Internat. J. Math. & Math. Sci.
oi. I0 No.
(1987)9-16
A *-MIXING CONVERGENCE THEOREM FOR
CONVEX SET VALUED PROCESSES
A. de KORVIN
Department of Computer and Information Science
Indiana University
Purdue University at Indianapolis
Indianapolis, IN 46223
and
R. KLEYLE
Department of Mathematical Sciences
Purdue University at Indianapolis
Indiana University
Indianapolis, IN
(Received May 27
ABSTRACT.
46223
1986 and in revised form August 27, 1986)
In this paper the concept of a *-mixing process is extended to multivalued
maps from a probability space into closed, bounded convex sets of a Banach space.
The main result, which requires that the Banach space be separable and reflexive, is
a convergence theorem for *-mixing sequences which is analogous to the strong law of
large numbers.
The impetus for studying this problem is provided by a model from
information science involving the utilization of feedback data by a decision maker
who is uncertain of his goals.
The main result is somewhat similar to a theorem for
real valued processes and is of interest in its own right.
KEY WORDS AND PHRASES.
Hausdorf f
Decision making, *-mixing process, multivalued maps,
metric.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
I.
28B20, 60G48, 94A15.
INTRODUCTION.
Our motivation for the present work arises from our efforts to extend a model of
a decision maker using feedback information to decide on an appropriate course of
action.
[2]
This model has been described by R. Alo, R. Kleyle and A. de Korvin in
and extended to include goal uncertainty on the part of the decision maker by
A. de Korvin and R. Kleyle in [12].
The role of the decision maker in this model is
to select an appropriate course of action from a finite set of possible courses of
action and to use the information obtained from implementing this course of action
to reassess the situation prior to selecting his next course of action.
tion process is determined by the decision
maker’s
utility associated with each course of action.
The selec-
current estimate of the expected
The case in which the decision maker
has a well defined utility function for each course of action (goal certainty) is
developed in detail in [2].
Some preliminary results for the case in which the decision maker is unable to
assign an explicit utility function to each course of action (goal uncertainty) are
obtained in [12]. In this paper goal uncertainty is expressed by interval-valued
functions.
In the present paper we extend the goal uncertainty case to the situation
A. de KORVIN and R. KLEYLE
i0
in which the decision maker considers a convex set of possible utility functions.
This leads us to consider multivalued maps from a probability space to subsets of a
Banach function space.
The first steps in that direction were taken by A. de Korvin
and R. Kleyle in [13].
The key to the results obtained in [13] is that for each course of action the
In the context of multi-
convex valued expected utilities form a supermartingale.
valued maps a supermartingale is a sequence {Fn, H
H
map and
n
where F
is a multivalued
n
n
an expanding sequence of -fields such that F
is H -measurable and
n
n
E[Fn+I Hn].
F
n
The formal definition for conditional expectations in the present context will be
given in the next section.
is defined in
The above property is a consequence of goal shaping which
[12] and in [13].
The purpose of the present work is to obtain a convergence result that would
replace the convergence result obtained in [12] for situations in which the goal
is a supermartingale.
Fn
Consequently we wish to remove the condition that
shaping condition is removed.
Another reason for removing the supermartingale condition is
that we do not want to tie the expected utility to the immediate past.
F
reasonable condition is to assume that the dependence of
n
A far more
on past history becomes
To accomplish this we will assume that
weaker as past history becomes more distant.
the process satisfies the *-mixing condition which will be defined in the next
section.
This condition is called *-mixing because of its analogy to the *-mixing
condition for real valued processes.
At each cycle of an ongoing decision process the decision maker hopes to improve
It is
his estimate of the expected utilities associated with each course of action.
reasonable to assume that he will in some sense want to average his estimates of the
sets of utility functions obtained so far.
In this paper we define such an average
and show that it satisfies a strong law of large numbers with respect to a metric to
be defined later.
2.
BACKGROUND AND PRELIMINARIES.
The main purpose of this section is to define the important concepts necessary for
understanding the results.
The most fundamental concept needed is that of Banach-
valued martingales.
Let (,E,P) be a probability space, and let
sequence of integrable Y-valued functions and
of E.
The sequence
Y
Fi
be a Banach space.
