679
Internat. J. Math. & Math. Sci.
VOL. 14 NO. 4 (1991) 679-682
A DISCRETE STOCHASTIC KOROVKIN THEOREM
GEORGE A. ANASTASSIOU
Department of tIathematical Sciences
Memphis State University
Memphis, Tennessee 38152 U.S.A.
(Received June 2, 1989 and in revised form October 16, 1989)
in
In this article we give a sufficient condition for the pointwise
the first mean Korovkin property on B0(P), the sDace of stochastic processes with
real state space and countable index Vet P and bounded first moments.
ABSTRACT.
KEY WORDS AND PllRASES. Positive linear operator, stochastic processes, pointwise
in the first mean convergence.
1980 AMS SUBJECT CLASSIFICATION CODE. (1985 P.evision): Primary 41A36, 60F25;
Secondary 60G99.
INTRODUCTION.
Let (O,A,Y) be a probability space and let P denote a fixed countable set.
Consider stochastic processes X with real state space and the expectation operator
E(X)(t)
foX(t,)(d), t e P. Define DO(P) {x" sup n[x[(t) < =}. Let T n
1.
toP
B0(P
n--=
B0(P
1,2,.?..
ETa= TnE,
all
be any sequence of positive linear operators such that
In Theorem i, under Korovkin type assumptions, we give a sufficient
condition such that for each X e
lim
n-oo
Do(P),
E[(TnX)(t,)
X(t,)] =0, for each
t e P.
In [3], see Theorem 3.2, was treated the continuous case, that is, when P is an uncountable compact space. There the sufficient condition is similar to ours, however,
it is produced under the additional assumption that T is a stochastically simple
n
operator.
Our result has as follows"
be a
{t
THEOREM I. Let (0__,A,Y) be a probability space and P
tj
countable set of cardinality > 2. Consider the space of stochastic processes with
real state space
{X: sup
teP
B0(P)
[X(t,0) l(d)
<
0
and the space
B(P)
{f" P +IR
fl I=
<
=},
where
[Ill Ioo
Let T
n" B0(P)
tative,
BO(P)
supl f(t) l; B(P) c
Bo(P
teP
be a sequence of positive liDear operators that are E-commu-
i?e.
(E (TnX) (t ,m)
(Tn(EX))(t,),
for all
(t,aO
P x 0_.
G.A. ANASTASSIOU
680
where
E(X(t,))
(EX)(t)
X(t
is the expectation.
Also assume that
(Tnl)(t,)
I, for all (t,)
E
P x _0_.
Xk(t,)}
c
Bo(P
{X 1(t,k0
For
assume that
P and all i
for all t.
X
E[(TnXi)(tj,w)
lira
k. (I.e.
-lim [(Tn(EXi))(tj)
P and i
In order that
k.)
for all t.
E[(TnX)(tj,w)
lira
Bo(P),
Sk such
P and all X
tj.
are real constants
O,
(EXi)(tj)
n
for all
O,
i(tj,m)]
$1
O,
X(tj,)l
it is enough to assume that each
p there
t.3
that
k
B0(P)
If there exists X
PROOF.
and t.
TXn and
TXn
Let
Lp(P,),
X(tj 0 ,w)]
O,
(EX)(tjo)[ > ,
for all n
>_
I.
(EX)(tj0)[ > ,
for all n
>_
1.
.
be a positive finite measure on P with
< p <
P such that
we get
[(TXn(EX))(tj0
B(P) c
{tj}.
P
> 0 such that
an
[(E(T x X))(tj
n
By E-commutativity of
g
J0
E[(TnX)(tj 0 ,m)
then there exist a subsequence
> 1, for all t
Xi(tj,)
iE[Xi(t,
i=l
({t})
> 0, for all t
P. Here
f. Hence Tn(f)
B(P), then E(f)
Tn(Ef) ETn(f) and Tn(f)
i.e.
maps B(P) into itself. Because each positive linear functional
on B(P) is bounded, by Riesz representation theorem, for the specific
Let f
Tn (’’tj)
Tn
there exists
gt
,
Jo
Lq(P,)
Tn(1)
where
J
p
1 and the positivity of
q
such that
f{t)gtj
(t)(dt), for all f
Tn (’’tjO)
one obtains
and
gtj (t)(dt)
Since EX
P
+
?
(Tn(f)) (tjo
By
B(P),
gtj(t) >_
O, for all t
B(P), we have
(T),n (EX)) (tj)
0
f
(EX) (t).
P
gtjo, (t)
n
B(P).
v(dt).
P.
J0’
DISCRETE STOCHASTIC KOROVKIN THEOREM
681
Thus
<
I
If
<
II
I(Tn (EX))(t)J0 (EX)(tj)[
0
(EX)(t)’gt.
