SmllARY
Some Extensions of the Hewitt-SavaGe Zero-one Law
(Short Title:
The Hewitt-Savage Zero-one Law)
Let M denote the class of infinite product probability l:leaSUres
J-1=J-1 xJ-l x ••• defined on an infinite product of
1 2
measurable space
(X~A),
replications of a given
and let H denote the subset of fl for which
J-1(A)=O or 1 for each permutation invariant event A.
Previous works
by Hewitt and Savage, [lorn and Schach, Blum and Pathak, and Sendler
(referenced in the paper) discuss very restrictive sufficient conditions under which a given member
J-I
of M belongs to H.
In the
present paper, the class H is shown to possess several closure
properties.
E. g., if J-1df and
lloxJ-li XJ-l
eH.
X •••
O < < lln for some n;o;l. then
Nhile the current results do not permit a
l-I
2
complete characterization of H, they demonstrate conclusively
that Ii is a r.lUch larger subset of
Mthan previous results indicated.
The interesting special case X={O,l} is studied in detail.
American Hathematical Society subject classification.
Key
~'Jords
and Phrases:
Primary-·60F20
zero-one laws, invariant events, orbits,
infinite product probability measures
SO~ffi
EXTENSIONS OF THE HEWITT-SAVAGE
ZERO-ONE LAW
by Gordon Simons
University of
1.
Introduction
~orth
Carolina
- Chapel Hill
The zero-one law described by Hewitt and Savage (1955)
asserts that an infinite product probability measure assigns the value
zero or one to each (permutation) invariant event when the component probability measures of the infinite product are identical.
with Kolmogorov's
event
zero-one law which makes the same claim for each tail
(a special kind of invariant event)
the components.
This contrasts
with no restrictions upon
The gap between these two laws was slightly bridged
when Horn and Schach (1970) showed that the assertion for invariant
events held when each component occurred infinitely often. in the infinite
product.
Blum and Pathak (1972) refined their argument and showed that
it was sufficient that each component be a limit point of the sequence
of components under the total variation norm.
The intent of the present
author's investigation was to discover whether the Blum-Pathak result
leaves much room for improvement.
In this regard, Sendler (1975) has
pointed out that the zero-one law holds for invariant events when each
component of the infinite product probability measure is degenerate, i.e.,
when each component assigns probability one to a single point.
there is at least some room for
in~rovement
Thus,
of the Blum-Pathak result.
The main conclusion to be dra",n from the present paper is that substantial
Ip..esearch SU;'l)()rte,l 1;y the National
No. MCS75-07556.
~.cience
Paunc.kti'):1 "mc18T FTCtnt
2
improvenents are possible.
It should be mentioned that a zero-one law
for invariant events does not hold without some restrictions upon the
components.
The reader can convince himself of this with very simple
counter-exmnples of his own making, or refer to any of several counterexamples given below.
co
Consider the infinite product measurable space (X , A ) which is
00
generated by a fixed measurable space
()~A)
replicated a cduntably
\
infinite nwnber of times.
Associated with each point x=(x ,x ... )
l 2
oo
E:)t
is the orabit o(x) consisting of all points y=(Yl'Y2' ... )sX
which can be obtained by
permutir.~
a finite number of components
oo
An event AcA is said to be (permutation) invariant if
of x.
o(A)=A.
Let M denote the collection of all infinite product probability measures
~=~lx~2x ...
Further, let H denote the collection of all
neasures on (XA).
probability measures
property·1:
whose components are probability
~E:M
which possess the lIHewitt-Savage zero-one
ll(A)=O or 1 for every invariant event Ael,co
we would like to be able to completely characterize
H.
Ideally,
This can
not be done at the present time except in the case of degenerate
measurable spaces (X,A).
Instead, we shall demonstrate that Ii
possesses certain closure properties, some of which lead to the
general conclusion that there are many members of i-I whose membership can not be established by applying Blum and Pathak's results.
A partial characterization of 11 is given in Section 3 for X={O,l} .
The simplest closure result is the following:
Closure property 1. Ii' '~EH and vdf -ts absoZ7;·teZy c.QntinuouB «)it?
vdf.
3
Proof.
This is immediate.
o
A set of necessary and sufficient conditions for v < <
~
has
In order
that v
< < ~
it is necessary and sufficient that (i) v
n:<:l, and (ii) ~ J Idv /d~
n=l X
n n
(1975) , page 44.)
dll
n
> O.
< < ~
n
for
(See, for instance, Neveu
Suppose X is the real line and each
Example 1.
n
~
n
is the l1standard"
normal distribution, whose mean is zero and whose variance is one.
