OF PATHS OF GAUSSIAN PROCESSES l
Stamatis Cambanis
University of North Carolina at Chapel Hill
ABSTRACT
This paper considers some path
cesses
~
pro~erties
of real separable Gaussian pro-
with parameter set an arbitrary interval.
are established, among others.
The following results
At every fixed point the paths of
continuous, or differentiable, with probability zero or one.
If
~
are
is meas-
~
urable, then with probability one its paths have essentially the same points
of continuity and differentiability.
If
continuous or
pc~_nt,
diffe~entiable
at every
~
is measurable and not mean square
then with probability one its
paths are almost nowhere continuous or differentiable respectively.
If
is
~
mean square contir.uous and 8tationary, then its paths are differentiable with
If
~
is
harmcnizable, then its paths are absolutely continuous if and only if
~
is
probability or.e if and only if
mean square differentiable.
~
is mean square differentiable.
Also a class of harmonizable processes is deter-
mined for which the following are true:
(i) with probability one paths are
either continuous or unbounded on every jnterval, and
(ii) path differentia-
bility with pr0bability one is equivalent to mean square differentiability.
ON SOME CONTINUITY AND DIFFERENTIABILITY
PROPERTIES OF PATHS OF GAUSSIAN PROCESSES I
Stamatis Cambanis
University of North Carolina at Chapel Hill
1.
INTRODUCTION AND STATEMENT OF RESULTS
In this paper local and global path continuity and differentiability properties of Gaussian processes are studied.
~
= {~(t,w),
rable Gaussian process on the probability space
mean and covariance functions;
a-algebra of subsets of
{~(t,w),
t€T}
a~e
S"2
T
tET}
is a real sepa-
(n, F, P)
with arbitrary
F
is the smallest
is any interval and
with respect to whi.ch the random variables
measurable.
It has been shown in [2] that the paths of
lowing properties on
~
have anyone of the fol-
T with probability zero or one:
tiability, absolute continuity, etc.
continuity, differen-
A difficult problem is to obtain necessary
and sufficient conditions for the two alternatives.
Partial answers to this
problem are gi.ven in Theorems 4 and 6, where necessary and sufficient conditions are obtained for almost sure path differentiability of stationary processes and almost sure path absolute continuity of harmonizable processes.
Zero-one type
~esults
for local path continuity and differentiability are
given in Theorems 1 and 3, where it is also shown that the set of points of
continuity and differentiability are essentially the same for abnost all paths.
Note that these results do not assume mean square continuity of
If in c..i.dition
continuity has
~
t;,
is mean square continuous on
bee~thoroughly
T,
~.
then its modulus of
studied by Ito and Nisio (1968).
A similar local
result is obtained under a local mean square continuity assumption (Theorem 1),
2
and corresponding local and global results for path differentiability are obtained under local and global mean square differentiability assumptions (Theorem 3).
Also Belayev's (1961) alternatives for mean square continuous stationary
Gaussian processes are shown to be valid for a slightly larger class of
Gaussian processes (Corollary 3).
A remarkable feature of the proofs of all results is the fact that even
when mean square continuity and stationarity are assumed, no use is made of the
available process representations:
spectral representation.
the Karhumen-Loeve representation and the
Instead a recently established zero-one law for
Gaussian processes is used [8, 9, 12].
All results are stated and discussed in this section and their proofs are
given in Section 2.
In should be pointed out that, for simplicity, no special care has been
taken for the endpoints of the interval
T,
when they belong to
T;
in
every particular case it is easy to see what is the precise meaning of the
results.
Finally. even though all results are stated and proven for
val in
RI
to the case where
and
~
~
and for
T
T an
inte~­
a real valued process, they can be easily generalized
is an olinterval ll in
takes complex values or values in
n
R
or a separable metric space,
Rn
(the n-dimensional case) or,for
some of the results. in a compact metric space.
1.1.
Continuity properties of paths of Gaussian processes.
uity properties
the path
~(·,w)
C2TI
&t
tv (t, w)
t;,
be studied by using the oscillation function
tET
=
Path continW~(t,w)
defined by
lim
sup
I~(u,w) - ~(v,w)1
E ~ ~ u,V E (t-E,t+E) n T
of
3
for every
tET
and
Clearly the path
= O.
The separability of
and only if
W~(t,w)
W~(t,w)
F-measurable, where
is
~(o,w)
WEQ.
~
is continuous at
t
if
implies that for each
tET,
-F- is the completion of F with respect to
P.
