Internat. J. Math & Math. Sci.
(1986) 29-37
29
Vol. 9 No.
AN EXTENSION OF A RESULT OF CSlSZAR
P. B. CERRITO
Department
University
Tan.
of Mathemat its
South Floricl
Flor Ida 33620
o
(Received July 22, 1985)
ge extend the results
ABSTRACT.
topological semigroup S.
o
a
sup
llm
sup
K
compact
n-e0
xS
zero or one.
S such that
o Cszszar (Z.
5(1966) 279-295) to a
Wahr.
Let p be a measure defined on S.
Hn(Kx-1).
We consider the value
First. we show that the value of a is either
1. we show that there exists a sequence of elements
I a
pn
converges vaguely to a probability measure
where
(an)
In
denotes
n
point mss.
In psrtcular, we apply the results to inverse and matrix
semlgroups.
KEY UORDS AHD PHRASES.
PRIMARY CLASSIFICATIOH.
1.
Topological semtgroup. Infinite convolutions.
60F17
INTRODUCTION.
Cslszar
[1]
proved the following result concerning a regular probability
measure p on a locally compact, second countable, l-lausdorff
sup
--/Jn(Kx-1)
O as n
for al]
convolution of p. or there exzsts a
sequence o
converges vaguely to a probability measure
where
elements
a
group G:
pn denotes
compact sets K. where
(an)
Either
the n-fold
such that
pn 5a
n
denotes point mass at a
n
n
We will extend this result to probability measures defined on certain tlpes of
locally compact, second countable. Hausdorff semzgroups which satisfy condition
-1
If A and B are compact then so are AB
and
where
A-1B
(c):
AB
-1
(y: there exzsts z e B such that yz
A).
30
P. B. CERRITO
We will also conszder
m x m mtrlces.
Wn(Kx -1)
p is defined on a semzgroup s
when
of
A matrix semigroup does not necessarilg satisfg condltion
(c).
To each regular probability measure /# on a semigroup S we associate the
value
a
a
sup
O
O then /#
0
we
0
Kx-l).
WE first show that
any sequence
O vaguely for
6an
(an)
f,nd
sup /#n,
IIm
K
cog)act
n
such that
/#n
of elements
O or
O
(an)
In S.
1.
If
If
converges to a probbilit measure
6
a
O
n
PRELIMIHARY RESULTS.
2.
In order to show the
main results, we need the following tentma.
proof s,nce t Is qute smtlar to an argument of Cslszar [1].
Assume S
LE]L 1.
such that
set
2
sup
/#l(kx
(d:endng on
/#2 (KX
/#1
an(K
Def,ne
a for a cx)npact set K c S.
/#l)
such that for
-1
-<
sup
/#
(Kx-1).
/#k
ly-1
a-
n
an
be a probablit9
measure
Then there exists a compact
other prbiltN sure on S.
-/J2(K2 x
(/2(1
/#1
-1
)).
n.
Then if k
x
n
Therefore
{an(K)}
and
sup
aO
Let
(c).
satsf,es condition
-t
We omit
is a nonincreas,ng
sequence.
Define
a(K)
lim
an(K
a(K).
K
compact
THEOREN 1.
PROOF.
0
Suppose 0
< a(l+a)/z
sup
a
pk(Zx-)
for some K
If S satisfies condition
a.
a
1.
then either a
0
a
n-k
/#
p
K2x-1
n( Kx
0
1.
For any compect set K there exists a
k(K) such that
a/2(1
suff,cientl large.
0 or a
Then there exists an a such that
to
W
/#1
k
and
2
n-k
/#
2
sup
But then
1.
0
Appll,ng Le
/#n(Kx-1)
If n is
cx
(c)
pn-k(Z2x-1 )).
pn-k
K2x-
a(Z2)
_( a
0
0w.(z-a)
a for all x since
a.
(
lields
the fact that
AN EXTENSION OF A RESULT OF CSISZAR
Therefore
((K) <_ (x(i+)12.
conclude that
0 or
(0
(x
Since
K
31
arbitrarg we have a contradiction.
is
We
!.
0
QED
Beore proceed,rig we present an example.
Define multiplication bg r-s
topolc.
subset of S.
Let S
[0.)
Let K
max(r.s).
w,th the usual
[O.n]
be a
Kx-I O=
{ 0K xx _< nn and /jn(Kx-1 { W{K) n
Therefore I p has compact support then
I.
0
x
n
x _< n.
Otherwise.
0
O.
MATRIX SEMIGROUPS
3.
Let S be the set of all m x m matrlces wlth probabilltg measure
S such that the support of p generates a subsemigroup S
Deflne G
the usual topology.
subgroup of S.
