26
Decomp Chapter osition of
Stationary Pro cesses into
Ergo dic Comp onents
This chapter is concerned with the decomposition of asymptoticallymean-stationary processes into ergodic components.
Section 26.1 shows how to write the stationary distribution as a
mixture of distributions, each of which is stationary and ergodic, and
each of which is supported on a distinct part of the state space. This
is connected to ideas in nonlinear dynamics, each ergodic component
being a dierent basin of attraction.
Section 26.2 lays out some connections to statistical inference:
ergodic components can be seen as minimal sucient statistics, and
lead to powerful tests.
26.1
Construction of the Ergo dic Decomp osition
In the last lecture, we saw that the stationary distributions of a given dynamical
system form a convex set, with the ergodic distributions as the extremal points.
A standard result in convex analysis is that any point in a convex set can
be represented as a convex combination of the extremal points. Thus, any
stationary distribution can be represented as a mixture of stationary and ergodic
distributions. We would like to be able to determine the weights used in the
mixture, and even more to give them some meaningful stochastic interpretation.
Lets begin by thinking about the eective distribution we get from taking
time-averages starting from a given point. For every measurable set B , and
every nite t, At 1B (x) is a well-dened measurable function. As B ranges over
the -eld X , holding x and t xed, we get a set function, and one which,
174
CHAPTER 26. ERGODIC DECOMPOSITION
175
moreover, meets the requirements for being a probability measure. Suppose we
go further and pass to the limit.
Denition 316 (Long-Run Distribution) The long-run distribution starting from the point x is the set function (x), dened through (x, B ) = limt At 1B (x),
when the limit exists for al l B X . If (x) exists, x is an ergodic point. The
set of al l ergodic points is E .
Notice that whether or not (x) exists depends only on x (and T and X );
the initial distribution has nothing to do with it. Lets look at some properties
of the long-run distributions. (The name ergodic point is justied by one of
them, Proposition 318.)
Prop osition 317 If x E , then (x) is a probability distribution.
Proof: For every t, the set function given by At 1B (x) is clearly a probability
measure. Since (x) is dened by passage to the limit, the Vitali-Hahn Theorem
(285) says (x) must be as well.
Prop osition 318 If x E , then (x) is ergodic.
Proof: For every invariant set I , 1I (T n x) = 1I (x) for all n. Hence A1I (x)
exists and is either 0 or 1. This means (x) assigns every invariant set either
probability 0 or probability 1, so by Denition 300 it is ergodic.
Prop osition 319 If x E , then (x) is an invariant function of x, i.e., (x) =
(T x).
Proof: By Lemma 275, A1B (x) = A1B (T x), when the appropriate limit exists.
Since, by assumption, it does in this case, for every measurable set (x, B ) =
(T x, B ), and the set functions are thus equal.
Prop osition 320 If x E , then (x) is a stationary distribution.
Proof: For all B and x, 1T 1 B (x) = 1B (T x). So (x, T 1 B ) = (T x, B ).
Since, by Proposition 319, (T x, B ) = (x, B ), it nally follows that (x, B ) =
(x, T 1 B ), which proves that (x) is an invariant distribution.
Prop osition 321 If x E and f L1 ((x)), then limt At f (x) exists, and is
equal to E(x) [f ].
Proof: This is true, by the denition of (x), for the indicator functions of
all measurable sets. Thus, by linearity of At and of expectation, it is true for
all simple functions. Standard arguments then let us pass to all the functions
integrable with respect to the long-run distribution.
At this point, you should be tempted to argue as follows. If is an AMS
distribution with stationary mean m, then Af (x) = Em [f |I ] for almost all x.
CHAPTER 26. ERGODIC DECOMPOSITION
176
So, its reasonable to hope that m is a combination of the (x), and yet further
that
Af (x) = E(x) [f ]
for -almost-all x. This is basically true, but will take some extra assumptions
to get it to work.
Denition 322 (Ergo dic Comp onent) Two ergodic points x, y E belong
to the same ergodic component when (x) = (y ). We wil l write the ergodic
components as Ci , and the function mapping x to its ergodic component as (x).
(x) is not dened if x is not an ergodic point. By a slight abuse of notation,
we wil l write (Ci , B ) for the common long-run distribution of al l points in Ci .