THEOREM 1.
where
T
(i)
Let
~
=
{~(t,w),
tET}
be a real separable Gaussian process,
is any interval.
At every fixed point
t
in
T
the paths of
~
are continuous with
probability zero or one.
(ii)
Let
T
c
be the set of points
continuous with probability one, and
uity of the path
~(o,w).
If
s
t
in
T (w)
c
T where the paths of
are
~
be the set of points of contin-
is measurable then with probability one
Leb{T (w)
c
~
T}
c
=
0
i,e., almost all paths have essentially the same set of points of continuity.
(iii)
If
s
is mean square continuous at the point
tET,
then with prob-
ability one
=
where
is a constant (extended real number).
The properties given in Theorem 1 are local continuity properties.
One
can similarly obtain the global property (the proof is similar to that of
Theorem 3 (iv»:
(iv)
If
~
is mean square continuous on
=
T,
for all
then with probability one
tET.
This result was proven by Ito and Nisio (1968) for compact intervals
T,
by
4
~,
employing a global property of
its Karhunen-Loeve representation; they
have also characterized the oscillation function
At a point of discontinuity it is important to know the value of the
oscillation and in particular whether this value is finite or infinite.
As
in Theorem 1, the following are seen to be true.
(v)
At every point
t
in
T
the paths of
~
have with probability one
either finite or infinite oscillation.
(vi)
Let
T
o
be the set of points
t
in
T where the paths of
T (w)
infinite (resp. finite) oscillation with probability one, and
o
set of points of infinite (resp. finite) oscillation of the path
is measurable, then with probability one
Leb{T (w) 6 T }
o
0
= 0,
~
have
be the .
~(·,w).
If
i. e. ,
almost all paths have essentially the same set of points of infinite (resp.
finite) oscillation.
By introducing appropriate special oscillation functions one can study
local properties such as left or right continuity at a point, existence of
left or right limits at a point, and whether a point is a simple discontinuity.
For
~nstance,
the oscillation functions appropriate for the study of left con-'
tinuity and of the existence of left limits are respectively
=
W~(t,w)
"
lim
s +
=
lim
s
+
sup
° uE(t-s,t)nT
l~(u,w) - ~(t,w)1
sup
° u,vE(t-s,t)nT
I~(u,w) - ~(v,w)l.
One C3.n thus obtain the results of Theorem 1 (i) to (vi) with "continuity"
repla,~ed
by:
left (or right) continuity
existence of left (or right) limits
5
simple discontinuity
and "oscillation" replaced by the corresponding special oscillation.
For Gaussian processes, almost sure path continuity on
square continuity on
T, but the converse is not true.
mean square discontinuity and path discontinuity on
T
implies mean
A relationship between
T
is given in the fol-
lowing theorem, whose inverse is not true.
THEOREM 2.
If a real separable and measurable Gaussian process on an in-
terval is not mean square continuous at any
point, then with probability one
its paths are almost nowhere continuous.
Here almost refers to the Lebesgue measure and also there is no fixed
point of continuity of the paths.
1.2.
Differentiability properties of paths of Gaussian processes.
For
the study of path differentiability properties we introduce the modulus of
6~(t,w)
differentiability
6~ (t ,w)
for every
tET
=
lim
sup
E + 0 U,VE[(t-E,t)U(t,t+E)]nT
tET
and
if and only if
for each
~(·,w)
of the path
tET,
WEn.
Clearly the path
= O.
6~(t,w)
6~(t,w)
at
tET
defined by
~(u,w)-~(t,w)
_
~(·,w)
v - t
is differentiable at
Also, the separability of
-
is F-measurable.
~(v,w)-~(t,w)
u - t
~
implies that
Differentiability properties simi-
lar to the continuity properties of Theorem 1 are given in the following
theorem.
THEOREM 3.
where
T
(i)
Let
~
=
{~(t,w),
tET}
be a real separable Gaussian process,
is any interval.
At every fixed point
with probability zero or one.
t
in
T
the paths of
~
are differentiable
6
(ii)
Let
T
d
be the set of points
t
differentiable with probability one, and
T where the path
~(',w)
in
T where the paths of
~
are
be the set of points
t
in
Td(w)
is differentiable.
If
~
is measurable, then with
probability one
i.e., almost all paths have essentially the same set of points of differentiability.
(iii)
If
~
is mean square differentiable at the point
tET,
then with
probability one
=
where
e~(t)
(iv)
If
is a constant (extended real number).
~
is mean square differentiable on
T,
then with probability
one
for all
=
tET.