Further, sUfpose v
n
is the normal distribution whose mean is a
whose variance is one.
n
and
Then, by applying (ii), it easily follows
00
(in fact v=~) iff I (a )2 < 00. Far such v, it follows
n=l n
from the classical Hewitt-Savage zero-one law and closure property
that v
< < II
1 that
vsH.
00
Supnose, instead, that a n
=n and
A=IR , the set of
-+
L
pointsx=(~,X2"")
all of whose components are positive.
Then,
00
by the Borel-Cantelli 1 emr.1a , v({XE:X :x <0 infinitely often})=O and,
n
2.
o
Consequently, v¢H .
hence, v(A)s(O,l).
Other closure properties.
Hewitt and Savage (1955) describe a proof of their zero-one
law due to Halmos which is appropriate for doubly infinite product
y
y
00
measures
~=
X ll.
This suggests the following question;
0=_00 n
Hewitt-Savage zero-one property order independent?
if
1l=~lx1l2xoooE:H,
does it follow that
Is the
For instance,
1l1=1l2xlllx~3x~4xOOO
and
ll"=llxll X~ Xll X~ x~ Xooo are members of H? The answer is in the
214 365
affirmative. Quite obviously ~ I sf{ because it can be obtained from
II through a finite permutation of the indices
result asserts that
~lIE:H
as well:
of~.
The following
4
Closure property 2. If llcH .'"nd vdl . is obt.ainabZ§ frOr(lll PH (a fini te
01' infin~te)
Proof.
pernrut-ing of the components in ll, then vcH.
If A is an invariant set, then v(A)=ll(TA) , where T is some
permutation operator on
o(TA)=TA.
V
OO
,l\
Since llEH, it suffices to show that
•
-1
This reduces to checking that T S T is a finite permu-
o
tation \ihenever S is a finite permutation.
Closure property 3. If lldl and vEtf, then llXVE H.
Remark.
The expression
"llxV"
will be taken to mean any arrangement
T=TIXT2xoooof the combined components of
11 and v.
Closure pro-
perty 2 provides a strong justification for this convenient abuse
of notation.
Proof.
For the sake of definiteness, the arrangement T=lllxVlx112xV2XOOO
will be used.
00
For each point x=(x ,x ,···)cX , define y=(YI'Y2"")
1 2
and z=(zl.z2"") by means of the identity x=(YI,zI'Y2,z2"")'
Let
A be a fixed but arbitrary invariant event, and let A denote the
Z
x=(YI,zl'~2"") EA},ZE:X
CXl
'section of A defined by {yeJt
•
Applying.
Fubini's theorem, one has
T(A) =
(1)
J
XOO
ll(A )v(dz) •
z
Since A is an invariant event, each set A is an invariant event as
z
well.
Consequently, ]l(A Z )=0 or I for each ZEX
oo
Moreover, Az is
unaltered by finite permutations of the components in z.
Thus
E={ z: II (A ) =I} is an invariant event. Hence, v ill) =0 or 1. In the
z
first case, it follows from (1) that T(A)=O, and, in the second case,
that T(A) =1.
Therefore T=]lXVEH.
The proof of the next closure property is similar to the last,
but is more complicated.
that is qbsoZuteZy continuous uitlz respect to one of the components
o
5
of
~, then
Proof.
T=vxlld/.
Because of closure property 2. it will be assumed, without
loss of generality. that \) < <
that
v=~l'
111,
Furthermore, it will be assumed
For. then, the general case \) < <.
~l
will follow from
this special case by means of closure property 1.
For any event
A~oo. Fubini's theorem yields
TCA)=! ~(A )V1(d~).
X
z
(2)
00
where A is the section {Y=(Y1JYZ"")€X :Cz,Y1,YZ •... )€A}.z€X.
z
Applying Fubini's theorem again. one obtains
(1)
(3)
v(A
wherel.l
(1)
)=J V
z X
=v 2 x v 3 x
z }'
(Y l' Y2' ... ) E A
(A.
z'Yl
1".
and A
Suppose
T(A)=~l(E),
where E={z€X:
It will now be shown that
that TEfl.
~1
{
00
is tho'sec!lon'(Y2'Y3"")cX::
z' 'Y 1
A is any
A is an invariant event, and
z
that
)1l1( dY1)'Z€X,
~(A
z
invariant event.
)=0 or 1.
Then' e"l~1t
It follows from (2)
~CAz)=l}.
(E)=O or 1. from which it will follow
Let EC denote the complement of E (relative to X). From
(3), it follows that
J
Ec xX
l.l(l)(A
.)
z;Yr
l.llX~ICd(z'Y1)) =
Thus
(4)
Similarly. one can show that
(5)
~ (1) (A. )=1 on ExEC
z'Yl
a.e.
