Metrika, Band 24, 19 77, Seite 35 43. Physica Verlag, Wien.

Moving Averages of Ergodic Processes

By A. del Junco, Toronto') and J.M. Steele, Vancouver 2)

Abstract: A necessary and sufficient condition for the almost everywhere convergence of the "mov
n
ing" ergodic averages (o (n))' E XE (74x) is given. The result is then generalized to ergodic
i=no(n)+1 flows, and finally constrasted with earlier results of Pfaffelhuber and fain.

1. Introduction

The ergodic theorem of Birkhoff states that for an invertible measure preserving transformation of the measure space (X, F, ju) the sequence

n
n' 7, f (Tx)
i=1

converges a.e. for all f EEL, (X, F, ju). A natural direction for generalization of the ergodic theorem is via a more general averaging process than (1). In particular one has the basic question: What are the necessary and sufficient conditions on the matrix (a,i) so that the sequence fn (x) = E anif (Tix) converges a.e. for eachf EL 1.

The present work tackles only a special case of the basic question where definitive results can be provided. In particular we consider the matrix (ani) defined by

1 /0 (n) n  0 (n) < i <, n
ani = 1 0 otherwise (2)

where 0 (n) is a positive nondecreasing function on the integers. For this choice of
(and we provide necessary and sufficient conditions for the a.e. convergence of the
fl (X)

1 A.del Junco, Department of Mathematics, University of Toronto, Toronto, Canada 2) J. Michael Steele, Department of Mathematics, University of British Columbia, Vancouver, Canada, 2075 Wesbrook Place.

36 A. del Junco, and LM. Steele

This particular choice of (and is motivated, first of all, by its relevance to the general problem as pointed out in Akcoglu and del Junco [ 19751 (see also Belley [ 19741). The more direct purpose of (2) is, of course, to study the average of the last 0 (n) of the values f (x), f (Tx), f (T'x), . . . , f JnX). With the motivation of studying such averages one can just as well consider

1 f (n) n + f (n)  1
n
b ni ~ (3)
0 otherwise

and this study has been carried out by Pfaffelhuber [ 19751 for ergodic transformations and by Jain [ 19751 for independent, Banach space valued random variables.
In the second section of this paper we prove our main result, Theorem 1, which provides the desired necessary and sufficient conditions for the ergodic theorem under the averaging process (2). In the same section we extend the result of Theorem 1 to the physically interesting case of measure preserving flows.
The third section applies Theorem 1, to the counterexample to Be.11ey,Is conjecture
and contrasts the present results to those 1 of Pfaffelhube r 19751 ~nd Jain [ 19,75 1. The
fourth section contains a result which contrasts two related surmnability methods of
the type (2) and (3) respectively.

2. Main Results

The basic result of the present paper 'is the following:

Theorem 1. Let 0 < 0 (n) < n be a nondecreasing integer valued function and suppose T is an ergodic measure preserving transformation on a nonatornic measure space. The sequence

0 (n)1 . 1 XE (T' x) (4)
1=no(n)+1
converges a. e. for all E E F if and only if

lim (n) > 0. (5)
n 00

Moreover, if is the limit a.e. of the sequence (4) then t (E) a.e.

Proof We will first suppose that c = 0 and proceed to construct a set E E F for which (4) fails po_converge on a set B of measure 1. By our hypothesis (5) we can select a

subsequence n, such that 7, 0 n, < 1/2.Next given a sequence e, with e, ~,0 we

i

i
1

Moving Averages of Ergodic Processes 37

can apply Rohlin's theorem (see Halmos [ 19561, Theorem 8.1) to obtain for each!
a set F C~ F such that TT are disjoint for i = 0, 1, 2, nj and such that
n/
U T F. > 1  c.. We now let
i=o 1 1

o(n 1 )1 n
E = U TT. and B. J T`*F.
1 i=o i=o(n

One notes that

(n) / n. so setting E = U E.