The results of Theorem 3 remain valid if "differentiability" is
by "left (or right) differentiability".
repl~ced
Also, one could possibly obtain a
characterization of the modulus of differentiability
characterization of the modulus of continuity
a~
S~
similar to the
given in [7].
An important question is the relationship between mean square differentiability and path differentiability.
differentiability on
T
For Gaussian processes almost sure path
implies mean square differentiability on
T.
The ex-
tent to which the converse is true for Gaussian processes is not known at
present (see also Theorem 7).
~
An interesting case where the converse is true
is that of stationary Gaussian processes.
7
THEOREM 4.
A real separable mean square continuous stationary Gaussian
process has everywhere differentiable paths with probability one if and only
if it is mean square differentiable, or equivalently, if and only if
where
F
is its spectral distribution.
For a real separable mean square continuous wide sense stationary (not
necessarily Gaussian) process it is known that mean square differentiability
implies absolute continuity of paths with probability one, and .hence almost
everywhere path differentiability with probability one [5, pp. 536-537].
Theorem 4 strengthens this result for Gaussian processes.
Theorem 4 is also valid for real separable mean square continuous Gaussian
processes with stationary increments.
In this case a necessary and sufficient
condition for mean square differentiability is
where
H is the spectral distribution as in [5, p. 552].
As it is clear from the proof of Theorem 4, if
~'(t,w)
ferentiab1e, the path derivatives
of
~
~(t,w)
is mean square difform a mean square con-
tinuous stationary Gaussian process with spectral distribution
dG(A)
= A2dF(A).
G such that
~'
According to Belayev's alternatives then, the paths of
are either continuous or unbounded on every interval with probability one.
It now follows from Theorem 4 and from Theorem 3 of [1] that a sufficient condition for the latter alternative is that
isfying
2
fOO A f(A)dA < +00
-00
f (A)
~
has a spectral density
and such that for some
C
for all
C > 0
and
f
sat-
8
Theorem 4 can be used to derive sufficient conditions for the derivatives
of the paths of a stationary Gaussian process to have certain properties,
from known sufficient conditions for the paths of a stationary Gaussian process to have the same properties.
For instance Hunt's (1951) sufficient con-
dition for almost sure path continuity:
for some
a > 1,
and Theorem 4 imply that a sufficient condition for almost all paths of
be n-times continuously differentiable
J:oo
(n = 1,2, •.. )
>-2n [log(l+I>-I)]a dF(A)
<
+00
~
to
is
for some
a> 1.
This condition is well-known [4] and only its derivation here is different.
Also, the sufficient condition for almost all paths of
Lipschitz condition of order
a,
~
to satisfy a
0 < a < 1,
[1], and Theorem 4 imply that a sufficient condition for almost all paths of
~
to be n-times differentiable
(n
fying a Lipschitz condition of order
The implication
ferentiability on
THEOREM 5.
T
= 1,2, •.. )
a
with n-th derivatives satis-
is
of mean square non-differentiability to path non-difis given in the following theorem.
If a real separable and measurable Gaussian process on an in-
terval is not mean square differentiable at any
one its paths are almost nowhere differentiable.
point, then with probability
9
Again, almost refers to the Lebesgue measure and there are no fixed
points of differentiability_
Theorem 5 includes as a particular case the well-known property of the
paths of the Wiener process.
1.3.
Som~
remarks on the absolute continuity of paths of Gaussian pro-
cesses. As it is pointed out in Section
1. 2, the implication of mean square
differentiability to path properties is not fully known at present even for
It is known however [3, pp. 186-187] that for a real
Gaussian processes.
separable (not necessarily Gaussian) process
~
on an interval
T,
mean
square differentiability plus some rather mild assumptions imply that with
probability one the paths of
subinterval of
T.
integrability (in
aR(t,s)
at
and
~
are absolutely continuous on every compact
These additional assumptions are
t
a 2 R(t,s)
atas
and in
t
and
s
(i) the local Lebesgue
respectively) of the derivatives
of the autocorrelation function
R of
~,
and
(ii) the existence of a measurable modification of the mean square derivative
of
~.
The converse of this result does not seem to be true for the general
Gaussian process.
path absolute
As it is seen from the proof of Theorem 6, almost sure
conti~uity
implies almost sure path differentiability and mean
square differentiability on
T
except on an at most countable set of points,
but we have not been able to strengthen this result.
An interesting class of
Gaussian processes for which the· converse of this result is true is the
harmonizable Gaussian processes.
THEOREM 6.