O.
S
(X
We consider the case where O
sl: that a
0
O or
If
locall ct.
then we need
defined on
We assume S has
Then O forms a
of O generated
Therefore we assume O
1.
aO
ls
p(O)
of S.
X is nonsingular).
S
We want to consider the subgroup O
topologlca! subgrOllp of S.
"
ccct
Then
Then O
onl appl
1.
M(G)
b
the set
becomes a
Csiszar
[1]
to
Define a measure
on O such that
p(B F G)/p(G) or B
H’(B)
Then
;
(p’)2(B)
p’(BX
S.
p’(dX)
S
o
P(Bx-
o o)/p(o) ’(dx)
O p(BX-1
1/p(O)
Ho Bx
-1)
=
F O)/p(O) p(dx)
Therefore.
-1
(p.)2(B)
If x (G then
(B
(p,)2(B)
I/p(O)
O)x
-I
2
O.
I/p(O)
#2((B F G)x
Thereore
2
p((B n
s
(B n o)(o,
Bg an induction argument.
(w,)n(B)
(dx).
(B n G)//J(G)
n.
(dx)
32
P. B. CERRITO
DeF;ne the
following notation:
sup
l=m
sup
KcO
n
xC-.G
sup
sup lzm
n
KeG
sup
0 or a
g
[sup
/jn(Kx-1)]//J(G)n.
xeS
n
Since
a
(/2’)n(Kx -1
xeS
lim
n
Kc-G
(’)n(Kx-t)
g
BI
Csiszar’s result
It
For groups, either
However.
1.
/#n(Kx -1) ]/p(G) n
llm[sup
ThzS is only possible if lira
assume that K is a
sup
Wn(Kx -1)
0 For
any K
c O.
HenceForth. we
coct
set cons]st,ng oF singular matr,ces. We w,ll also
-1
exclude the zero matrlx from our d,scusslon since 0 0
S reduces the problem to
a triviality and it s obvious that a
a
0
sup
Im
sup
KcS
n
xeS
xO
We glve an exanple.
1/m.
Then
n+l
(Zx-)
p
K(Yn..-ytx)
where
Wlj
-1
e K(y
n
(Zij)
I.
Hence for any
0
[0.l/m).
everl entrtj in X is contained
Wll
wlm
wml
Wnn
ccct
O.
K)
0
mn-16n for all
mn-ln zij e K
set
and j.
Therefore for
.
so that as n
zij
0 for all
set K.
/n(Kx-l)
1.
,n the
and
0
n
By a similar argument, if ever N
then a
nonnective
I P(K(Yn’’’Ylx)-I)/J(cIJI)’’’U(dIn)
(zS:ZYn-..ytx
llX
lim
(x
consists oF matr,ces wth
has mininl value
Z
and
Suppose S
/
yn-..yl x
where
(Kx).
For any X e S
entries such that
where 6
n
/
That is. we define
1.
0
entrN
of X
S
is contained in
AN EXTENSION OF A RESULT OF CSISZAR
order to state a mre general result,
Let
notation.
zk
necessary to
it is
define some
Let
ldempotent matrzx of rank K.
be the dzagona!
Then
$
"tln
33
_Wlj n
0
0
n-1
where j(1...-.m
and
Wjn
w
We need to conslder the
represents the product of n real numbers.
zjn
dlstrzbution of
$
W
n
,jn
(I
where j
2
..-.m
Let F
).
be the
,jn
J
d,str,butzon funct,on of the random variable
W
1.2.--..m:
,in
j
n
1.2,....m
..
mag applg
the matrices then we
array
{W,jn )3 n
for
THEOREM Z.
cond,t
6
the Lzndberg-Feller Theorem
[2]
to the double
every
Suppose the {W
sat,sf
defined above
jn jn
the followng
ons for each
I.
If
If we assume ndependence between the entr,es in
1.2....
2
E(W
g2
dF
0 for
then a
By
PROOF:
everg
j.n.
<_ M)
ever
,.
nl
converges
.
Therefore for n and II
for all
/(x
Ig21
8 for each
I.
0
the Lindberg-Feller Theorem. S
standard normal for
where e
0 as 11
KI,
(xij): xj1-..Jnll
Therefore
[-I.I,,]}
J xtjSni
_< w
(x:
12 xij l" -<
K for all j)(1 -e)
for
in distrlbutlon to the
sufflcently large.
/(x:
az.z J)
m
)(K.).
>_ (
Mote that k"
everl n.
