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SCIENCE IN CHINA (Series A)
VOI. 40 NO. 1
January 1997
Strong uniform convergence of composition sequences of probability
measures on locally compact topological semigroups *
LIU Jin'e (jCI]%%)
(Department of Statistics, Shandong Institute of Economics, Jinan 250014, China)
Received July 23, 1996
Abstract
Weak convergence of convolution products of non-identical probability measures on topological semigroups is investigated. Firstly, the dependence relationship between the semigroup N of measures and suppN is studied; secondly, a necessary and sufficient condition that insures the strong uniform composition convergence of sequence
1 p. I is presented; thirdly, a more extensive class of semigroups is found, on which 1 p, I must be of strong uniform
composition convergence, if it is of composition convergence.
Keywords:
locally compact semigroup, probability measure, composition convergence.
Throughout this paper, S is always a locally compact Hausdorff second countable topological
semigroup. If the conditions are more restrictive, they will be stated. P( S ) denotes the Bore1
regular probability measures on S . For A S S and p E P ( S ) , E( A ) denotes the set of idempotents in the set A . S, denotes the support set of measure ,u .
The convolution of measures p l , p2 E P [ ( S ) is defined by
where E is any Borelset of S, E x 1 = I y E S : y x E E / and x - ' E = l y E S ; x Y E E I .
Dejinition 1.
A sequence p,
1
of measures on S is called composition-convergent if for all
k>O,
W
(as n +oo),
pk,,-&
where
If, in addition, Ak -A
gent.
W
(as k
+
03
), then
1 p, I
is called uniform composition-conver-
In particular, if A = w H , where wH is the Haar measure on subgroup H of S , then { p,
called strong uniform composition-convergent.
I is
* Project supported by the National Natural Science Foundation of China and the Natural Science Foundation of Shandong
Province. China.
38
SCIENCE IN CHINA (Series A)
Vol. 40
We always assume that { pk, : O<k < n \ is tight. Then it is known'" that for any given sequence ( pi ) C ( n ) , there exists a subsequence ( n i) C ( pi ) such that for all k 2 0 ,
and
This paper investigates the weak convergence of convolution products of non-identical probability measures on semigroups. Many authors have analyzed the limit behavior of the sequence
1 pk I of convolution of probability measures on a topological group or semigroup. The basic proband
] MukherjeaL3]considered, relems concern the conditions of weak convergence. ~ a x i m o v ' ~
spectively, probability measures p k , on a compact group and Abelian semigroup S such that for
each k>O, pk, converges weakly to Ak(as n + a ) . It then follows that A h converges weakly to
wH(as k - + a ) . Recently, Xu an') considered this problem on a compact H-semigroup. Other
main contributions in this area include references [4-71.
In this paper our research work has three main points. First, we got a general method to
give the dependence relationship between the semigroup N of measure S and its support set supp
N on locally compact semigroups; second, we presented a necessary and sufficient condition that
insures the strong uniform composition convergence of sequence 1 p, \ of measures, if I p, 1 is
composition-convergent; third, using these results, we found a more extensive class, namely the
class of locally compact L-X semigroups, so that if a sequence { p, of measures is compositionconvergent, then { ,un\ must be of strong uniform compositon convergence. It is well known that
locally compact L-X semigroups include compact or locally compact groups, Compact or locally
compact Abelian semigroups, compact or locally compact Clifford semigroups and so on. So related results obtained on compact groups and compact Abelian semigroups are special cases of our
theorems.
Our main results can now be stated as follows:
Theorem 1.
Let N be a simple semigroup of measures on S and E( N ) # 0.Denote that
Corollary 1. Let N be a left zero ( or right zero) semigroup of measures on S . Then GI
= Go and they are both completely simple semigroups .
Corollary 2. Let N be a completely simple semigroup of measures on S . Then G1 = Go
and they are both completely simple semigroups.
1)Xu Kan, Limit behaviors of convolution sequences of probability measures on compact topological semigroups, Acta Math.
Sinica (in Chinese), preprint.
No. 1
39
STRONG UNIFORM CONVERGENCE
Corollary 3.
-
pn 1 be a sequence i n P( S ) such that for k 2 0 , p k , - pk +
Let
+
pk +
*
W
pn +Ah
( n + w ) a n d let F be the set o f all weak cluster points o f the sequence Ak 1 . Denote S 1 = U S , and S o = 31. Suppose that pk, n 1 is t i g h t . T h e n S 1 = S o and they are both
+
PEF
completely simple semigroups o f S .
Theorem 2.
1 pn 1
Let
be a sequence i n P( S ) such that f i r all k 2 0 , the sequence pa,,
W
T h e n for a n y open set U 3 S 1 ( S 1 as i n Corollary 3 ) , one w i l l have (i) limAk ( U )=
-Ak.
k--
1 and ( i i ) l i m i n f p k , n ( ~ ~ - l ) =where
l,
UU-' =
k-mn>k
U (UX-I).