But, since A is an invariant event. A
=A y z for each
z')'1
l'
z and Y , in X. This leads to an inCOMpatibility of (4) and (5)
1
6
c
~l(E)~l(E
unless
~l(E)=O
)=0, i.e., unless
o
or 1.
Closure property 4 provides a condition under which membership is preserved when a component probability measure is added
to a member of
H.
Some such condition is necessary as the next
example demonstrates.
Example 2.
Suppose X={ 0, l} and A is the power set of X.
]In({O})=l foT.
n~l.
= 1/2 and T=VX~.
Then
~=lJ{]J2x
T
t
Suppose v({O})=v({l})
Let A denote the invariant event ,0((1,0,0, ... ))
(the orbit of (1,0,0, ... )).
ly,
00,£11.
Let
Obviously, T(A)=1/2 and, consequent0
H.
Closure property 4 can be used repeatedly to add any finite
number of components to a probability measure
removing it from
H.
II
in
H without
This fact is stated precisely with the
appropriate assumptions as follows:
Closure property 4' .
I,f ~l£H anc?
v=vlx 00 ox v
n
is
0.;
1)1>oduct
probability measure on (Xn,A n ) (foY' 'some finite; n ~ 1)
eaeh
component of which if] ahsoluteZy continuous u.n:th respect to some
(Xn I"
An).lS t h e measurabl e space
component ofll, then VX l 111£ 1 .(Ilere,
1
generated
by
n replicates of (X,A).)
This result can be extended to include probability measures
v containing a countably infinite number of components.
One simply
must combine closure properties 4' and 5 (the latter given below).
Before we state closure property 5, we must introduce some
new notation and a convention.
11=1l xp x
1
2
and v=v 1 Xv 2 x
000
cODponent of
II
000
For two probability measures
in 1'1, we shall write
if every
is a COMponent of v with the frequencies of the
repetitions in v as great as those in p.
II
IlS;V
Expressed alternatively,
can be produced from v by (possibly) deleting SOrle of the com-
7
ponents of v and rearranging those which remain.
then
If llS;:V and
VS:ll,
can be produced from v by ;erm':.ctinS the components of v.
II
Then closure property 2 says that llE:.H iff 'JEff. Since membership or
nonmembership in H is our sole focus of attention, there is no harm
done by treating
II
Now suppose ]l
(.)
1
E:. M and ]l
this convention,
(i)
~ v for some
If]l is the snallest such rlcl:tber v of V,i..oJ.
vdA, iE:!.
a v and
~1ith
and v as equivalent.
l.!~V
for every such v, then we shall write
II =
>
Lff.!'is such
U]l
(i)
.
id
It is easily seen that such a ]l always exists when I is nonempty
and countable, and that it is unique up to an equivalence.
Closure property
II
= .~O
n-
Proof
5•
If]l
(n)
.
df mui]l
(n)
'n+l)
~ II ~
foY'
nzO, then
]l (n) £ H,
The essence of the general argument can be seen in the
Let j.l (0) = v=V{V2X
proof of the following special case;
000
and, for nzl, ll(n) = vlxTlxv2xT2xOOOX v XT xv
Xv
Xoco
n n n+l n~'2
where the v. 's and T.' S
1
II
=n=Xl
1
(v x T).
n
n
are probability measures on (X, A) •
Let A be any invariant event.
Then
For each n>-O,
Fubini's theorem yields
(6)
=J
II (n)
where each section A
(A
)
zn+l,zn+2""
"',
X
k=n+l
T
J(
(d(z n+ l'z n+ 2" .. )),
co
zn+ l' zn+ 2; ...
= {(yl·zl' .. ··y,z.y
n n n+ l'yn+ 2····)E:.X
is always an invariant event.
Let En
co
= {(zn+ 1'z n+ 2"")£ X : j.l(n)(A
)=l}, n>-O.
zn+ 1'zn+ 2""
co
(the complement of E relative to X ).
n
8
, • • •
n
)= f
n
V(A)
,XTk(d(zl~···)zn) )
,Z
l,::;
X
z
=
Thus.'
Similarly,
J
X nxE
fn
x EC
n
X
v (AJ T(d ) :: 0, n2:0, where T= T x T x
l 2
z
t..
(l-v(A ))T(d ) = 0 n~O.
z
n
Z
v(A ) = 0 on
z
•
n
e
X x E
n
= 1 on X\n
Therefore T(EOA (Xn
E
n
Hence. for n~O,
a.e.T
a.e.T.