(Xi,Fi)
is called a martingale if
E[XI+
F i]
X
i
Let
X
i
be a
an expanding sequence of sub -flelds
X
i
is
Fi-measurable
and
(2.1)
a.s.
For properties of real-valued martingales the reader is referred to [8] and for
the Banach-valued case to [6].
We now focus attention on multivalued functions defined on
closed, bounded, convex subsets of Y.
define a new map
(F
S)()
Let
F
and
G
whose values are
denote any such maps.
cl{a + b / a e F(), b e G()}
norm.
where cl denotes the closure with respect to the
R
We now
CONVERGENCE THEOREM FOR CONVEX SET VALUED PROCESSES
Ii
We set
where
Of course
6
6(F,G)()
Max{sup d(x,G()), sup d(y,F())}
x e F()
y e F()
d(x,G())
x-t
inf
t e G()
II
In
F
i=l
E
F
i=l
II
d(x,F())
inf
x-t
t e F()
is just the Hausdorff distance of F() to
confusion, we will write 6(F,G).
and
I
i
G(), and when there is no
F
For a finite sequence of maps
$ F $
F
i
E
n
F
i=l
L
i
$ F
n
2
denotes the limit, if it exists, of
We define the analog of an
II.
in the
i
6-metric.
distance by
I 6(F,G)dP
A(F,G)
In this context the notations
6, A
and
$
were first introduced by Debreu in
[7]..
Finally we define
IFI
Note that
IFi
6(F,{0}).
R.
is a non negative function defined on
A multivalue
F
map
E-measurable if for any open set
is
B
Y, F
-I
(B)
E
E
where
F-I(B)
{ e R / F() n B # @ }.
It is shown in Himmelberg [II] that
sequence {f n
of
D(F)
{ e R / F() # @} and there exists a
E-measurable functions such that
F(m)
cl{f (m)
n
for all
m e
D(F)}.
In fact the above property can be used to define measurability for Banach spaces.
For an equivalent definition of measurability the reader is referred to [II], [5],.and
[7]. We define F to be integrable if
I
The space
L2wk(Y)
IFI
dP <
will refer to all multivalued maps
is a weakly compact non empty convex subset of Y and
I
F: R
2
FI
2
Y
such that
F()
dP <
By a selector of F we mean a function f: R
Y such that f() e F() for all
F
If
(F) will denote all F-measurable and integrable
is integrable S
F
selectors of F.
will simply be written as
For related kinds of
e
SIF
SIF(E)
selectors see Alo, de Korvin and Roberts [I].
F E-measurable
I
From now on
E(F)
FdP
I
f dP / f e S
Following Aumann [3] we define, for
IF
}.
will denote the Aumann integral of F.
12
A. de KORVIN and R. KLEYLE
We now define the concept of conditional expectation for an integrable, multiWe assume that Y is separable, and the values of F are
valued function F.
Following Hial and Umegakl [I0] (Th. 5.1, p. 169), the
Y.
weakly compact subsets of
F
conditional expectation of
SIE{FIF]
relative to a sub o-fleld
(F)
cl{
E[flF]
F
E
of
is defined by
/ f e S F
The lhs of the above llne indicates that the conditional expectation of
F
respect to
is the set of integrable selectors of this set which are
F
with
F-measurable.
For notational convenience we denote this set as
Now that the conditional expectation of multivalued functions has been defined,
the concept of a martingale can be extended to these functions by replacing Y-valued
in (2.1) with convex set valued functions F
i. For convergence theorems
i
multlvalued
pertaining to
martingales the reader is referred to the work of Hia and
X
functions
Umegakl [I0] and for multivalued supermartingales to A. de Korvln and R. Kleyle [13].
Let F
be a sequence of integrable multivalued functions whose values are weakly
i
compact subsets of Y where Y is separable. We say that the sequence is *-mixing
if there exists some positive integer
that
N
and some function
we have
(n)E]Fn+ml.
Fm]], E[Fn+m]]
A[E[Fn+m
defined on [N, ) such
#
N and m
is strictly decreasing to 0, and for all n
(2.2)
A law of large numbers was shown for a somewhat similar real-valued process by Blum
et al
[4].
The concept of a *-mixing sequence is central to our result.
says is that on the average the dependence of
EIFn+ml
provided
is reasonably bounded.
