Internat. J. Math. & Math. Sci.
VOL. 14 NO. 3 (1991) 435-438
435
A REMARK ON THE WEIGHTED AVERAGES FOR SUPERADDITIVE PROCESSES
DOGAN COMEZ
Department of Mathematics
300 Minard Hall, P.O. BOX 5075
North Dakota State Unlverslty
Fargo, ND 58105
(Received November 24, 1987 and in revised form August 22, 1988)
ABSTRACT.
A decompostion of a superadditlve process into a difference of an additive
This result Is used to prove
and a pos[tlve purely superaddltlve process is obtained.
an ergodic theorem for weighted averages of superaddltlve processes.
KEY WORDS AND PHRASES.
Ergod[c theorem, bounded sequences, superaddltlve process,
additive process.
|980
I.
AMS SUBJECT CLASSIFICATION CODES.
INTRODUCTION.
Let (X,
Primary 47A35, Secondary 28D99, 60GlO.
) be a probablllty space and let
,),
L (X,
P
l’p" be
P
Let T:L
p
L be a linear operator.
A family of L
p
functions
p
F
{F
n n>l
<.
is called a
T-superaddltlve process if
Fn+m
> F
n
+
TnFm
for all n, m
"
(1.1)
1,
and F is called a T-additlve process if equality holds in (I.I).
let
fl Fi+l-Fi’
a sequence A-
for i>O we have F
[an}n>O of
n-I
v
n
i0=
f
i’
where F
o
Notice that if we
m O, for all n
1.
complex nnbers and a T-(super) additive process F.
Consider
We
n-!
define a family of
Lp-fUnctlons (F,A)={anfn}nO,
and set
Sn(F,A)
i-O
anin"
If A is
436
C6EZ
D.
the constant sequence
I=(I,I,....), then Sn (F,A)
F
n
In the following, we observe that the weighted and subsequential ergodic theorems
for
T-superaddit ire
processes
are
consequences
direct
of
their
T-additive
counterparts.
THE DECOMPOSITION OF F.
2.
In this section a decomposition of a T-superadditive process F into a difference
of a T-additive process G and a positive, purely T-superadditive process H (that is, H
is a positive T-superadditive process that does not dominate any non-zero positive T-
process) is obtained.
additive
CASE p=l.
Let T be a positive Dunford-Schwartz operator i.e., T is an
Ll-Contraction
is
S
Sf
Tf d
ITII
with
In
di.
I.
We will also assume that T is Markovian, that
case,
this
-I
if
sup
n
nl
result is
all
n
M.A.
by
obtained
Akcoglu
[I].
Sucheston
<
,
Namely,
then
they
the
decomposition
obtained
that
for
1,
F
n-1
where
L.
and
IIFnll
Gn
Ti6
}"
i=O
n
G
n
n-1
n/
the same result that lira
n
<
p
<
.
(2.1)
n
for some 6eL I, and
result they showed that lira
CASE
H
!
--"
n
--"
H
n n
F
H
=i --0 hl
n
with
hi fi -Ti6"
Using this
exists a.e., and moreover it is a consequence of
n
0 a.e.
In this case we let T be a positive L -contraction and F a TP
n
<
conditions B. Hachem [2] showed that llm
I__
n
F
n
oo.
Under these
exists a.e. by reducing the problem to
a problem in an appropriate
p=l above.
Ll-space and employing Akcoglu-Sucheston’s result in case
Here we observe that the same technique can be applied to yield to a
decomposition result similar to (2.1).
Using a result of M.A. Akcoglu and L. Sucheston [3] one can decompose X uniquely
c
where:
into disjoint union of sets E and E
+,P
(i) E is the support of a T-invarlant function heL
invariant geL
+
P
L (E) and L (E c) are invariant subspaces for T.
P
P
Then the following results are obtained [2,3]:
(i i)
and supp g c E for all T-
WEIGHTED AVERAGES FOR SUPERADDITIVE PROCESS
437
F
IEC I)
0 a.e., so one can
The operator P:L (m)
L (m) defined as Pf
P
P
positive L (m)-contraction and
p
In particular,
operator on
Ll(m).
PI--I
fdm for all
Pfdm
assize
Consequently
where
fL+(m).
P
F’={h-IFn
that
(2.2)
,
T(fh)
fgL
P
(m)
is a
(2.3)
m--hP..