Obviously, the ergodic components partition the set of ergodic points. (The
partition is not necessarily countable, and in some important cases, such as
that of Hamiltonian dynamical systems in statistical mechanics, it must be
uncountable (Khinchin, 1949).) Intuitively, they form the coarsest partition
which is still fully informative about the long-run distribution. Its also pretty
clear that the partition is left alone with the dynamics.
Prop osition 323 For al l ergodic points x, (x) = (T x).
Proof: By Lemma 319, (x) = (T x), and the result follows.
Notice that I have been careful not to say that the ergodic components are
invariant sets, because weve been using that to mean sets which are both left
along by the dynamics and are measurable, i.e. members of the -eld X , and
we have not established that any ergodic component is measurable, which in
turn is because we have not established that (x) is a measurable function.
Lets look a little more closely at the diculty. If B is a measurable set,
then At 1B (x) is a measurable function. If the limit exists, then A1B (x) is also
a measurable function, and consequently the set {y : A1B (y ) = A1B (x)} is a
measurable set. Then
(x) =
B X
{y : A1B (x) = A1B (y )}
(26.1)
gives the ergodic component to which x belongs. The diculty is that the
intersection is over al l measurable sets B , and there are generally an uncountable
number of them (even if is countable!), so we have no guarantee that the
intersection of uncountably many measurable sets is measurable. Consequently,
we cant say that any of the ergodic components is measurable.
The way out, as so often in mathematics, is to cheat; or, more politely,
to make an assumption which is strong enough to force open an exit, but not
so strong that we cant support it or verify it1 What we will assume is that
1 For instance, we could just assume that uncountable intersections of measurable sets
are measurable, but you will nd it instructive to try to work out the consequences of this
assumption, and to examine whether it holds for the Borel -eld B say on the unit interval,
to keep things easy.
CHAPTER 26. ERGODIC DECOMPOSITION
177
there is a countable collection of sets G such that (x) = (y ) if and only if
(x, G) = (y , G) for every G G . Then the intersection in Eq. 26.1 need only
run over the countable class G , rather than all of X , which will be enough to
reassure us that (x) is a measurable set.
Denition 324 (Countable Extension Space) A measurable space , F is
a countable extension space when there is a countable eld G of sets in such
that F = (G ), i.e., G is the generating eld of the -eld, and any normalized,
non-negative, nitely-additive set function on G has a unique extension to a
probability measure on F .
The reason the countable extension property is important is that it lets us
get away with just checking properties of measures on a countable class (the
generating eld G ). Here are a few important facts about countable extension
spaces; proofs, along with a much more detailed treatment of the general theory,
are given by Gray (1988, chs. 2 and 3), who however calls them standard
spaces.
Prop osition 325 Every countable space is a countable extension space.
Prop osition 326 Every Borel space is a countable extension space.
Remember that nite-dimensional Euclidean spaces are Borel spaces.
Prop osition 327 A countable product of countable extension spaces is a countable extension space.
The last proposition is important for us: if is a countable extension space,
it means that N is too. So if we have a discrete- or Euclidean- valued
random sequence, we can switch to the sequence space, and still appeal to
generating-class arguments based on countable elds. Without further ado,
then, lets assume that , the state space of our dynamical system, is a countable
extension space, with countable generating eld G .
Lemma 328 x E i limt At 1G (x) converges for every G G .
Proof: If : A direct consequence of Denition 324, since the set function
A1G (x) extends to a unique measure. Only if : a direct consequence of Denition 316, since every member of the generating eld is a measurable set.
Lemma 329 The set of ergodic points is measurable: E X .
Proof: For each G G , the set of x where At 1G (x) converges is measurable,
because G is a measurable set. The set where those relative frequencies converge
for all G G is the intersection of countably many measurable sets, hence itself
measurable. This set is, exactly, the set of ergodic points (Lemma 328).
CHAPTER 26. ERGODIC DECOMPOSITION
178
Lemma 330 Al l the ergodic components are measurable sets, and (x) is a
measurable function. Thus, al l Ci I .
Proof: For each G, the set {y : (y , G) = (x, G)} is measurable. So their
intersection over all G G is also measurable. But, by the countable extension
property, intersection this is precisely the set {y : (y ) = (x)}. So the ergodic
components are measurable sets, and, since 1 (Ci ) = Ci , is measurable.
Since we have already seen that T 1 Ci = Ci , and now that Ci X , we may
say that Ci I .