0 where the ntegral is taken over the set
0 as n
P(JSn,
for
Var(Wijn)
nds onl on the ce o M and K and t on the cice o n.
it s clr tt
incrse
S we g al let
k
Therefore as K
0
sup
im
n(
It ,s clr that ndtons
Var(W jn
J
M for
QED
1.
()
and
M and (w
jk
(2)
be relaxed
for all
k
tt
P. B. CERRITO
34
We
suppose
present an example,
Sp
x
x2
x3
x4
0
0
0
0
0
0
0
0
0
0
0
0
of the measure
support
the
x
R
E
Then
x
0
0
0
0
0
0
Therefore we need onltj be concerned wth the probability dxstrxbuton of the
Suppose X..
corner element.
xJ
I/2)
P(Xij
Then
0 and
E(X J
..x
var(Xtjx2j
P(Xij
wj
zxj.
1j2
zx1jx2j’
n.
0 and
E(XljX2j..-Xnj
j
1.2.3.4
j
2XljX2j..-Xnj.
E(WIjn) O AlsO
dFljn(g
above theorem,
aO
The Case Jhere a
Also for an 9
9.
Define
Ulj n
4.
I/2 for all i.
=-I/2)
I/4.
Var(Xij
I/4.
nj
a random variable such that
ts
j
1.2.-...4
0 if n s
n-1
Then
and
var(Uljn)
sufficientIN
BN the
larcje.
I.
0
If
clear that for
an
sequence
..(an).
n
p
converges vagueIN to
8n
Therefore we concentrate on the case where
second countable. Hausclorff
LEMMA 2.
that for
any
If
0
a
ct.
semgroup
and S is abetan then there exists a
0
zero measure.
Let S be a locally
i.
aO
the
there exzsts a
ct
set
K
such that
sequence {x n
such
__un(KXn-I)
a for
at! n.
PROOF:
n.
For
I/2 there exists
$mlarly. for each a
n
sequence
Therefore there exists a
(KaXna
sO there
-I
a.
$nce
must exzst w e
a K
(Xn}
I/2 there exists
a
{K2x n
such that
K and a
a
I/2. the sets
-I
such that
2
{Zxr
K2xn-I
-1
).
and
sup
n
@n(K2x-t
(K2x n
sequence
KaXncx
I/2 for all
I/2 for all
(Xncx)
n.
such that
cannot be disjoint
Ths ples that
AN EXTENSION OF A RESULT OF CSISZAR
x
K w
na
y
E
a
K x
-I c
-I
zz"
yx n e
Z’Xn
and
Therefore K
Ka[Ka(K2xl)]
aK-(K2Xn -1)-1
z
Also
-1 -1
y
Then
n
Ka.
Ka
yz
Ka( K2x n
(Ka2)-IK 2K
a
K
and
2
(Za2%,- 1Ka2K2. pn(KaXn-1
y e ((Ka
c
Ka{Ka( K2x n
so there exists z
11
Suppose
)]
-I -I
K
(Kzx n
a
K2Xn-1
Ka2K2"
implies there exists z’
(yz)(zz’)(Z’Xn)
Therefore
2.
-I
x
ana
35
-1Ka2K2Xn-I ).
2
such that
such that
Slnce S s abellan.
By redeflning K
to be
a for all n.
QED
group we can
If $ ,s a
Uk+l
k
6
n
x
u
Uk+2
n-k
6
n
6
H
-1
nlng tn a semtgroup and the u "s st
sen .(Xn
Un
an
such tt for
x
n
(Zx
then
)S
k
u
n
of
)S
s
k
seguence.
of th,s
n
Uo(Ka
U
Kae
o(Zs)
Z
prov,d
such tt
0
an
lmit
,ntlt
en
elnt
sets fln in S and
s ntain in S.
of
resct
to G.
for anN a.
[] can
n
lnt o
--(KXn-1
ntained In S.
the
reactive
w,th
Le 2 there exists
Let
AI
b
alied
l
%o his
s ve
k
u
((Ka
U
s an abelan nverse sigro.
or an x
F provided
then we can flne u
u
$
S there exists a
U K
n (Za
n
6
x
e
e.
Fe
Therefore all llmt po,nts
unite x"
S such
p
n-1
6
x e
IF S ntains a minil
with x
n
ae)
Kj) (Xne )-I
’>(Jn-)
S Is a
A tural orrlng n be Fn on the
x’.
elents of S: e
n
x
let
b
srt
wzth
srt
O
"s are
therefore be nsder a sure on S.
x and x’xx’
,tent e
K
pn (Kx n
n
Next nsder the case where
itent
for the
since the
t is al clr tt
that xx’x
)O
n
other wa N.