(t) .(et)
jO,Xn
(El()
P
[(F.X) (t)
P-{t.
JO
[[EX
P
(tjo) gtj 0 ’;&n (t) "l(dt)
(EX)
(tjo )]’gtj O’n (t) -l(dt)
(f
(gx)(tj)[[
0
jp_{tj 0 gtj 0
n
(t)v(dt)),
so that
P-{tj 0 }gtj0
(t)(dt) >
Xn
I[EX
1
flk
Xi(tjo,m)
>
There cannot be real constants
6 > O, for all n > 1.
(EX)(tj)[I
0
with
k
iE[Xi(t,c0)
i=l
for all t
P- {t
J0
}.
Since, otherwise, we would have
k
[ 8iE[Xi(t’m)
i=l
Xi(tjo ’w)]’gt jO,Xn" (t)
>
gt
(t), for all t
P
Jo’Xn
{t.
Jo
and therefore
[(EX i)(t)
P-{t.
i--i
]0
>
j
P-{tjo
(EX
gt
i)(tjO
x
j0 ’n
]’gtjO,An(t)’(dt)
(t) .(dt) >
(Note that
(T x
n
(EXi))(tj)0
P
k.)
(t)’(dt), i
(EXi)(t)’gt
0
However from the assumptions of the theorem, we have
lira (T x (EXi))(t
n
0
n
(EXi)(t.),
.10
all i
k.
ttence
k
0
Thus
lira
n
i= 1
i [(TXn (EXi))(tj 0
6 <0, contradicting d > 0.
To show that the assumptions of Theorem
(EXi)(tj)])
0
> 6.
are not empty and they are power-
ful, we present
),
where a > 0,
EXMPLE 2. (i) Consider the probability space ([-a,a], 8,
8 the Bore1 o-algebra on [-a,a],
the Lebesgue measure on [-a,a]. Since
{_+1, _+2,
is a probability measure on [-a,a]. Let also P
i,
([-a,a])
_+T} be a finite set of integers. That is here m 0_ [-a,a] and t P.
682
G.A. ANASTASSIOU
Consider the sequence of operators
B 0(p)
Tn" B 0(P)
such that
(TnX)(t,)
X(t,)(1
e
-nIt])
+
X(-t,)e -nIt[
for all n >
If X > 0 then T X > 0 that is T is a positive operator
n
n
for all n > I. It is obvious that T is linear.
n
Observe that
(ECTnX))(t,w)
furthermore Tn (1)
-nIt[) (EX)(-t).e-n[t[
(EX)(t).(1- e
(Tn(EX))(t,m),
+
i.e., ET
TnE, that is T is E-commutative for all n > I. Therefore T fulfills
n
n
n
the assumptionof Theorem i.
From
(EX)(t)-(l- e -n’t’)
(E(TnX))(t)
(EX)(-t)’e
/
it is clear that
lim
n-+oo
E[(TnX)(t,)--
X(t,)]
0,
for all t e P and all X e
B0[P)._ Thus Tn fulfills the conclusion of Theorem
(ii) Continuing in the stting of part (i)" Let Xl(t,
1, X2(t,)
2t[[/a
and X3(t,
Then (EXl)(t)
1, (EX2)(t)
t and (EX3)(t)
t
It is
obvious that XI,X2,X
We would like to find 81,82,83 such that
3
3t22/a 2.
2.
B0(P
3
Y. 8 i[(Exi)(t)
i--1
For that we can pick
81(1for all t e P
I)
+
81
_>_
an arbitrary real number,
(-2tj)(t- tj)
{tj}.
(EX i)(tj)]
Hence Xi, i
+ (t
t)
2-
1, for all t
2 -2tj
(t-
P
and 8
3
{tj}.
1. We have
tj)2> I,
1,2,3 fulfill the sufficient condition of
Theorem I.
Trivially
TnXi
X.
EX
x’ giving us ETn X i
(TnX 2) (t,)
X2(t,m? (1
e
for i
i’
-nltl)
/
And
,I,3
X2(-t,m
e
implying
(E(TnX2))(t)
t(1
2e-n[t[).
Clearly
lira
n-o
We have seen how X i, i
(E(TnX 2))(t)
(EX 2)(t).
1,2,3 fulfill the assumptions of Theorem I.
REFERENCES
DUDLEY, R.M. Real Analysis and Probability, Wadsworth Brooks/Cole, Pacific
Grove, California, 1989.
2. HEWITT, E and STRObSERG, K. Real and Abstract Analysis, Springer-Verlag, New
i.
York/Berlin, 1965.
3. WEBA, M. Korovkin Systems of Stochastic Processes, Mathematische Zeitschrift,
192 (1986), 73=80.