A real separable harmonizable Gaussian process has with prob-
ability one paths absolutely continuous on every compact interval i f and only
if i t is mean square differentiable, or equi.valently, i f and only if
where
r
is its two-dimensional spectral measure.
10
Since a mean square continuous stationary process is harmonizable (with
a spectral measure supported by the diagohal of the plane), Theorems 4 and 6
give the following result.
COROLLARY 1.
For a real separable mean square continuous stationary
Gaussian process, everywhere differentiability of its paths with probability
one is equivalent to the absolute continuity of its paths on every compact
interval with probability one.
1.4.
Some remarks for non-Gaussian processes.
It is of interest to
study the extent to which the results of Theorems 1 to 6 remain valid for
non-Gaussian processes.
It should be first remarked that the word "Gaussian"
cannot be altogether deleted from the statements of these theorems.
This is
obvious for Theorems 1 and 3; for Theorems 5 and 6 it follows from the existence of a mean square continuous stationary process with almost surely absolutely continuous paths, which is not mean square differentiable [5, p.537];
and for Theorems 2 and 4 it is conjectured.
The following property opens a way of determining classes of non-Gaussian
processes for which the results of Sections 1.1 to 1.3 apply.
THEOREM 7.
Let
~
and
n be two stochastic processes defined on the
same probability space and time interval and let
lation processes defined in Section 1.1.
so are
w~
and
W,
n
and if
W~
and
H'~
and
~
and
Then if
W be their osciln
n are equivalent,
Ware singular, so are
n
~
and
n.
Clearly the result of Theorem 7 is true for the left oscillation process,
etc., and also for the modulus of differentiabi.lity
1.2.
t:.~
defined in Section
The following is an immediate consequence of Theorem 7.
11
COROLLARY 2.
~
If a real separable process
n
separable Gaussian process
is equivalent to a real
satisfying the assumptions of anyone of
Theorems 1 to 6, then the theorem applies to
~.
As a practical method of determining non-Gaussian processes for which the
results of the previous sections are valid, Corollary 2 is of limited interest
because very little is known at present concerning characterizations of nonGaussian processes equivalent to fixed Gaussian processes.
In fact, the only
complete result available in the literature is the characterization of all
processes equivalent to the Wiener process.
On the other hand complete characterizations are known for all Gaussian
processes equivalent to a fixed Gaussian process.
Using these characterizations
we obtain from Belayev's alternatives, Theorem 4 and Corollary 2, the following result, which in fact generalizes Belayev's alternatives and Theorem
4 to a subclass of the harmonizable Gaussian processes.
COROLLARY 3.
Let
~
be a real separable harmonizable Gaussian process
with zero mean and two-dimensional spectral measure
Borel sets
Band
=
q(BnC) +
f
B
and
q
I K(A,~)dq(A)dq(~)
c
is a finite nonnegative measure on the Borel sets of the real line
K is a symmetric function in
eigenvalue.
(i)
such that for all
C of the real line
r(BXC)
where
r
L (qxq)
2
that does not have
-1
as an
Then
the paths of
~
are either continuous or unbounded on every inter-
val with probability one, and
(ii)
the paths of
if and only if
if
~
~
are everywhere differentiable with probability one
is mean square differentiable (or equivalently, if and only
fro-00 A2dq(A) < +00).
12
Implicit in the proof of Corollary 3 is the interesting fact that a
zero mean Gaussian process is equivalent to a zero mean stationary Gaussian
process with spectral measure
tra1 measure
q
if and only if it is harmonizab1e with spec-
as described in Corollary 3.
r
The case of nonzero mean
can be included in Corollary 3; (i) and (ii) remain valid if
~
~
has mean
m
of the form
m(t)
for all
t€T,
where
J:oo
-
exp(itA) f(A) dq(A)
f€L (q).
2
Corollary 3 raises the problem of characterizing the classes of Gaussian
processes, and in particular the classes of harmonizab1e Gaussian processes,
for which Be1ayev's alternatives and Theorem 4 are valid.
2.
PROOFS
The following facts are repeatedly used in the proofs of the theorems.
Let
T
R
be the set of all real functions on
subsets of
where
mation
T
R
generated by sets of the form
n
B
t , ••• ,t € T and
1
n
41:
(~2,
F,
T
P) -+ (R , U (R
T
U(R )
»
T
defined on the probability space
l.I
subset of
where
=P
0
~.
41-
1
Hw)
on
~.
= ~(o ,w)
T
U(R ).
