24
The Chapter Almost-Sure Ergo dic
Theorem
This chapter proves Birkho s ergodic theorem, on the almostsure convergence of time averages to expectations, under the assumption that the dynamics are asymptotically mean stationary.
This is not the usual proof of the ergodic theorem, as you will nd in e.g.
Kallenberg. Rather, it uses the AMS machinery developed in the last lecture,
following Gray (1988, sec. 7.2), in turn following Katznelson and Weiss (1982).
The central idea is that of blocking: break the innite sequence up into nonoverlapping blocks, show that each block is well-behaved, and conclude that
the whole sequence is too. This is a very common technique in modern ergodic
theory, especially among information theorists. In pure probability theory, the
usual proof of the ergodic theorem uses a result called the maximal ergodic
lemma, which is clever but somewhat obscure, and doesnt seem to generalize
well to non-stationary processes: see Kallenberg, ch. 10.
We saw, at the end of the last chapter, that if time-averages converge in the
long run, they converge on conditional expectations. Our work here is showing
that they (almost always) converge. Well do this by showing that their lim inf s
and lim sups are (almost always) equal. This calls for some preliminary results
about the upper and lower limits of time-averages.
Denition 294 For any observable f , dene the lower and upper limits of its
time averages as, respectively,
Af (x) lim inf At f (x)
(24.1)
Af (x) lim sup At f (x)
(24.2)
t
t
Dene Lf as the set of x where the limits coincide:
Lf x Af (x) = Af (x)
161
(24.3)
CHAPTER 24. THE ALMOST-SURE ERGODIC THEOREM
162
Lemma 295 Af and Af are invariant functions.
Proof: Use our favorite trick, and write At f (T x) = t+1 At+1 f (x) f (x)/t.
t
Clearly, the lim sup and lim inf of this expression will equal the lim sup and
lim inf of At+1 f (x), which is the same as that of At f (x).
Lemma 296 The set of Lf is invariant.
Proof: Since Af and Af are both invariant, they are both measurable with
respect to I (Lemma 262), so the set of x such that Af (x) = Af (x) is in I ,
therefore it is invariant (Denition 261).
Lemma 297 An observable f has the ergodic property with respect to an AMS
measure if and only if it has it with respect to the stationary limit m.
Proof: By Lemma 296, Lf is an invariant set. But then, by Lemma 287,
m(Lf ) = (Lf ). (Take f = 1Lf in the lemma.) f has the ergodic property with
respect to i (Lf ) = 1, so f has the ergodic property with respect to i it
has it with respect to m.
Theorem 298 (Almost-Sure Ergo dic Theorem (Birkho )) If a dynamical system is AMS with stationary mean m, then al l bounded observables have
the ergodic property, and with probability 1 (under both and m),
Af = Em [f |I ]
(24.4)
for al l f L1 (m).
Proof: From Theorem 291 and its corollaries, it is enough to prove that all
L1 (m) observables have ergodic properties to get Eq. 24.4. From Lemma 297, it
is enough to show that the observables have ergodic properties in the stationary
system , X , m, T . (Accordingly, all expectations in the rest of this proof will
be with respect to m.) Since any observable can be decomposed into its positive
and negative parts, f = f + f , assume, without loss of generality, that f is
positive. Since Af Af everywhere, it suces to show that E Af Af 0.
This in turn will follow from E Af E [f ] E [Af ]. (Since f is bounded, the
integrals exist.)
Well prove that E Af E [f ], by showing that the time average comes
close to its lim sup, but from above (in the mean). Proving that E [Af ] E [f ]
will be entirely parallel.
Since f is bounded, we may assume that f M everywhere.
For every > 0, for every x there exists a nite t such that
At f (x) f (x)
(24.5)
This is because f is the limit of the least upper bounds. (You can see where
this is going already the time-average has to be close to its lim sup, but close
from above.)
CHAPTER 24. THE ALMOST-SURE ERGODIC THEOREM
163
Dene t(x, ) to be the smallest t such that f (x) + At f (x). Then, since
f is invariant, we can add from from time 0 to time t(x, ) 1 and get:
t(x, )1
n=0
K n f (x) + t(x, )
t(x, )1
K n f (x)
(24.6)
n=0
Dene BN {x|t(x, ) N }, the set of bad x, where the sample average fails
to reach a reasonable ( ) distance of the lim sup before time N . Because t(x, )
is nite, m(BN ) goes to zero as N . Chose a N such that m(BN ) /M ,
and, for the corresponding bad set, drop the subscript. (Well see why this level
is important presently.)