I€ U
Theorem 3 .
Let S be a locally compact L - X toPological semigroup ( S is called a n L-X
semigroupifex=xe foreveryeEE(S) a n d x E S ) and,u,EP(S), n=1,2,3,... . I f f o r k >
W
0 , pk, n-Ak,
then S1 is a closed subgroup o f S ( S 1 as in Corollary 3 )
Theorem 4.
W
+
W
1
be a sequence i n P ( S ) such that for k 2 0 , p k , ,,-Ak.
Then A
a Haar measure i f and only i f S 1 is a subgroup o f S ( S 1 as i n Corollary 3 ) .
Theorem 5 .
2,3,
Let { pn
....
I f for k>O,
Corollary 4.
-
Let S be a locally compact L-X topological semigroup and pn E P( S ) , n = 1,
W
pk, n+Ak,
Let { p ,
1
then Ak
W
a Haar measure.
W
be a sequence i n P( S ) such that for k 2 0 , p k , ,-Ak
. Whenever
W
one o f the following cases holds, Ak
a Haar measure. ( i ) S is a compact g r o u p ; ( i i ) S is
a compact Abelian semigroup ; ( i i i ) S is a locally compact group ; ( i v ) S is a locally compact Abelian semigroup ; ( v ) S is a compact L-X semigroup.
+
1
Proof of Theorem 1
Lemma 1"'.
T h e support o f a n idempotent probability measure on a locally compJct
(second countable H a u s d o r f f ) semigroup is a closed completely simple subsemigroup.
Lemma 2. Let S be a semigroup. T h e n S is simple i f and only i f S a S = S for each a E S
(see Lemma 3 . 1 4 i n reference [9] ) .
Let S be a completely simple semigroup and e € E( S ) . T h e n eSe is a subgroup
of S ( see Theorem 3 . 2 0 i n reference [ 9 ]) .
Lemma 3.
Lemma 4.
Let N be a simple semigroup o f measures on S . Denote by G 1 = suppN P
U S A. Suppose that A = A2 E N and e = e2 E S A. T h e n for a n y x € G I , there exist y E eG1 and
AEN
z E G l e s u c h that y x z = e .
Proof. Fix an element x E G I . Choose a measure p E N such that x € S, . Since t h e set
N is a simple semigroup, b y Lemma 2 there exist measures v l , vz E N such that vl + p * vz =
A . Let a 1 € S v l and a2 E S v 2 . T h e n
SCIENCE IN CHINA (Series A)
40
Vol. 40
and ea xu 2 e € eSAe. Since the set S, is a completely simple semigroup ( see Lemma 1) , hence
eSne i s a group with i d e n t i t y e . Therefore, there exists the element b E SAsuch that ( ebe )
.
( e a l x a 2 e ) = e . Put y = e b e a l E e G 1 and z = a 2 e E G l e . Then y x z = e .
Let T be a subsemigroup of S a n d I a n ideal of T . For A = A 2 E P ( S ) , if
G T a n d S a n I # O , then SaGl.
Lemma 5 .
Proof.
SA
T, we obtain
Since I is an ideal of T and S A C
. I C T I G I and Si . SAC SA.
fl
I ) G (S, . S,) n ( S A I ) G Si n I . Similarly, (S, n
S,,
.
Therefore, S, (S,
I. In view of Lemma 1, we see that S, fI I = S, and S,CI follows.
n
I)
. S, C S,
Lemma 6 . Let N be a completely simple semigroup of measures on a locally compact topological semigroup S. Denote by G1 = s u p p N 4 U S,, . Suppose that A = A 2 E N a n d e = e2 E SA.
AEN
E G I , there exist y E eGl e a n d
exz = e ; (ii) eGle is a subgroup of S.
Then ( i ) for any x
E eGl e such that yxe
z
= e and
Proof.
( i ) See the Lemma on page 191 in ref. [ 6 ] . ( ii) It is clear that eGl e is a semigroup with identity e . Therefore, for each x E eGl e we have x = xe = e x . In view of (i) , there
exists y E eGl e such that yx = yxe = e . Similarly, there exists z E eGl e such that xz'= exz = e .
We conclude that eGl e is a subgroup.
Lemma 7 .
"'.
Let S be a locally compact topological semigroup a n d ,u, E P( S ) , n = 1 , 2 , 3 ,
If for any k>O,
W
(as n + a ) ,
then F is a left zero semigroup, where F
= ~ A € P ( S ) : Ai s a weakcLusterpointofAai.
Proof.