The transforis measurable
T
U(R )
The stochastic process
T
(R , U(RT),~)
same mean and covariance functions as
~
the a-algebra of
T
n
{x€R : (x(t ), .. ,x(tn » € B },
1
defined by
with respect to
kernel Hilbert space of
T
U(R )
is an n-dimensional Borel set.
and induces a probability measure
completion of
T and
denotes the
{x(t),t€T}
is clearly Gaussian with the
Let also
H(~)
be the reproducing
(or of its autocorrelation function), i.e., the
RT which consists of all functions of the form
E[~(t,w)n(w)],
n is a random variable in the L2 (Q,F,P)-c1osure of the linear span
13
{~(t,w),
of the random variables
t€T},
and
E denotes expectation.
The
following facts are known [8, 9, 12].
- T
T
(i) If G is a U(R )-measurable subgroup of R then ~(G) = 0 or 1- T
T
(ii) If g is a U(R )-measurable real function on R and g(x-tm) =
g(x) a.e.
for all m € H(O,
[~ ]
then
PROOF OF THEOREM 1. If S is
separant of
then .for every
~,
=
lim
£
For
x€R
T
and
t€T
t€T,
a.e.
[~]
a countable dense subset of
t€T
0
T which is a
we have
sup
1~(u,w)-~(v,w)1
0 U,V€(t-£,t+£)nS
lim
£
Then for every
= constant
a.s. [P]
0
set
=
W(t,x)
~
g(x)
~
sup
Ix(u)-x(v)l.
0 U,V€(t-£,t+£)nS
W(t,x)
T
is
U(R )-measurable and
W~(t,w)
= W(t,~(w»
a.s. [P].
(i)
Then
For fixed
S u {t}
F
t
=
{W€Q:
G
t
=
{x€RT : H(t ,x)
is continuous at t},
0 or 1 and by
(ii)
Since
~
F =
t
W~(t,w)
E € B(T)xF,
its sections
~
G
t
-1
Gt
where
=
(G ),
t
P(F )
t
{(t,w) € TXQ:
B(T)
= O}
= O} •
= {x€RT :
=0
w~.
the restriction of x to
T
R •
is a subgroup of
is measurable, so is
E
Then
define the sets
T
is U(R )-rneasurable and since
G
t
~(Gt) =
t€T
It follows that
or 1.
Let
W~(t,w)
= a}.
is the a-algebra of Borel subsets of
T,
and
14
satisfy
o}
=
E
W
::
{tET: Wt; (t,W)
=
E
t
=
{WEn:
= o},
E E B(T)
w
for all
W~(t,w)
WEn,
and
T (w),
c
W€ n
t ET
EtEF
for all
tETe
Also
and
T E B(T), since E E B(T)xF implies that P(Et) is B(T)-measurab1e.
c
Now let F = T XQ. Then (EtlF) t = E tI F and this is equal to Q - E for
c
t
t
t
tET
and to
c
Et
for
t
Tc .
~
Thus
=
(LebxP) (EtlF)
=
P{(EtlF)t} dt
IT
I
T
c
=
by the definition of
Now from
T
(EtlF)
o =
it follows that
(iii)
Since
t;
W(t,x)
w
F
w
c
=
=
(Lebxp) (EtlF)
c
=0
T (w) tI T
c
c
= W(t,x)
W(t,x)
~
W(t,x+m) + W(t,m).
= Ctt;(t)
a.s.
[~].
xER
Thus
T
t
E
T-T ,
c
T
m,x E R
W(t,x) + W(t,m)
for all
c
a.s. [Pl.
~
W(t,m)
P(Et)dt
and
W(t,x+m)
and thus
T-T
IQ Leb{(EtlF) W}dP(w)
is mean square continuous at the point
t
I
which implies that for
= Ew tI
Leb{T (w) tI T }
+
0
It is easily seen that for all
continuous at
W(t,x+m)
and (i),
c
P(Q-E )dt
t
= O.
every m E H(t;)
It follows that for all
and thus
Wt;(t,w)
tET,
W(t,x)
= Ctt;(t)
m E H(t;),
is a constant a.s.
a.s. [Pl.
is
[~l,
15
PROOF OF THEOREM 2.
of
~
Fix a point
are continuous at
t
t
in
T.
By Theorem 1. (i), the paths
with probability zero or one.
If they are con-
tinuous with probability one, since almost sure convergence of a sequence of
Gaussian random variables implies convergence in the mean square, it follows
~
that
is mean square continuous at
tinuous at
t,
Hence, if
set
~
~
is mean square discon-
with probability one its paths are discontinuous at
is not mean square continuous at any
Leb{T (w)} = 0
c
a.s. [P]'
point of
T,
t.
then the
It follows, again from Theorem
i.e., with probability one the paths
are continuous on a set of zero Lebesgue measure.