Well nd it convenient to not deal directly with f , but with a related func
tion which is better-behaved on the bad set B . Set f (x) = M when x B ,
(x, ) to be 1 if x B , and t(x, )
and f = (x) elsewhere. Similarly, dene t
elsewhere. Notice that t(x, ) N for all x. Something like Eq. 24.6 still holds
for the nice-ied function f , specically,
t(x, )1
n=0
K n f (x)
t(x, )1
K n f (x) + t(x, )
(24.7)
n=0
If x B , this reduces to f (x) M + , which is certainly true because f (x)
M . If x B , it will follow from Eq. 24.6, provided that T n x B , for all
n t(x, ) 1. To see that this, in turn, must be true, suppose that T n x B
for some such n. Because (were assuming) n < t(x, ), it must be the case that
An f (x) < f (x)
(24.8)
Otherwise, t(x, ) would not be the rst time at which Eq. 24.5 held true. Similarly, because T n x B , while x B , t(T n x, ) > N t(x, ), and so
At(x,
)n f (T
n
x) < f (x)
(24.9)
Combining the last two displayed equations,
At(x, ) f (x) < f (x)
(24.10)
contradicting the denition of t(x, ). Consequently, there can be no n < t(x, )
such that T n x B .
We are now ready to consider the time average AL f over a stretch of time
of some considerable length L. Well break the time indices over which were
averaging into blocks, each block ending when T t x hits B again. We need to
make sure that L is suciently large, and it will turn out that L N/( /M )
suces, so that N M /L . The end-points of the blocks are dened recursively,
starting with b0 = 0, bk+1 = bk + t(T bk x, ). (Of course the bk are implicitly
dependent on x and and N , but suppress that for now, since these are constant
through the argument.) The number of completed blocks, C , is the large k such
164
CHAPTER 24. THE ALMOST-SURE ERGODIC THEOREM
that L 1 bk . Notice that L bC N , because t(x, ) N , so if L bC > N ,
we could squeeze in another block after bC , contradicting its denition.
Now lets examine the sum of the lim sup over the tra jectory of length L.
L1
n=0
L1
bk
C
K n f (x) =
K n f (x) +
k=1 n=bk1
K n f (x)
(24.11)
n=bC
For each term in the inner sum, we may assert that
t(T bk x, )1
n=0
K f (T x)
n
bk
t(T bk x, )1
K n f (T bk x) + t(T bk x, )
(24.12)
n=0
on the strength of Equation 24.7, so, returning to the over-all sum,
L1
n=0
C
K n f (x)
=
bk 1
k=1 n=bk1
bC +
K n f (x) + (bk bk1 ) +
bC 1
K n f (x) +
n=0
bC 1
L1
L1
K n f (x(24.13)
)
n=bC
K n f (x)
(24.14)
M
(24.15)
n=bC
L1
K n f (x) +
bC +
bC + M (L 1 bC ) +
bC + M (N 1) +
L + M (N 1) +
n=0
n=bC
bC 1
bC 1
K n f (x)
K n f (x)
(24.17)
n=0
L1
K n f (x)
(24.18)
n=0
where the last step, going from bC to L, uses the fact that both
non-negative. Taking expectations of both sides,
E
L1
K n f (X )
n=0
L1
E K f (X )
n
n=0
LE f (x)
E
(24.16)
n=0
L + M (N 1) +
L1
and f are
K n f (X )
(24.19)
E K n f (X )
(24.20)
n=0
L1
L + M (N 1) +
L + M (N 1) + LE f (X )
n=0
(24.21)
CHAPTER 24. THE ALMOST-SURE ERGODIC THEOREM
165
using the fact that f is invariant on the left-hand side, and that m is stationary
on the other. Now divide both sides by L.
E f (x)
M (N 1)
+ E f (X )
L
2 + E f (X )
+
(24.22)
(24.23)
since M N/L . Now lets bound E f (X ) in terms of E [f ]:
Ef
=
f (x)dm
(24.24)
f (x)dm +
=
Bc
f (x)dm
M dm
f (x)dm +
=
(24.25)
(24.26)
B
Bc
E [f ] +
B
M dm
(24.27)
B
= E [f ] + M m(B )
(24.28)
E [f ] + M
(24.29)
= E [f ] +
M
(24.30)
using the denition of f in Eq. 24.26, the non-negativity of f in Eq. 24.27, and
the bound on m(B ) in Eq. 24.29. Substituting into Eq. 24.23,
E f E [f ] + 3
Since
(24.31)
can be made arbitrarily small, we conclude that
E f E [f ]
(24.32)
as was to be shown.
The proof of E f E [f ] proceeds in parallel, only the nice-ied function
is set equal to 0 on the bad set.
f
Since E f E [f ] E f , we have that E f f 0. Since however it is
always true that f f 0, we may conclude that f f = 0 m-almost everywhere.
Thus m(Lf ) = 1, i.e., the time average converges m-almost everywhere. Since
this is an invariant event, it has the same measure under and its stationary
limit m, and so the time average converges -almost-everywhere as well. By
Corollary 292, Af = Em [f |I ], as promised.
Corollary 299 Under the assumptions of Theorem 298, al l L1 (m) functions
have ergodic properties, and Eq. 24.4 holds a.e. m and .
Proof: We need merely show that the ergodic properties hold, and then the
equation follows. To do so, dene f M (x) f (x) M , an upper-limited version
CHAPTER 24. THE ALMOST-SURE ERGODIC THEOREM
166
of the lim sup. Reasoning entirely parallel to the proof of Theorem 298 leads to
the conclusion that E f M E [f ]. Then let M , and apply the monotone
convergence theorem to conclude that E f E [f ]; the rest of the proof goes
through as before.