Fn+
on
Fm
What (2.2) really
grows weaker as
m
If the interpretation of
is the estimate of the average utility at trial
n
Fn+m
n+m when goal uncertainty
is
present, then it is reasonable to expect that the dependence of this average on the
early history of the process (i.e.
the history up to trial m) grows weaker as
For technical reasons we will need the following
o-flelds.
Let
w
n
n
.
be a family
of E-measurable selectors of F; then
F[Wn(.)
GF(Wn)
/ n(’) e N
That is, we consider the o-fleld generated by all functions obtained from
allowing
If
F
denote
G
is a family of measurable multlvalued maps,
t
F
(w
n
t
Gt(Wn,t)
will be used to
k
F I, F
F
Ft2
E-measurable
of
2
E
o
t
E U U
i--I
3.
by
t).
By the o-fleld generated by
a sequence
w
n
to be a variable index.
n
Ftk
we mean
o[ U
G
(Wn,tl)].
t
i=l
i
multivalued maps, we will replace
E
Given
by
Gi(wn i) ].
RESULTS.
From now on
sequence with
Y
F
is a separable, reflexive Banach space and
e L
n
the previous section.
2
wk(Y).
We replace
E
F
n
by the larger o-fleld
denotes a *-mixing
E
as defined in
We start by listing a result on martlngales known to be true for
the real valued case [9].
CONVERGENCE THEOREM FOR CONVEX SET VALUED PROCESSES
LEMMA I.
S
b
Let
be a sequence of Y-valued random variables such that
fi
n
13
E f
is a martingale relative to the expanding sequence of o-fields F n
i
i=l
Then if
be an increasing sequence of positive reals such that limb n
n
n
.
E
b
i=l
it follows that
-2m[llfill2
i
Sn/b n
lim
Fi_l]
,
<
0 (a.s.).
For details the
The proof is essentially the same as for the real case.
PROOF
Let
reader is referred to [9], Theorem 2.18, pp. 35-36.
F
F I, F 2,
be any finite
q
are
weakly compact subsets
of integrable multivalued functions whose values
Let
We now obtain an important inequality.
sequence
Y, and let
of a separable Banach space
H
H 2,
H
denote an arbitrary
q
sequence of expanding o-fields in E’.
LEMMA 2.
fi
> O, there exists a sequence
For every
f
fl’ f2’
is an integrable selector of
F i, and such that for each
A( .q F i,
f
i,
fi
such that
q
Hi-measurable,
is
and
i=l
PROOF.
2
.E q
i=l
{v
m
-Eq
E[FilHi_l])
i=l
RI
Let
1" Here
E[FilHi_ I] is
I i=lEq(f i m[filHi_l])II dP +
E q F i)
{m/Sup d(s,
q
ranges over .E
s
Sup d(t,.
i=l
s
i=l
E[Fi]Hi_I],
q E[FilHi_I])},
and let
i=l
t
q
and t ranges over .E F
i.
i=l
Since
a measurable multivalued function, it has a sequence of selectors
such that
cl
{Vm
The sequence
"Eq
{Vm ()
.E q F i)
()
a.s.
Vm(.)(’),
Now there exists functions
E-measurable.
is
which we continue to denote by
d(Vm’
m[FilHi-i
i=l
vm, such that
-->
Sup d(s, .l
i=l
q
i=l
s
F i)
e
Thus
I 6( r. q Fi, .E q
I
i=1
i=1
E[FilHi_I])dP
<
I d(vm, .I q
i--1
i
Fi)dP
+
e
By Theorem 5.1 of [I0] there exists H i measurable functions gmi which are
selectors of F
such that the right hand side of the above inequality is dominated
i
by
I
i
I vm
Eq
i=l
m[gmilHi_l II de
+ d(E q
q
m[gmilHi_l ],.E
Fi)de
i=l
+
i
i=l
-< I
II i=Eq (E[gmilHi_ I] gmill dP +
and where, moreover,
i=l
Hence
(. Eq F i,
i=
Eq
i=
E[FilHi_I])dP
2
.