PROOF OF THEOREM 3.
every
~
Thus if
defined in Theorem 1. (ii) is empty.
T
c
1. (ii), that
of
t.
This theorem is proven as Theorem 1.
We have for
t€T,
lim
E~O
and if we define for
lI(t,x)
=
lim
then for every fixed
T
x€R
E~O
lI(t,q,(w»
~(u,w)-~(t,w)
sup
U,V€[(t-E,t)U(t,t+E)]nS
and
v - t
a.s. [P]
t€T,
sup
U,V€[(t-E,t)U(t,t+E)]nS
t€T,
~(v,w)-~(t,w)
_
u - t
A(t,x)
x(v)-x(t)
v - t
x(u)-x(t)
u -
t
is U(RT)-measurable and
lI~(t,w) =
a.s. [Pl.
Parts (i) and (ii) are shown exactly as parts (i) and (ii) of Theorem 1.
For part (iii), it is easily seen that for all
If
~
T
m,x € R ,
lI(t,x+m)
~
lI(t,x) + A(t,m)
lI(t,x)
~
lI(t,x+m) + lI(t,m).
is mean square differentiable at
tET,
then every
m€
H(~)
is
16
differentiable at
t
[10, p.303] and so
= O.
~(t,m)
The proof of (iii) is
completed as in Theorem 1.
(iv)
val
We introduce the modulus of differentiability on the closed inter-
[s,t],
s < t,
~~([s,t],w)
by
l~
=
l~
wp
nt+ oo mt+oo
1
1
u,v,wE(s--,t+-)nT
n
n
u < v < W
lu-wl
where
6~(t,w).
s
= ~l (v,u,w;w).
~3(u,v,w;w)
and right continuous in
t.
mean square differentiable on
Since
s
t,
$
T then for fixed
=
S~(s,t):
s
is left continuous in
t
$
in
= S~(s,t)
s
s
t
$
in
T}
and right continuous in
6~([s,t],w)
= S~(s,t)
for all
s
$
t
in
T}
and thus
This argument is
is
constant
for all rationals
is left continuous in
P{W€Q:
~
T,
follows that
P{W€Q:
=
Hence
~~([s,t],w)
~~([s,t],w)
~~([t,t],w)
Also, it follows as in (iii), that if
~~([s,t],w)
with probability one.
~2(u,v,w;w) =
and
We also define
~~([s,t],w),
It is easily seen that
P{WEQ:
!
m
I~(U,W)-~(V,W) _ ~(U,W)-~(W'W)I
u - v
u - w
~l(u,v,w;w) =
~l(w,u,v;w),
<
6~(t,w)
s~ilar
= S~(t)
for all
to the one given for
W~
t E T)
in [7].
=
1.
=
I
=
t,
1.
it
17
PROOF OF THEOREM 4.
Assume first that
s
is mean square differentiable.
Then with probability one its paths are absolutely continuous on every compact subinterval of
[Leb].
T
[5, pp.536-537] and thus differentiable on
T a.e.
Hence with probability one
=
6. (t,w)
s
0
on
T a.e. [Leb].
On the other hand, Theorem 3. (iv) and the stationarity of
It follows that
=0
Ss
differentiable on
s
imply that
and thus with probability one the paths of
T.
Let
s'(t,w)
s'(t,w)
Since for every fixed
t,
t,
be the path derivative at
s(s,w)-E;(t,w)
lim
=
s
converging to
-+
t
s - t
1;'
i.e.,
a.s. convergence of Gaussian random variables
t
in
s
is mean square
T.
is defined except on a zero probability set and it is
clearly stationary and Gaussian.
a.s. [P]
tET,
are dif-
the convergence is almost sure along any sequence
and since
differentiable at every
s
a.s. [P].
implies convergence in the mean square, it follows that
The process
are
T.
Conversely, assume that with probability one the paths of
ferentiable on
s
Also for every fixed
the mean square derivative of
s
at
t.
t,
I;'(t,w)
equals
Thus its spectral distri-
bution
G is equal to the spectral distribution of the mean square derivative
of
Le.,
1;,
dG(A) = A2dF(A).• ·
Theorem 5 is proven as Theorem 2, by using parts (i) and (it) of
Theorem 3.
18
PROOF OF THEOREM 6.
If
~,
harmonizable process
then for all
R(t,s)
where
r
R is the autocorrelation function of the
=
t
and
s
in
T we have
J~ooJ exp(i(tA-s~» dr(A,~)
is its spectral measure, a finite signed measure on the plane which
is nonnegative definite on the measurable rectangles.