According to the tightness of
1 ,uk,
:O<
k<n
1 , F#@[']. For O<k < m < n , we
have
Letting n - + a we obtain
Ak = j * k S r n t A r n
(O<k<
m).
If A ' € F , then there exists a subsequence ( mi ) C ( m ) such that
Taking m = mi in ( I ) , we have Ak = p k
* rn,
For A"
+
A, . As i + a ,
E F , there exists a subsequence ( k, ) C ( k ) such that Ah,-
we have, in particular,
W
A". Taking k = k, in (2), we
No. 1
get Ah. = Ak,
STRONG UNIFORM CONVERGENCE
* A'.
* A ' . So F is a left zero semigroup of
As i--+m, A"= A"
41
measures. The proof
of Lemma 7 is completed.
Proof of Theorem 1 .
Step 1 . Fix an idempotent measure A E N . By Lemma 1 , S , is a completely simple semigroup of S . Let e be a primitive idempotent in S , . We will show that M 4 GleG1 is a minimal
ideal of G 1.
Let I be an ideal of G I and IC G l eG1. For x
and z E G l e such that
E I, in view of Lemma 4, there exist y E eG1
Therefore, G l eGl S G1IG1E I . Thus I = G l eG1. Consequently M = G l eGl is a minimal ideal
of G I . Thus, M is a simple semigroup (see Proposition 1 . 5 in ref. [ l o ] ) and it is clear that e €
M.
Step 2.
We show that G ~ C
M
CG~
e E 4 n M # @ and M is an ideal of G I . In the meantime S A G G 1 .It follows from Lemma
5 that S A C M . Note that N is a simple semigroup of measures. For A € N , we have N * A * N
=N.
Therefore
where M is a minimal ideal of G I . The proof of Theorem 1 is completed.
Proof of CoroLLary 1. Note first that N is a left zero semigroup of measures. For any A E
N , we have N * A * N = N . Thus, N is a simple semigroup of measures. In view of Theorem
1, there exists a simple semigroup M such that G ~ L M CGo and e E E( M ) . In view of Lemma
2, it is easily verified that eMe is also a simple semigroup with identity e . Then for any x EeMe
there exist a , b E eMe such that axb = e . It is evident that ( s b a )' = xba , ( bax )' = bax and xba
E eMe S ( S A G I )C S A U S, = U ( S A S F )G U SA.. = S A , since N is a left zero semi-
.
PEN
PEN
PEN
group of measures.
Similarly, bas E SA. On the other hand, since e is the identity element of eMe, it follows
that xba = ( x b a )e = e (xba ), where e is a primitive idempotent in S, and this then implies xba
= e . Similarly, we have bax = e . This means that eMe is a subgroup of S with identity e .
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42
Vol. 40
For f = f 2 E M , i f f = f e = e f , then f = ( e f ) e E e M e . Therefore f is an idempotent element of a group eMe. It implies f = e . It means that e is a primitive idempotent of the simple
semigroup M and M is completely simple. Therefore M is closed (see Theorem 2. 14, in ref.
[81) and M L G ~ E M =M . We conclude that G 1 = M = Go.
Proof of Corollary 2 . Since N is completely simple, there exists A = A 2 E N and SAis a
completely simple semigroup of S . Therefore, there exists a primitive idempotent element e E
SA. By Lemma 6, eGle is a subgroup of S with idemtity e . Therefore, for any f = f 2 E M , if f
=fe- ef, then f = efe E eMe
eGl e . It follows that f = e, so e is a primitive idempotent in
c
simple semigroup M . Thus, M is completely simple and therefore, closed. By Theorem 1, we
obtain G1 = M = Go. Thus both G1and Go are completely simple.
Proof of Corollary 3 . In view of Lemma 7, F is a left zero semigroup of measures. It is
immediate from Corollary 1 that both S1 and S o are completely simple semigroups.
2
Proof of Theorems 2
-5
Proof of Theorem 2.
First, we prove ( i ) .
In fact, if this is not true, then for some U 3 S1 and
C ( k ) such that
A, ( u )
In view of the tightness of
P ( S ) (as i + m ) .
€0
>0, we can find a subsequence ( ki )
< 1 - EO.
I ,uk,,:O<k < n I ,
(3)
W
there exists ( n , ) C ( k i ) such that A,,---tA'G
Therefore S A , C S I C
U . Thus,
lim jnfA,,(U)
A'(U)
>, A f ( S A , )= 1.
It follows that lim A, ( U ) = 1. It means that A, ( U ) > 1 - eo for i>,io. This contradicts ( 3 ) .
i
1
Next we prove ( i i ) . Since ,uk,,,= ,uk,m* ,um,.(O< k
tain
So we have V E >0, 3 ko such that, for k>ko,
where
Therefore
It implies that there exists at least one
xoE U, such that
< m < n).