Remark: Because Ci is a (measurable) invariant set, (x, Ci ) = 1 for every
x Ci . However, it does not follow that there might not be a smaller set, also
with long-run measure 1, i.e., there might be a B Ci such that (x, B ) = 1.
For an extreme example, consider the uniform contraction on R, with T x = ax
for some 0 a 1. Every tra jectory converges on the origin. The only ergodic
invariant measure the the Dirac delta function. Every point belongs to a single
ergodic component.
More generally, if a little roughly2 , the ergodic components correspond to
the dynamical systems idea of basins of attraction, while the support of the
long-run distributions corresponds to the actual attractors. Basins of attraction
typically contain points which are not actually parts of the attractor.
Theorem 331 (Ergo dic Decomp osition of AMS Pro cesses) Suppose , X
is a countable extension space. If is an asymptotical ly mean stationary measure on , with stationary mean m, then (E ) = m(E ) = 1, and, for any
f L1 (m), and - and m- almost al l x,
Af (x) = E(x) [f ] = Em [f |I ]
(26.2)
so that
m(B ) =
(x, B )d(x)
(26.3)
Proof: For every set G G , At 1G (x) converges for - and m- almost all
x (Theorem 298). Since there are only countably many G, the set on which
they all converge also has probability 1; this set is E . Since (Proposition 321)
Af (x) = E(x) [f ], and (Theorem 298 again) Af (x) = Em [f |I ] a.s., we have
that E(x) [f ] = Em [f |I ] a.s.
Now let f = 1B . As we know (Lemma 289), E [A1B (X )] = Em [1B (X )] =
m(B ). But, for each x, A1B (x) = (x, B ), so m(B ) = E [(X, B )].
In words, we have decomposed the stationary mean m into the long-run
distributions of the ergodic components, with weights given by the fraction of
the initial measure falling into each component. Because of Propositions 313
and 315, we may be sure that by mixing stationary ergodic measures, we obtain
an ergodic measure, and that our decomposition is unique.
2 I dont want to get into subtleties arising from the dynamicists tendency to dene things
topologically, rather than measure-theoretically.
CHAPTER 26. ERGODIC DECOMPOSITION
179
26.2
Statistical Asp ects
26.2.1
Ergo dic Comp onents as Minimal Sucient Statistics
The connection between sucient statistics and ergodic decompositions is a very
pretty one. First, recall the idea of parametric statistical suciency.3
Denition 332 (Suciency, Necessity) Let P be a class of probability measures on a common measurable space , F , indexed by a parameter . A -eld
S F is parametrically sucient for , or just sucient, when P (A|S ) =
P (A|S ) for al l , . That is, al l the distributions in P have the same distribution, conditional on S . A random variable such that S = (S ) is cal led a
sucient statistic. A -eld is necessary (for the parameter ) if it is a sub -eld of every sucient -eld; a necessary statistic is dened similarly. A
-eld which is both necessary and sucient is minimal sucient.
Remark: The idea of suciency originates with Fisher; that of necessity, so
far as I can work out, with Dynkin. This denition (after Dynkin (1978)) is
based on what ordinary theoretical statistics texts call the Neyman factorization criterion for suciency. We will see all these concepts again when we do
information theory.
Lemma 333 S is sucient for if and only if there exists an F -measurable
function ( , A) such that
P (A|S ) = ( , A)
(26.4)
almost surely, for al l .
Proof: Nearly obvious. Only if : since the conditional probability exists,
there must be some such function (its a version of the conditional probability), and since all the conditional probabilities are versions of one another, the
function cannot depend on . If : In this case, we have a single function
which is a version of all the conditional probabilities, so it must be true that
P (A|S ) = P (A|S ).
Theorem 334 If a process on a countable extension space is asymptotical ly
mean stationary, then is a minimal sucient statistic for its long-run distribution.
Proof: The set of distributions P is now the set of all long-run distributions
generated by the dynamics, and is an index which tracks them all unambiguously. We need to show both suciency and necessity. Suciency: The -eld
3 There is also a related idea of predictive statistical suciency, which we unfortunately
will not be able to get to. Also, note that most textbooks on theoretical statistics state things
in terms of random variables and measurable functions thereof, rather than -elds, but this
is the more general case (Blackwell and Girshick, 1954).