~
If
is mean square differentiable at a point
shown in [11, p. 282] that
00
1_ 001 iA exp(itA)
t
in
T,
then it is
(-i~) exp(-it~) dr(A,~)
exists
as an ordinary Lebesgue integral and thus
Conversely, the finiteness of this integral implies mean square differentiability
of
~
at
t.
Indeed
=
and it follows from
for all
A and
1
,
exp(ihA)-l - i~
1 1m
h
- 1\
h+O
h f 0,
lim
h,h' -+0
for all
A,
and the bounded convergence theorem that
h~'· b.hb.h ,
R(t,t)
i.e., the limit exists and is finite.
at
R(t+h,t) - R(t,t+h') + R(t,t)}
hh' {R(t+h, t+h')
=
Thus
Joo_oof exp(it(A-Il» Alldr(A,Il),
~
is mean square differentiable
to
Hence a harmonizable process is mean square differentiable at one point
if and only if it is everywhere mean square differentiable, and a necessary
and sufficient condition is the r-integrability of the function
All.
19
Assume now that
~
is mean square differentiable.
Then
aR(t,s)
at
2
a R(t,s)
atas
are both continuous functions, hence locally Lebesgue integrable.
since
of
is the autocorrelation of the mean square derivative
g
~,
Also,
has a measurable modification.
paths of
It follows that with probability one the
are absolutely continuous on every compact interval [3, pp. 186-
~
187] •
Conversely, assume that almost all paths of
on every compact interval of
differentiable on
=0
Leb{T-Td(w)}
in
T •
d
T
T.
excp.pt at most on a countable set of points.
and by Theorem 3.(ii),
Leb{T-T }
d
is mean square differentiable at
~
~(·,w)
Then, with probability one,
Then with probability one the paths of
and thus
are absolutely continuous
~
~
t,
= O.
is
Thus
Now fix a point
t
are differentiable at
t
and also on
T
from the
previous remark.
PROOF OF THEOREM 7.
(Q,
-F,
let
l.l
P),
1;
If
~ = {~(t,w),
-
(Q,F,p)
define the map
= poll>
-1
~
tET}
~ (R
T
and
W~
Similarly for
~
W( ° ,x )
every
.
1.S
W(t,x)
and
~~(w) = ~(o,w)
by
(RT , U(RT) ) •
are equivalent:
mutually absolutely continuous) and singular if
x
T
,U(R »
be the induced probability measure on
are called equivalent if
Note that
is a stochastic process on
l.l~
~
and
and
(1. e. ,
and
are singular:
w.
n
as defined in the proof of Theorem 1 is such that
a measura bl e map f rom
we have
(RT,U(RT»
T
C = {xER : W(o,x) E B}
to
E
(RT,U(RT
T
U(R ).
».
Also
Hence for
n
ZO
fl
W
=
(B)
~
po~-l (B)
= P{WEQ:
W~
= P{WEQ:
W(.,~~(W»
= P{WEQ:
~~
and since
0
similarly
fl~
~
W
.L fl
~
n
n
i. e. ,
fl W «fl W '
~
fl
fl (C)
fl '
E B}
fl~
(C) .
T
fl W (B)
B E U(R ),
~
= 0 and ]JW (B) = o.
n
fl
W
~
fl
~
n
implies
W
n
=
If for some
fl n ·
fl~ ~
E B}
(w) E C}
(C)
= poq,-l
~
Assume now
W~(.,w)
W
n
= 0 then
fl~ (C)
=
fl W and
t;
n
In a similar way it is seen that
fl W «
Thus
fl~ .L fl n •
PROOF OF COROLLARY 3.
Let
{n(t,w),
-00
+oo} be a mean square con-
< t <
tinuous wide sense stationary Gaussian process with zero mean, autocorrelation
function
R and spectral measure
n
Rn(t,s)
Let
LZ(n)
iables
J~oo
=
Then for all
t
and
s,
exp(i(t-s)A) dq(A).
-
be the
{n(t,w),
q.
LZ(Q, F, P)-closure of the linear span of the random var-
-00
t
<
+oo}
<
B(R)
R is the real line and
LZ(R, B(R), q)
and denote
the Borel sets of
R.
by
denoted by
Now let
n
++
such that
= {n(t,w),
tET}
n(t,w)
and let
++
~
exp(itA)
is equivalent to
n
if and only if for all
Rt; (t, s)
where
eigenvalue;
K may be taken self-adjoint.
and
-00 < t < +00.
be a Gaussian
Rt;(t,s).