we have ME 1/2. Also we have

MB. > (1  c.) (n,  (n (n, + 1) so

for B= lim sup B n U B. it follows that p (B) 1.
k=l l=k
Finally we come to the crucial observation. If x E B, then there is an n with
(nf):li~ n < n, such that TnX, n1x, . . . , Tno(n)+1X are elements of
o(n) n~5(n)+lx
U T'T Since 0 (n) < 0 (ni) this implies that Tnx, Tn lX, T are ele,
i=o i
ments of E C E. We have thus established that for x EE B1, there is an n >, (nf) so that
1
n
(n)  i=no(n)+j XE (Tlx) 1. (6)

Now if (n) is unbounded (6) implies that for each x EE B we have,

n
lim sup 1 X (Tx) = 1. (7)
p
n i=no(n) E

But, by the ergodicity of T, if the series (6) converges on a set P of positive measure then the limit of (4) must be equal to g (E) < 1 'a.e. on P.
We have thus established the first half of the theorem under the assumption that (n) is unbounded.
To deal with the case when 0 (n) is bounded we note that 0 (n) must be constant, say equal to k, for all n > no for some no. Then by Rohlin's theorem one can choose a
k1 set E such that the functionf (x) defined byf (x) k~1 1 XE (Tx) is larger than 314

J

38 A. del Junco, and J.M. Steele

on a set of positive measure and less than 114 on a set of positive measure. Now to show
(1) fads to converge we note that for n > no

n
1 x. (Tx) = 0 (no)' . E XE (Tx)
i=nO(n)+1 f=n0(n')+1

n
= k' 1 XE (Tx)
i=nk+l f (Tn +k1 X).

But since T is ergodic one has by the Poincarg recurrence theorem that

lim sup f (Tn *+ lx) >, 3/4 a.e.
n

and

lim inf f (Tnk+1x):E~ 114 ax.

Consequently the sequence (1) fails to converge a.e.
To prove the second half of the theorem, we may assume. by symmetry that

lim XE (Tx) > IA (E) + 8 (8)
n~ 1=n1n)+1

for some 8 > 0 and a.e. x. By the ergodic theorem one naturally has

lim (n  0 (n)  1)1 ntn)1 XE (Tlx) = IA (E) (9)
n > ~i=,1

for a.e. x. Moreover for fixed x we can choose a subsequence so that (8) and (9) hold along nj and such that 0 (ni) In, converges to d, d > c 0. But inequalities (8) and (9) immediatly show that

lim. E X (Tx) >, d (IA (E) + 8) + (1  d) IA, (E),
(10)
ni=l E n

This shows that for a.e. x one has

E XE (Tx) >,IA (E) + 6C (11)
im n

i

Moving Averages of Ergodic Processes

and this is in contradiction to the ergodic theorem. We have thus proved that for c :P' 0 we have ax. convergence in (4).
The first application of Theorem 1 will be to show that a completely analogous result holds for ergodic flows.

Theorem 2. Let 0 < 0 (t) < 1 be a nondecreasing real valued function, and suppose Tt is an ergodic, measure preserving flow on a nonatomic Lebesque space X Then as
t >. 00

t
f XE (Ttx) dt
(t) tO(t)

converges for a.e. x and all E GE F if and only if

lim 0(t)lt=C>O.
t

#4

39

(12)

(13)

Proof If f (x) is a real valued function on [0, 1 ] we can define a flow on
{(x, y) : 0 < x < 1,0;!Q y < f (x)} by allowing the point (x, y) to move vertically at unit
speed until (x, f (x)) and then jump to (Sx, 0) where S is an ergodic transformation of
[0, 11, According to the Ambrose and Kaku~ani [ 1942] representationtheorem, any
ergodic, measure preserving flow on,a Leb ' esgue space is isomorphic to a flow "built
under a function" as just described. Further by Rudolfs Theorem [ 19,75,1, the function
can be assumed to be a step function f (x) which takes on only values cl and P where
0 We can thus assume that T, is a flow built under a step function f (x) as above. Now by Theorem 1 we can construct a set E C [0, 11 such that E has measure less than 6 and such that

lim
n ~,

n
1; ^ XE WX)
n.[o(n)1+1

fora.e. xG[O, 1I.NowletE'= {(x,y) :xEE, 0:liQyIf 0 (t)   as t  , we easily obtain

