5
Stationary Chapter One-Parameter
Pro cesses
Section 5.1 describes the three main kinds of stationarity: strong,
weak, and conditional.
Section 5.2 relates stationary processes to the shift operators introduced in the last chapter, and to measure-preserving transformations more generally.
5.1
Kinds of Stationarity
Stationary processes are those which are, in some sense, the same at dierent
times slightly more formally, which are invariant under translation in time.
There are three particularly important forms of stationarity: strong or strict,
weak, and conditional.
Denition 49 (Strong Stationarity) A one-parameter process is strongly stationary or strictly stationary when al l its nite-dimensional distributions are
invariant under trnaslation of the indices. That is, for al l T , and al l
J Fin(T ),
L (XJ )
= L (XJ + )
(5.1)
Notice that when the parameter is discrete, we can get away with just checking
the distributions of blocks of consecutive indices.
Denition 50 (Weak Stationarity) A one-parameter process is weakly stationary or second-order stationary when, for al l t T ,
E [Xt ]
and for al l t, T ,
E [X X +t ]
= E [X0 ]
= E [X0 Xt ]
23
(5.2)
(5.3)
CHAPTER 5. STATIONARY PROCESSES
24
At this point, you should check that a weakly stationary process has timeinvariant correlations. (We will say much more about this later.) You should
also check that strong stationarity implies weak stationarity. It will turn out
that weak and strong stationarity coincide for Gaussian processes, but not in
general.
Denition 51 (Conditional (Strong) Stationarity) A one-parameter process is conditional ly stationary if its conditional distributions are invariant under time-translation: n N, for every set of n + 1 indices t1 , . . . tn+1 T ,
ti < ti+1 , and every shift ,
L Xtn+1 |Xt1 , Xt2 . . . Xtn
= L Xtn+1 + |Xt1 + , Xt2 + . . . Xtn + (5.4)
(a.s.).
Strict stationarity implies conditional stationarity, but the converse is not
true, in general. (Homogeneous Markov processes, for instance, are all conditionally stationary, but most are not stationary.) Many methods which are
normally presented using strong stationarity can be adapted to processes which
are merely conditionally stationary.1
Strong stationarity will play an important role in what follows, because it
is the natural generaliation of the IID assumption to situations with dependent
variables we allow for dependence, but the probabilistic set-up remains, in a
sense, unchanging. This will turn out to be enough to let us learn a great deal
about the process from observation, just as in the IID case.
5.2
Strictly Stationary Pro cesses and MeasurePreserving Transformations
The shift-operator representation of Section 4.2 is particularly useful for strongly
stationary processes.
Theorem 52 A process X with measure is strongly stationary if and only if
is shift-invariant, i.e., = 1 for al l in the time-evolution semi-group.
Proof: If (invariant distributions imply stationarity): For any nite collec
tion of indices J , L (XJ ) = J 1 (Lemma 25), and similarly L (XJ + ) =
1
J + .
J +
1
J +
1
J +
L (XJ + )
1 For
= J
= 1 J 1
= 1 J 1
J 1
=
= L (XJ )
more on conditional stationarity, see Caires Ferreira and (2005).
(5.5)
(5.6)
(5.7)
(5.8)
(5.9)
CHAPTER 5. STATIONARY PROCESSES
25
Only if : The statement that = 1 really means that, for any
set A X T , (A) = ( 1 A). Suppose A is a nite-dimensional cylinder
set. Then the equality holds, because all the nite-dimensional distributions
agree (by hypothesis). But this means that X and X are two processes
with the same nite-dimensional distributions, and so their innite-dimensional
distributions agree (Theorem 23), and the equality holds on all measurable sets
A.
This can be generalized somewhat.
Denition 53 (Measure-Preserving Transformation) A measurable mapping F from a measurable space , X into itself preserves measure i, A X ,
d
(A) = (F 1 A), i.e., i = F 1 . This is true just when F (X ) = X , when
X is a -valued random variable with distribution . We wil l often say that
F is measure-preserving, without qualication, when the context makes it clear
which measure is meant.
Remark on the denition. It is natural to wonder why we write the dening
property as = F 1 , rather than = F . There is actually a subtle
dierence, and the former is stronger than the latter. To see this, unpack the
statements, yielding respectively
A X , (A)
A X , (A)
= (F 1 (A))
= (F (A))
(5.10)
(5.11)
To see that Eq. 5.10 implies Eq. 5.11, pick any measurable set B , and then
apply 5.10 to F (B ) (which is X , because F is measurable). To go the other
way, from 5.11 to 5.10, it would have to be the case that, A X , B X such
that A = F (B ), i.e., every measurable set would have to be the image, under
F , of another measurable set. This is not necessarily the case; it would require,
for starters, that F be onto (surjective).
Theorem 52 says that every stationary process can be represented by a
measure-preserving transformation, namely the shift. Since measure-preserving
transformations arise in many other ways, however, it is useful to know about
the processes they generate.
Corollary 54 If F is a measure-preserving transformation on and X is a valued random variable, then the sequence F n (X ), n N is strongly stationary.
Proof: Consider shifting the sequence F n (X ) by one: the nth term in the
shifted sequence is F n+1 (X ) = F n (F (X )). But since L (F (X )) = L (X ), by
hypothesis, L F n+1 (X ) = L (F n (X )), and the measure is shift-invariant. So,
by Theorem 52, the process F n (X ) is stationary.
Exercise 5.1 (Functions of Stationary Pro cesses) Use Corol lary 54 to show
that if g is any measurable function on , then the sequence g (F n (X )) is also
stationary.
CHAPTER 5. STATIONARY PROCESSES
26
Exercise 5.2 (Continuous Measure-Preserving Families of Transformations)
Let Ft , t R+ , be a semi-group of measure-preserving transformations, with F0
being the identity. Prove the analog of Corol lary 54, i.e., that Ft (X ), t R+ ,
is a stationary process.
Exercise 5.3 (The Logistic Map as an M.P.T.) The logistic map with a =
4 is a measure-preserving transformation, and the measure it preserves has the
density 1/ x(1 x) (on the unit interval).
1. Verify that this density is invariant under the action of the logistic map.
2. Simulate the logistic map with uniformly distributed X0 . What happens to
the density of Xt as t ?