Then [lZ]
~
t,s E T
n
{n(t,w), tET},
LZ(n)
tET}
R (t,s) + {Kn(t),n(s»L (Q
K is a Hilbert-Schmidt operator in
the linear span of the set
~
=
for all
= {~(t,w),
process with zero mean and autocorrelation function
where
Then it is well-known
that there is an inner product preserving isomorphism between
LZ(q),
LZ(q),
Z
LZ(n,T),
FP)
' ,
the L (Q,F,P)-closure of
2
that does not have
-1
as an
Because of the isomorphism
21
L (n,T)
2
c
L (n)
2
++
L (q),
2
and the fact that Hilbert-Schmidt operators in
spaces are of integral type, there exists a symmetric function
L (qxq)
2
= L2 (R 2 ,B(R2 ),qxq)
with kernek
for all
T.
€
where the measure
f:oof
is defined on
r
=
B(R).
q(BnC) +
Clearly
2
2
(R , B(R ))
by
J I K(A,~)
dq(A)
B
r
c
is equivalent to
sure
r
n
r.
~
is harmo-
if and only if it is harmonizable with spectral mea-
as described above.
n
It follows from Corollary 2 and Theorem 4 that the paths of
and Theorem
~
ferentiable with probability one if and only if almost all paths of
differentiable, i.e.,
definition of
r
that
1: 1 IA~ldr(A,~)
< +00,
00
of
~.
dq(~)
Hence a Gaussian process
Now (i) is a consequence of Belayev's alternatives for
7.
dq(A)
dq(~)
is a finite measure and thus
nizable with two dimensional spectral measure
~
K(A,~) exp(i(tA-s~))
exp(i(tA-s~)) dr(A,~)
r(BxC)
€
L (q)
2
as an eigenvalue, and
-1
exp(i(t-s)A) dq(A) +
= I:oof
B,C
in
Hence
R~(t,S) = f:oo
for all
such that the integral type operator in
K does not have
t, s
K(A,~)
L
2
f
2
-00 A dq(A) < +00.
foo A2dq(A) < +00
00
-00
are dif-
n are
Moreover, it is easily seen from the
is equivalent to
and from Theorem 6, to the mean square differentiability
22
REFERENCES
[1]
BELAYEV, YU.K. (1961). Continuity and Holder's conditions for sample
functions of stationary Gaussian processes. Proc. Fourth Berkeley
Symp. Math. Statist. Prob. 2, 23-33. Univ. California Press.
[2]
CAMBANIS, S. and RAJPUT, B.S. (1972). Some zero-one laws for
Gaussian processes. Ann. Math. Statist.~ to appear.
[3]
GIKHMAN, 1.1. and SKOROKHOD, A.V. (1969). Introduction to the Theory
of Random Processes. Saunders, Philadelphia •
[4J
CRAMER, H. and LEADBETTER, M.R. (1967).
chastic Processes. Wiley, New York.
[5]
DOOB, J.L. (1953).
Stochastic Processes.
[6]
HUNT, G.A. (1951).
Soc. 71, 38-69.
Random Fourier transforms.
[7]
ITO, K. and NISIO, M. (1968). On the oscillation functions of
Gaussian processes. Math. Scand. 22, 209-223.
[8]
KALLIANPUR, G. (1970). A zero-one law for Gaussian processes.
Amer. Math. Soc. 149, 199-211.
[9]
..
Stationary and ReZated StoWiley, New York.
Trans. Amer. Math.
A
(1969) .
Lecture Notes.
Trans.
Indian Statistical Institute, Calcutta.
[10]
PARZEN, E. (1959). Statistical inference on time series by Hilbert
space methods, I. Dept. of Statistics, Stanford Univ. Tech Rep.
No. 23. Also in Parzen, E. (1967). Time Series Analysis Papers.
Holden-Day, San Francisco, 251-382.
[11]
ROZANOV, YU.A. (1959).
Spectral analysis of abstract functions.
Theor. ProbabiZity Appl. 4, 271-287.
[12]
(1971). Infinite-Dimensional Gaussian Distributions.
Amer. Math. Soc., Providence.
FOOTNOTES
American Mathematical Society 1970 subject classification:
Primary 60G15, 60G17, 60GlO.
Key words and phrases: Gaussian processes, path properties, stationary
processes, zero-one laws.
1. This research was supported by the Air Force Office of Scientific Research
under Grant AFOSR-68-l415.