This says that EO is
(See Blum and Pathak (l972) for-the meaning of
tail approxirnability.)
Thus E is T-equivalent to a tail event
o
and, hence, T(EO)=O or 1.
J v (A
E
o
3.
x
En)) = 0 for n2:0.
x
T-tail-approximable.
~ (A) =
1
An application.
z
Then for the case n=O in (6),
) T (d13)
= T (EO)
:: 0
o
or 1.
We are now in a position to say much more
about independent Bernoulli random variables than previous
description of the Hewitt-Savage zero-one property permitted.
The
study of such random variables is equivalent to the study of
probability measures IJ=J.l { IJ X
2
00 0
power set.
{-f
E
tl when X={ 0, I} and A is its
Clearly the membership or nonmembership of
~
in
depends completely upon the sequence of parameters
p
n
=
~
n
({I}), n2:l.
We shall demonstrate the followinL;:
print,
pE (0,1),
then
If' {Pn} possesses a 'limit
~d{.
Proof.
n.
By the classical Hewitt-Savage zero-one law, vdf,
assumption, some subsequence Pn,of Pn converges to p.
Now, by
There
exists a further subsequence Pn ,! which converges to p so rapidly
that the corrcsiJondin'!,
l'rocluct -o)roljability
T=T 1XT 2'"
x
,",
i
con-
()
(l
0
9
sisting of components
~
n
" taken from ]..I» is absolutely continuous
(It is sufficient that III (p-p II)
with respect to v.
n
n
2
< "".
follows from Kakutaniis theorem» referred to previously.)
ing to closure property 1, Tdl.
This
Accord-
In turn, it folloh's from closure
properties 4' and 5 that ]..IsH.
0
The following is a partial converse:
)<
If 0< E min(p n ~l-D
~n
at,
n=l
then
Proof.
"" min (Pn' (l-P ))< "", then it follows from the Bore1If nh
n
Cantelli lemma that ]..I has a countable support.
point x
= (x 1 ,x 2 , ... )
in the support is such that for some large
N, depending on the point, and every
x =1 if P
n
n
> 1/2.
Specifically» every
n~N,
x =0 if P
n
n
~
1/2» and
Let x be a point in the support and A=o(x)
co
A point Y E X
(the orbit of x).
belongs to A iff the sum
co
1:
(y -x ) has a finite number of nonzero terms and equals zero.
n= 1
n n
00
•
If E mm(p ,(l-p ))>0, there exists an index m for which p is
n= 1
n
n
m
neither zero nor one.
In such a case. the point y=(x l ,x 2 ' ...•
xm- l.(l-xm).x.m+ l'xm+~"), ... ) must be in the support of]..l but
outside the set A.
Then
O<]..I({x})~]..l(A)$I-]..I({y})
< 1.
Thus, jltH
If ""E min(p ,(l-p ))=0, i.e., if every p is zero or one,
n= 1
n
n
n
then it is easily checked that ]..IdL
a member of
4.
H if., for instance.
Conclusions.
VJe do not know whether ]..I is
Pn = l/n,n~l.
~
We view the collection of results in this paper
as a modest beginning.
There remain many other issues that need
to be settled before a very clear picture of H can emerge.
For
instance, it is (implicitly) suggested by all previous papers
concerned with the Hewitt-Savage zero-one law that the following
should be true:
If ]..l=]..Ilx]..IZXo.o sH, then]..l 1= ]..I2X]..I3xaoosH.
no idea whether this is true.
i'~or
We have
do we know whether the condi-
0
10
tions
~, VE
~(i)E::H.i=1,2 imply J l Jl(i)d-l" or whether the conditions
H and
~ s; T ~
v imply
TE:iI.
In spite of the nany unanswered
questions, our results demonstrate conclusively that the Hewitt-Savage
zero-one property helds much more \ddely than previous literature
on the subject has suggested.
REFERENCES
11
[1]
Slum, J.R. and Pathak, P.K. (1972). A note on the zero-one law.
Ann. Math. Statist. 45, 1008-1009.
[2]
Hewitt, E. and Savage, L.J. (1975). Symmetric measures on Cartesian
products. Trans. Amer. 'Math. Soc.) 80; 470- 501.
[3]
Horn, S. and Schach, S. (1970). An extension of the Hewitt-Savage
zero-one law. Ann. Math. Statist. 41, 2130-2131.
[4]
Neveu, J. (1975). Discrete-Parameter MartingaZes.
American Elsevier.
[5]
Sendler, \'1. (1975). A note on the proof of the zero-one law of Blum
and Pathak. Ann. FPobabiZity, 3: 1055-1058.
North-Hol1and/