Letting n + w ,
we ob-
No. 1
Since
E
43
STRONG UNIFORM CONVERGENCE
pk,m(uu-l)
is arbitrary, we have
>
> 1 - 2~
( M
> k ),kO)
lim inf pk,, ( U U - ~) = 1.
k-mm>k
Proof of Theorem 3 . By Corollary 3, S1and S o are both completely simple subsemigroups
of S and S1= S o . Let e be a primitive idempotent of S1 and lo= { x E S1:ex = x } . It is clear
that e E I o , so that Io#@.Note that S is an L-X semigroup. For € S l , x € Io, we have yx
- y ( e x ) = e ( y x ) , so that yx E I o , and thus S I I o C
lo.Similarly I o S l cl o . This means that
I. is an ideal of S1, and this then implies that I. = S1. For any f = f 2 E S1, we have f = ef = fe
and this then implies that e = f , since e is a primitive idempotent in S 1 . This means that the
idempotent in S1is unique. Therefore, S1is a subgroup of S (see page 487 of ref. [ 11] ) and S1
= S1 - S o , so S is closed.
W
Proof of Theorem 4 . "a"
Suppose that hk
w H Then F = 1 wH 1 is a set of simple
points; therefore, S - U Sn= SwH = H. The conclusion follows.
---+
'-AEF
+" Suppose that S is a subgroup of S . By Lemma 7, F is a left zero semigroup. Let A E
F. Then h = h2 ; therefore, Snis a completely simple semigroup and S n ES1. Since there is only
one idempotent element in group S1, the idempotent elements in Snare unique and this then implies that Snis a subgroup of S . Therefore, Sn.S1= S1. SA= S 1 . By Lemma 7, for any p E F,
we have h * ,u = h .
"
Thus
s, = s , . s l = s,.us,
= u s , . s , r ~ s , , =,yFsn
;
= sn.
QEF
F
C EF
PC
It follows that
S 1 = S,.
It is known that the idempotent measure whose support set is a subgroup is a Haar measure on the
subgroup. Therefore, A = US,, where wsl is the Haar measure on the subgroup S1. Since the
Haar measure is unique on a group, F is a set of simple points. On the other hand, notice that
W
J*~,~--)hk
( v k >0)
and {,uk,*
:O<k< n 1 is tight. It is easily verified that { hk 1 k > O is tight. By Prohorov's TheoW
rem, { hk 1 k>o is relatively compact. This assures the weak convergence of { hk 1 and Ak --+A
US,.
Proof of Theorem 5.
Theorem 5 follows immediately from Theorems 3 and 4 .
In view of Theorem 5, we obtain Corollary 4 .
=
44
SCIENCE IN CHINA (Series A)
Acknowledgement
Vol. 40
The author would like to thank Dr. Zheng Hengwu and Prof. Xu Kan for their discus.
sion.
References
1 Budzban, G . , Mukherjea, A. , Convolution products of non-identical distributions on a topological semigroup, Journal of Theoretical Probability, 1992, 5(2) : 283.
2 Maximov, V. M . , Composition convergent sequences of measures on compact groups, Theory Probub . Appl . , 1971, 16 :55.
3 Mukherjea, A. , Convolution products of non-identical distributions on a compact Abelian semigroup, Lecture Notes in Math. ,
1988, 1379:217.
4 Csiszar, I . , On infinite products of random elements and infinite convolutions of probability distributions on locally compact
groups, Wahrsch 2. Verw . Gebiete . , 1966, 5 :279.
5 Mukherjea, A. , Limit theorems: Stochastic matrices, Ergodic Markov chains, and measures on semigroups, Probabilistic
Analysis and Related Topics, 1979, 2 :143.
6 Mindlin, D. S . , Convolutions of random measures on a compact topological semigroup, Journal of Theoretical Probability,
1990, 3(2) :181.
7 Liu Jin' e, Xu Kan. LiAt behaviors of composition convergent sequences of probability measures on compact L-X semigroups,
Advance in Math. (in Chinese), 1991, 20(3) : 397.
8 Mukherjea, A . , Tserpes, N. A . , Measures on Topological Semigroups, Lecture Notes in M a t h . , Vol. 547, Berlin:
Springer-Verlag, 1976.
9 Carruth, J. H . , Hildebrarant, I. A . , Koch, R . J . , The Theory of Topological Semigroups, New York: Marcel Dekker Inc.,
1983.
10 Howie, J . M. , An Introduction to Semigroup Theory, London: Academic Press Inc. , 1976.
11 Liu Jin'e. Fang Junling. Yu Shaohong, Essential point set of convolution power of a probability measure on a locally compact
topological semigroup, Chinese Science Bulletin. 1995, 40( 14) :1155.