CHAPTER 26. ERGODIC DECOMPOSITION
180
generated by is the one generated by the ergodic components, ({Ci }). (Because the Ci are mutually exclusive, this is a particularly simple -eld.) Clearly,
P (A| ({Ci })) = ((x), A) for all x and , so (Lemma 333), is a sucient
statistic. Necessity: Follows from the fact that a given ergodic component contains al l the points with a given long-run distribution. Coarser -elds will not,
therefore, preserve conditional probabilities.
This theorem may not seem particularly exciting, because there isnt, necessarily, anything whose distribution matches the long-run distribution. However,
it has deeper meaning under two circumstances when (x) really is the asymptotic distribution of random variables.
1. If is really a sequence space, so that X = S1 , S2 , S3 , . . ., then (x)
really is the asymptotic marginal distribution of the St , conditional on
the starting point.
2. Even if is not a sequence space, if stronger conditions than ergodicity
known as mixing, asymptotic stability, etc., hold, there are reasonable senses in which L (Xt ) does converge, and converges on the long-run
distribution.4
In both these cases, knowing the ergodic component thus turns out to be necessary and sucient for knowing the asymptotic distribution of the observables.
(Cf. Corollary 337 below.)
26.2.2
Testing Ergo dic Hyp otheses
Finally, lets close with an application to hypothesis testing, inspired by Badino
(2004).
Theorem 335 Let , X be a measurable space, and let 0 and 1 be two innitedimensional distributions of one-sided, discrete-parameter strictly-stationary valued stochastic processes, i.e., 0 and 1 are distributions on N , X N , and
they are invariant under the shift operator. If they are also ergodic under the
shift, then there exists a sequence of sets Rt X t such that 0 (Rt ) 0 while
1 (Rt ) 1.
Proof: By Proposition 314, there exists a set R X N such that 0 (R) = 0,
1 (R) = 1. So we just need to approximate B by sets which are dened on
the rst t observations in such a way that i (Rt ) i (R). If Rt R, then
monotone convergence will give us the necessary convergence of probabilities.
Here is a construction with cylinder sets5 that gives us the necessary sequence
4 Lemma 305 already gave us a kind of distributional convergence, but it is of a very
weak sort, known as convergence in Ces`ro mean, which was sp ecially invented to handle
a
sequences which are not convergent in normal senses! We will see that there is a direct
corresp ondence b etween levels of distributional convergence and levels of decay of correlations.
5 Intro duced in Chapters 2 and 3. Its p ossible to give an alternative construction using the
Hilb ert space of all square-integrable random variables, and then pro jecting onto the subspace
of those which are X t measurable.
CHAPTER 26. ERGODIC DECOMPOSITION
of approximations. Let
Rt R
t
181
(26.5)
n=t+1
Clearly, Rt forms a non-increasing sequence, so it converges to a limit, which
equally clearly must be R. Hence i (Rt ) i (R) = i.
Remark: R is for rejection. Notice that the regions Rt will in general
t
depend on the actual sequence X1 , X2 , . . . Xt X1 , and not necessarily be
permutation-invariant. When we come to the asymptotic equipartition theorem
in information theory, we will see a more explicit way of constructing such tests.
Corollary 336 Let H0 be Xi are IID with distribution p0 and H1 be Xi are
IID with distribution p1 . Then, as t , there exists a sequence of tests of
H0 against H1 whose size goes to 0 while their power goes to 1.
Proof: Let 0 be the product measure induced by p0 , and 1 the product
measure induced p1 , and apply the previous theorem.
Corollary 337 If X is a strictly stationary (one-sided) random sequence whose
shift representation has countably-many ergodic components, then there exists a
t
sequence of functions t , each Xt -measurable, such that t (X1 ) converges on the
ergodic component with probability 1.
t
Proof: From Theorem 52, we can write X1 = 1:t U , for a sequence-valued
random variable U , using the pro jection operators of Chapter 2. For each
ergodic component, by Theorem 335, there exists a sequence of sets Rt,i such
t
t
that P (X1 Rt,i ) 1 if U Ci , and goes to zero otherwise. Let (X1 ) be the
t
set of all Ci for which X1 Rt,i . By Theorem 331, U is in some component with
probability 1, and, since there are only countably many ergodic components,
t
with probability 1 X1 will eventually leave all but one of the Rt,i . The remaining
one is the ergodic component.