1 t
(t) f XE' (Tt Z) > a
to(t)

1

i
i
1

i
11

(14)

(15)

for a.e. z EE {(x, y) : 0 gEQ x :E~ 1, 0 E~ y < f (x)}. By the ergodicity of Tt if (12) converges on a set P of positive measure then the limit must be less than the measure of E' for a.e. z EP. Since 8 can be chosen so that 80 < ot, we have by (15) that (12) cannot converge on any set of positive measure.
To, complete, the proof we observe that, if 0 (t) is bounded, Theorem 2 is proved essentially as in Theorem 1. Also the sufficiency of (13) for convergence of (12) is proved almost without change from the proof of Theorem 1.

1 1

i i

i

i

i i

40 A. del Junco, and LM. Steele

3. An Application of the Main Result

As was noted in the introduction, one would certainly like to know conditions on (ail) such that f (x) = 2: a
n i= 1 n if (T'x) converges a.e. On the basis of spectral considerations the following conjecture has been advanced:
If p, (z) = E ani Zi are uniformly bounded and pointwise convergent on the unit circle thenfn converges a.e.
An attempt to prove this conjecture was made by Belley [ 19741 but the question
was settled by A kcoglu and del Junco [ 19751 where the coice
1 z [,~fnl + n
[\fn] + 1
ani = 1 0 otherwise (16)
was shown to be a counterexample to the conjecture. n
We note here that for the choice (a,i) given in (2) we have p. (z) 1 Z"
(n)i=no(n)1
These functions are also bounded uniformly by 1 on lz 1 = 1 and converge pointwise to
the function on, lz 1 = 1 which is 1 at z = 1 and 0 for z 4 1. Hence we have that the class
of matrices (a, ),given by (2) provides an uncountable class of counter examples to the
n
conjecture quoted above.

4. Related Results

As we have already mentioned the summability method (3) represents an average
over a moving block as well as (2). If f in (3) is taken to be an increasing positive func
tion defined for all real x > 0, the 0 (n) which gives the corresponding method in (2)
is f (n), or more precisely [f (n)l, since f (n) is not an integ er in general. Thus we
have the two sequences

f(n)+n1
IS = 7
XE (T'x),
n n i=f(n)

(17)
n+,b(n)1
E XE (Tx)
gn
(n)

and one might suppose that Sn converges a.e. if and only if gn converges a.e. In fact Sn is a subsequence of gn, so the convergence of gn implies the convergence of Sn. However the converse is not true as can easily be seen from the following result by taking T to be the appropriate Bernoulli shift.

i

Moving Averages of Ergodic Processes 41

Theorem 3. Suppose Xi, i = 1, 2. . are i.i.d. random variables talcing values 0 and 1
with probability .1 . Letf (n) = 22" and 0 (n) = [109 2 109 2 n]. Then
2

1 n+o(n)1 1
lim w (18)
n) Xi = 1 and lim E Xi 0,
n t=n n) =n

but

n+f(n)1
lim 2; Xi = 2
n~ n i=f(n) Proof.. Let nk = 2k and note that

P {X~= 1,n 1
k k + (nk) k

Furthermore since n k +0(n k) 1 < n k+l, the events

A k {Xi 1,n <,i<,n +0(nk)1}
k k

are independent. Thus by the BorelCantelli lemma for independent events one has
P (Ak i.O.) = 1, which shows that the first equality in (17) holds. The second equality
of (17) is proved identically. To prove the third equality note that
n f(n)+n1 In n
P fi  1 Xi 1 >En} =P fl  Y, X. 1 > C n Pn
2 i=f(n) 2 i.1