So P can be etended to a Markovlan
is a bounded P-superaddltlve process
[2]
Ll(m).
Now by applying the kcoglu-Sucheston’s result [1] we can decompose F’
into a difference of a P-addltive process and a positive, purely P-superadditlve
-I
process as h F =G -H
n a I. Also we see that:
in
n
lira
n/
n
--In
n
(h-IFn)
exists -a.e. and lira
_I
n
H
n
hG-hH
and that Off{hG
and H={hH
are T-addltlve
n
n
n
n
n
processes respectlvely by (2.3). Consequently (2.4) gives that
so F
lira
!
F exlsts
n n
(2.4)
0 m-a.e.,
and
T-superaddltlve
-a.e. X and
(2.5)
lira
!
n
H
0 -a.e. X
n
by (2.2) and (2.3)
3.
WE IGHTED AVE RAGE S
Given a linear operator T on L
numbers if
", and a sequence A-{a
of complex
n nO
p
p
n-I
n /n
k-O
for all feL
then we say that A is a good weight for T [4], or (A,T) is Sirkhoff [5].
P
R. Sato [6] showed that the uniform sequences are good for lp<
C. RyllNardzewski [7] proved that the bounded Beslcovltch sequences are good for T induced by
a
measure
.
preserving
transformation
:X
X by Tf(x)-f((x)) for
any fcL
I.
This
result combined with the remarkable theorem of J.
2.19]
imply
that
bounded
Besicovltch
Baxter and J. Olsen [5, Theorem
sequences are good for Dunford-Schwartz
operators.
Now we observe the following:
superaddltlve process.
If F mG -H
n n n
S (F,A)
n
so
$
n
(G,A)
Let T be an operator on
L
P
and F be a T-
then for any sequence A
S (H,A)
n
(3.1)
D. COMEZ
438
Also
0
where M
sup
n )
a
n
I"
sup
lira
n
--In H
lira
sup
0 a.e., lira
n
n
H
_I
n
n
S (A,H)=0 a.e. for
n
We summarize this discussion as
T is Markovian and
inf
II In
sup
REMARK 3.4
Fnlll
<
(Fi-TFi- l)l{ p <
i
I__
n
S
n
<
,
and
when p=l,
(A,T)
(F 20) when l<p<.
o
is Birkhoff, then
(3.3)
(F,A) exists a.e.
The limit in (3.3) exists a.e. when A is a uniform sequence or a
bounded Besicovitch sequence of AW
[5].
4
4] is valid for superadditive processes.
ACKNOWLEDGEMENTS.
p
n
If A is a bounded sequence such that
llm
P
Assume also that
F be a T-superaddltive process.
lira
M
Let T be a positive Dunford-Schwartz operator on L
THEOREM 3.2.
(ll)
A,H)
Therefore if llm
any bounded sequence A.
(i)
In ISn
The author would like
In particular the subsequence theorem [5,
to
thank Professsor J.H.
Olsen of North
Dakota State University for his valuable comments on the subject.
REFERENCES
I.
2.
3.
4.
5.
6.
7.
AKCOGLU, M.A. and SUCHESTON, L., A Ratio Ergodic Theorem for Superadditive
Processes, Z, Wahr., 44, (1978), 269-278.
l<p< C.R. Acad. Scl
HACHEM, B., Un Theorem, Ergod.ique Sur-additif dans Lp
Paris, Serie. I, 292, 531-534, (1981).
AKCOGLU, M.A. and SUCHESTON, __On_Convergence of Iterates of Positive
J. Approx. Theory, 13, (1975), 348-362.
Contractions inLp-spaces
BELLOW, A. and LOSERT, V., ..__._e Weighte.d Ergodic Theorem and the Individual
Ergodic Theorem Along_Subsequences, Trans. A.M.S., 288, (1985), 307-3435.
BAXTER, J. and OLSEN, J., Wei.ghted and Subsequential Ergodic Theore.ms, Can. J.
Math., 3__5, (1983), 145-166.
R., _Operator Averages for Subsequences, Math. J. Okayama Univ. 22,
(1980), 161-168.
RYLL-NARDZEWSKI, C., Topics in Ergodic Theory, Proc. of the Winter School in
Prob., Karpacz, Poland, 131-156, Lecture Notes in Math., #472, SprfngerSATO,
Verlag, Berlin, (1975).