Then
be a prbllt sure and Csiszar
ntaned n S and
s,group
-1
Therefore a
Then
n
(Kx
and
ever
)G.
(Zx
Uk +I
lnt
-1
pn((Kx-t)G ).
pn((Kx-t)S
n
c
(Kx
where the x
n
Unfortutel 6 x -1 s
x
n
If we write
Of O.
x
to
a there exists a
g ass tt H Is a sure deflned in G
is well fined in O ,f we
6
p
-1
x
[1]
s edble n an abelan group O.
Supse
We
6
flned In
n
a
p
-1
n-1
Then we can apply Cslszar
deflned In lem%a 2.
n
deflne u
;o a, n.
of
n
YR
are probabl,ty
measures.
0
e.
Then
P. B. CERRITO
36
If
product
a
$ contains
nunC>er of ldempotents, sag e
fin,re
Therefore Csiszar
is minimal in S.
ele2.’’en
then the
1.e 2.-..,en
[I]
can be a[:)lied to ang
abellan nverse semlgroup with a fnlte number of dempotents.
Suppose nstead that S Is an inverse semigroup such that the set of
idempotents can be ordered
is. suppose s
the following manner:
in
Let x
an w-semlgroup.
is
Gven ang x
detlned in lenm%a 4
the
b
there exists
some
e
>
j
6
n
/J
x
n
-1
I
n
n-X
u
n
n
2
n
lm
gk
J
J
sequence
{Xn)
e
n
e
J
if
If
n
n
j
x. e.
n n
u
Ow n
Bg CsIszar [1]
PROOF.
n
n
K. gi
That
n or
for all j
such that e
Xil .x,2
S sat lsfg,ng the hgpotheses of Cslszar
such that for each
>
is true then S has a m,nimal Idempotent.
(a)
Uk+l
gk
n
j
n-1
THEOREM 3.
e
x x
nn
j
not. there exists a subsequence x 0
Delne u
f2
f!
and consider the
n such that e.
If there exlsts some n for which
>
ether
n
x
idetent x jj
a.
f0
0
f0
n
is
a sequence of pr(X>ab, litg masures on
then there exists a
[I]
sequence
(Wn)
in S
converges vaguelg to a probabil,tg measure as
sequence o ,ntegers
there exists a
such that
A
k
lm k
and
k
n
where the limits are defined with
.
probab, l,tg measure for all k
respect to the vague topolog9
and k
is Idempotent and k
Also k
a
is
k
k
k
k
for
all K.
The
H denote the support of
is
k.
a Haar ,asure on H and H
Since S s abelan. H
is
a
is
a
group.
Furthermore.
Is
k
ct group
The remainder of the proof, dealing with the choice of a suitable
{Wn},
Let
support of ang idempotent probabltg measure Is con01etelg simple.
gute slmlar to the argument
in
Cslszar
[4]
sequence
and will be cnItted.
QED
We define a
If
$
lim
n
is
n
gO
n
x w
n n
x
where
embeddable or an inverse
n
w
lim
n
n a
k
n
0
Is defined In lemm 2 and
semgroup
w
n
s deflned above.
with a minimal ioempotent then
which is a probabiltg
,L=asure.
In the other
two cases, the same argument can be applied to an infinite subsequence.
AN EXTENSION OF
,
RESULT OF CSISZAR
37
REFEREHCES
1.
CSISZAR. I. On In1nite Products oF Random Elements and Infinite
Convolutions oF Probability Distributions on Loclly Coe)act Groups. Z. Wahr.
ye. Oe. 5_. 1966. 279-295.
A Course In Probability Theorg. Academic Press. New York. 1970.
2.
CHUIIO. K. L.
3.
CLIFFORD. A. H. and PRESTOH. O. B. The Algebraic Theory oF Semlgroups. Vol.
I. Mathematical Survegs 7. Amr. Math. Soc.. Provldience. R.I.. 1961.
4.
CEITER. B. and MUKHERJEA. A. More on Limit Theorems For Iterates of
Probabllty Measures on Semgroups and Groups. Z. Wahr. verw. Oeb. 46. 1979.
259-275.
A. Lmt Theorems or Convolution Iterates o a PrcW:wblility
Measure on Completely Smple or Conl)act $emgroups. Trans. Amer. Math. 5oc..
225. 1977. 355-3?0.
5.
MUI{HERJEA.
6.
YEAOER. D.
Imbedding Compact Semgroups
Se_m_1 _r._o_u.p__ Fo_ru_ 10. 1925.
"/6-83.
in
Coct
Inverse $emgroups.