Since the Pn are summable, by the usual estimates (e.g. Bahador, Rao 19601) one has the third equality by the (unrestricted) BorelCantell! lemma.
A niost. notable virtue of the surnmability method (b. j) of (3), is the number of
occasions it produces convergence. This is reflected in ' a small way by (17), and more
powerfully by the result of Jain [ 1975 ], where simple necessary and sufficient conditions
are given for independent Banach space valued random variables to converge a.e. under
(bni).
The possibility that the functions Xi = XE (T'x) constructed in Theorem 1 are in~, dependent cannot be ruled out a priori. This would, in fact, correspond to the situation where T is a Benoulli shift With respect to the partition (E, Elc).'Nevertheless, we have the following fact.

n
Theorem 4. ' If (h)~log n as n and 1 1; XE (T'~) fails to con
i 0( 1 n), i=,,,.p(n)+1
verge a.e., then the functions X, XE (T x) cannot be independent.

42 A. del Junco, and LM. Steele

Proof We let Yi = Xi  IA (E) and note Y, are bounded with mean 0. We will suppose for now that the Xi are independent. By elementary estimates of the binomial distribution, we have

n
c 0 (n)) < Ce`0(') (19)

for some constants C and a > 0, (for even more precise estimates see (Bahadur, Rao
[19601 or Cramer [19381). Now, easy estimates and theBorebCantelli lemma show
n
that E XE (Tx) converges.
~(n) i=n.O(n)+1
One reason for noting this property of the XE (T'x) is to contrast the present
method with that of Pfaffelhuber [ 19751, where the following is proved:

If 0 (n + 1)  0 (n) >, n for infinitely many n then there is a measure space (2, F, p) and an ergodic, measure preserving transformation T sucht that E bni f (Tix) converges at most on a set of p measure 0.

This result was obtained by taking T so that f (Tkx) are independent with finite mean
2
and infinite variance. The particularly sim le choice 0 (n) = n will thus provide an
1 p
example where the Xl = XE (Tix) of Theorem 1 cannot be taken independent and the procedure of Pfaffelhuber 1975] be to take the Xi = f (Tix) independent.

5. A Valuable Problem

One is more often confronted with a plethora of problems than a paucity. We are fortunate in the present circumstance to . be able to pinpoint a single problem of particular significance.
Quite without hesitation one can now point to the value of providing necessary and sufficient conditions'for the a.e. convergence of 1 ani ~(Tix). The results given here, and contrasted to earlier ones, show that this problem contains cases of variety and interest. In fact, the~purpose of this report iswell served,'!f greater attention is brought to focus on this central open problem.

References

A kcoglu, M.A., andA. del JuncoXonverge cc of Averages of Point Transformations, Proc. Amer.
n
Math. Soc. 49 (1), 1975, 265266.
Ambrose, W.,and S. Kakutani: Structure and Continuity of measurable flows, t)hke Math. J. 9, 1942, 2542.
Bahadur, R.R., and R. Ranga Rao: On deviations of the sample mean, Annals of Mathematical Statistics 31, 1960, 10151027.
Belley, JX: Invertible measure preserving transformations are pointwise convergent, Proc. Amer. Math. Soc. 43, 1974, 159162.

Moving Averages of Ergodic Processes 43

Cramb, H.: Sur un Nouveau thiore~mlimite de la theorie des probabilds. Actualitgs Scientifiques et Industrielles, No 736, Herman et Cie, Paris, 1938.
Jain, N. C: Tail Probabilities for Sums of Independent Random Variables, Z. flir Wahrscheinlichkeitstheoric verw. Gebiete 33, 1975, 155166.
Halmos, P.R.: Lectures on Ergodic Theory, Chelsea Publishing Co., New York 1956.
Pfaffelhuber, E: Moving Shift Averages for Ergodic Transformations, Metrika 22, 1975, pp. 97101.
Rudolf, D.: Some Problems of Ergodic Flows, Stanford University Thesis, Stanford C.A. 1975.
Smorodinky, M.: Ergodic theory, entropy, Lecture Notes in Mathematics, Vol. 214, New York 1971.

Eingegangen am 4. Mai 1976
Revidierte Fassung am 8. September 1976