14
A. de KORVIN and R. KLEYLE
Now if
2’
m
.E q
.E
of selectors of
Um
q
F
i=l
E[FilHi_I])
i=l
Um(.)(’),
E-measurable, we can pick a sequence
is
i=l
which we denote by
d(Um,
.E q F i
since
q
d(t, .E
> sup
t
such that
i
E[Fi[Hi_I])
i=l
Hence
q
I 6(.E q F i, .E q E[F i[Hi_ l])dP =< I d(um, .E E[F i[Hi_ l]dP +
i=l
i=l
2
2 i=l
I i=lEq (Umi
Y
<
n2
where
are
Uml
E[ Uml
Hi- ])II dP
selectors of F
Hi-measurable
fl
Thus the lemma is proved by picking
+
E
i.
I’
on
gml
and
fl
Uml
2"
on
We are now ready to prove the main result.
Let
THEOREM.
F
be a *-mixing sequence with
n
separable and reflexive Banach space.
E
(1)
(ll)
where
b
n
EIFn 12
-2
b
n
n=l
sup b
n
n
-I
En
L2wk(Y)
n
-I
n
is a
<
is a sequence of positive constants increasing to infinity.
A(b
Y
where
<
mlFi
i=l
F
Assume
.En Fi
i=l
b
-I
n
E
n
i=l
E(FI))
Then
O.
By the *-mixing property, there exists N such that the *-mixing
N large
inequality (2.2) holds for n >= N and m >= I. Given E > 0, pick n o
PROOF.
enough so that
and
j
(n 0) <
Thus since
we have by theorem 5.4 of
Y
is reflexive for all positive integers
[I0],
(3.1)
(E(Fin0+j Fn0+J F (i_l)n0+j ),) E (Fino+j ))
F(i-1)no+J G(i_l)no+j],E[E(Fino+J G(i_l)no+J )]}
j.
A{E[E(Fino+
A
where
(i-l)no+J
(i_l)n0+j
is the
o-fieid generated by
Fn0+J, F2no+j
is generated by
F1
F2
F(i_l)n0+j
and
F(i_l)n0+J.
By theorem 5.2 of [i0] the right hand side of (3.1) is dominated by
A[E(FIn0+ F(i_l)n0+J), E(Fin0+j)]
j
-< (n 0)
E
Fin0+
The last inequality is a consequence of the *-mixing inequality.
.zn F i,
A(bn
A(bn
i=l
-I
.Eno
i=
Fi
(3.2)
j
Now
bn-I .zn E(Fi))
i=l
b
-i
n
.Eno E(FI))
i=
+ A(b n-I
.Eq-I
i=l
.Eno-I Fino+j
j=l
CONVERGENCE THEOREM FOR CONVEX SET VALUED PROCESSES
b
.lq-i .lno -I
-I
n
+ A(b n -i
where for any
0
n
r
n
F
-I
.lr
E(F
j=l
qn0+J
where
q
n
qn 0 + r
n
O
b
qn0+J’
(3.3)
))
and
are positive integers such that
r
The inequality (3.3) holds because
I.
o
.Er
j=l
n
E(Fin0+J ))
j=l
i=
15
B, C $ D)
A(A
A(A,C) + A(B,D).
(See p. 162 of [I0])
The first term of the right hand side in (3.3) can be written as
A(.7.n0 Fi .zn0 E(Fi))
b-I
n
i=l
i=l
and goes to zero when
since n
n
O
is fixed and
It remains to show
bn
that the second and third term of the right hand side of (3.3) goes to zero as
We give the proof for the second term, the third term is handled similarly.
n
.
By the triangular inequality the second term is dominated by
A (b -I
n
q-i
Z
i=l
n0-1
=i
+
b
-I
n
n0-1
.E
.l
i=l
b
b
Fin0+
j=l
q-I
+ A(D n -I
E
Z
n0-1
E(Fin0+
j=l
q-i
-I
Fin0+
i=l
(n0)
E
no-i
Z
Z
i=l
j=l
G(i_l)n0+j)
E(Fin0+ G(i-l)no+j
b
n
-I .E q-i .Z
i=l
no-i
j=l"
E
(Fino+j
q-i
A(.E
n
q-i
-i
n
n0-1
j,
i=l
E[Fin0+j G(i-l)no+J])
q-i
E
j=l
i=l
EIFin0+Jl
(3.4)
The last inequaltiy follows from (3.1) and (3.2).
The second term on the right hand side of (3.4) can be made arbitrarily small by
the *-mixing condition and condition (il) of the theorem. It remains to show that the
first term on the right hand side of (3.4) goes to 0 as r (and therefore q) goes to
infinity.
By Lemma 2, this term is dominated by
Z
no-i
j=l
where
fin0
+ j
b
-i
f
n
is a
I
q-i
(fin0+ E[fin0+J G(i_l)n0+J])ll
Z
i=l
Gin0+j-measurable
integrable selector of
de
+ 2
Fino+j.
e
Furthermore,
it is easy to show that
q-i
Sq-i
Z
i=l
is a martingale with
(fin0+j E[fin0+J G(i_l)no+J]
respect
{G(q_l)no+j} q ->
to
Condition (i) of the theorem implies that the hypothesis of Lemma
holds, so that
A. de KORVIN and R. KLEYLE
16
for each fixed
n
b-i
II
n
and j,
o
q-I
(f
7.
i=l
G
E[fin0+
in0+J
(i-l)n0+J
])li
0
(3.5)
a.s.
The left hand side of (3.5) is dominated by
b
-I
q-i
q-I
[I
(fin0+j E[fln0+J G(i_l)no+J])ll
7.
i=
q-I
-I
+
bq_l
Since the map
bq_
q-I
-i
i=17.
i=iI Fin0+
Fin0+ E[Fin0+J G(i_l)n0+J]
j
IE[
Fin0+ G(i-l)n0+J
j
is non-expanslve, (see Th. 5.2 of
[I0]), it follows that the right hand side of the above inequality has
L
norm less
than or equal to
2b
q-I
-i
Z
q-I
E
i=
Fin0+Jl.
Thus
q-l
bn-I f I i=IZ (fin0+ E[fin0+J G(i_l)n0+J])ll
j
by condition (ii) and the dominated convergence theorem.
dP
0
This completes the proof of
the theorem.
ACKNOWLEDGEMENT.
This Research was supported in part by national Science Foundation
Grants IST-83-03900 and IST-8518706.
REFERENCES
I.
ALO, R., de KORVIN, A., and ROBERTS, C. "p-lntegrable Selectors of Multimeasures," Internat. J. Math. and Math. Scl. (1979), pp. 209-221.
2.
ALO, R., KLEYLE, R., and de KORVlN, A.
Times in a Generalized Information
"Decision Making Mechanisms and Stopping
System," Math Modellln (1985),
pp. 259-271.
AUMANN, R.J. "Integrals of Set-Valued Functions," J. Math. Anal. Appl. I_2(1965),
pp. 1-12.
4. BLUM, J.R., HANSON, D.L., and KOOPMANS, L., "On the Strong Law of Large Numbers
for a Class of Stochastic Processes," Z. Wahrsch. Verw. Geblete 2(1963),
pp. I-II.
3.
5.
CASTAING, C. "Le Theoreme de Dunford-Pettls Generallse," C.P. Acad. Sci. (Paris)
Ser. A. 268(1969), pp. 327-329.
6.
CHATTERJI, S.D.
7.
Spaces," Math. Scand. 22(1968), pp. 21-41.
DEBREU, G. "Intergratlon of Correspondences," Proc. Fifth Berkeley Smp. on Math.
8.
DOOB, J.
9.
HALL, P. and HEYDE, C.C. Martingale Limit Theory and Its Application, Academic
Press, New York, 1980.
I0.
HIAI, F. and UMEGAKI, H. "Integrals, Conditional Expectations, and Martingales
of Multivalued Functions," J. Multivariate Anal. 7(1977), pp. 149-182.
Ii.
HIMMELBERG, C.J.
12.
de KORVIN, A. and KLEYLE, R. "Goal Uncertainty in a Generalized Information
System: Convergence Properties of the Estimated Expected Utilities,"
Stochastic Anal. and AppI. 3(1984), pp. 437-457.
13.
de KORVIN, A. and KLEYLE, R.
"Martingale Convergence and the Raydon-Nikodym Theorem in Banach
Stat. and Prob. 2(1966), pp. 351-372.
Stochastic Processes, Wiley, New York, 1953.
"Measurable Relations," Fund. Math. 87(1975), pp. 53-72.
Supermartingales,"
"A Convergence
Theorem for Convex Set Valued
Stochastic Anal. and Appl. 3(1985), pp. 433-445.