On the Construction of Almost Uniformly Convergent Random Variables with Given Weakly
Convergent Image Laws
Author(s): Michael J. Wichura
Source: The Annals of Mathematical Statistics, Vol. 41, No. 1 (Feb., 1970), pp. 284-291
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2239738
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Statistics
TheAnnalsofMathematical
1970,Vol. 41, No. 1,284-291
ON THE CONSTRUCTION OF ALMOST UNIFORMLY
CONVERGENT RANDOM VARIABLES WITH GIVEN WEAKLY
CONVERGENT IMAGE LAWS'
BY MICHAEL J. WICHURA
of Chicago
University
d, and
metricspace,withdistancefunction
Let S be an arbitrary
1. Introduction.
distributions
letY' be itsBorela-algebra.DenotebyY(S) theclass ofall probability
on (S, 9). A net(Py)y of probabilitiesP e (S) is said to convergeweaklyto a
probabilityP eG (S) if P(f) = lim Pf(f) foreach real-valuedboundedcontinuous
functionf on S; here P(f) = ffdP, PY(f) = ffdPY. Let gs(S) denote the subclass of CA(S)consistingof thoseprobabilitiesP forwhichthereexistsa separable
one. gS(S) includestheso-calledtightprobabilities
subsetofS in f ofP-probability
i.e. probabilitiesP such thatsup {P(K): Kcompact} = 1 ([5] page 29). The chief
resultofthispaperis statedin thefollowing.
be a netofprobabilities
THEOREM1. Let (S, d) be a metricspace and let (Py)yEr
P E gs(S). Thenthereexistsa probability
weaklytoa probability
PYG i7(S)converging
on Q
X andX/(yE F) defined
S-valuedfunctions
space (Q, ., ji) andA-9- measurable,
- 1 ofXyare respectively
uX- ' ofX and luXy
P andPy(yE F)
suchthatthedistributions
toXalmostuniformly.
andsuchthatXyconverges
One sometimes([1], [8]) has occasion to considerthe weak convergenceof
distributions
probability
Pywhichare definedonlyon certainsub-c-algebrasof9,
in Theorem1 thatthe
of interestto knowthatthe requirement
and it is therefore
PY belongto 3d(S) can be weakened.To make thisprecise,let us say thata net
ofprobabilities
(Py)yEr
Pydefinedon sub-c-algebras-4yof " convergesweaklyto a
P E ^(S) iflimyPY(f) = Pff) = limyPY(f) foreach real-valuedbounded
probability
theupperand lower
continuousfunctionfon S; here Pyand Pydenoterespectively
associatedwithPY:
probabilities
PY(f) = inf{Py(g):f < g, Py(g)defined}
PY(f)
sup { Py(g):f?> g, P,(g) defined}
of thisdefinition
see Theorem1 of [8]). It is clearthat
(forequivalentformulations
thisdefinition
of weak convergence
reducesto theusual one ifall the -4yequal Y.
Let Y' denotethesub-c-algebraof S9 generatedby the open balls of S. We then
extensionofTheorem1:
havethefollowing
THEOREM2. Let S, 9, and Yo be definedas above and let (Py)yerbe a net of
in 9, which
probabilities
g'o and contained
Py,definedon a-algebras 4y containing
ReceivedOctober23,1968;revisedAugust7, 1969.
l Thisresearch
was supported
in partbyResearchGrantNo. NSF-GP-8026fromtheDivision
of Mathematical,
Physicaland Engineering
Sciencesof theNationalScienceFoundation,
and,in
partfromtheStatistics
Branch,Office
ofNavalResearch.
284
ALMOST UNIFORMLY CONVERGENT RANDOM VARIABLES
285
Pe Ys(S). Thenthereexistsa probability
convergesweaklyto a probability
space
onQ suchthat:
(Q, X, ji) andS-valuedvariablesX andXy(yeF) defined
(1)
X is . -J9 measurable, Xy is M -.y
(2)
=x-
(3)
- P,
Xy-X
measurable (yeF)
uX,-1 = PY(yeFr)
almostuniformly.
withall theusual axiomsof settheoryto assume
We remarkthatit is consistent
in Theorems1
thatYS(S) = Y(S) (see [2] page 252). In thissense,therequirement
and 2 thatPG Y,(S) can be replacedbythetrivialone thatP Y
g(S).
In the construction
used to validateTheorems1 and 2, Q0is theproductspace
Sx HlyerSy,
whereeach Sy is a copy of S, M is a a-algebra whichcontainsthe
1uis theprolongationto (Q, X) of a mixture
sl = 9 x Hlyerdy,
product F-algebra
of productprobabilitieson (Q, d), and X and Xyare thecanonicalprojectionsof
LI onto S and Sy(ye F). When F is countableand S separable,one has -=a.
havebeenusedto validatespecialcases ofTheorem1. Working
Otherconstructions
is equivalentto almostsure
withsequences(forwhichalmostuniform
convergence
convergenceby Egoroff'stheorem)instead of nets, Skorokhod ([7], Theorem
3.1.1) has provedTheorem1 for S separableand complete;in his construction
Q = [0, 1], theunitinterval,4 is thea-algebraof its Borel sets,and p is Lebesgue
measure.Again workingwith sequences,Dudley ([3], Theorem 3) has proved
Q is a countableproductof copies
Theorem1 forS separable;in his construction,
of S x [0, 1], a is the producta-algebra on Q2,and p is a mixtureof product
of almostsurely
probabilitieson (Q, X). For applicationsand otherconstructions
see
convergent
processeswhichare of interestin thetheoryof weak convergence,
thesurveypaperbyPyke[6].
2. Proof for F countableand S finite.The simplicityof our constructionis
obscuredin thegeneralcase by severaltechnicalconsiderations;in orderto illustratethe generalidea we willin this sectionproveTheorem1 underthe assumptionsthat F is countableand S is finite.To this end, let (Sy,fy) be a copy of
(S, S9) foreach y, and let (Q, A)=(S xHlyr Sy,Y x flyrJS"y)be the productof
the measurablespaces (S, f) and (SY, Vy)(yeF). Let the canonical coordinate
mappingsX and Xy(yE F) be definedon Q by
(4)
X( (s,(so)oer)) = S,
Xy((s, (SO)oer))
The requiredmeasurability
properties
clearlyhold.
Let k: y-- k(y)be anyfunction
fromF to {0, 1, 2, ..,
(5)
=
sy(y F).
o} suchthat
limye r k(y)= oo
(k(y) should be thoughtof as a measureof the largenessof y and later will be
further
specified).For I < k < oo, set
(6)
= X}.
Uk = nyy:k(y)>k{XY
286
MICHAEL J. WICHURA
Observethat each UkE X since F is countableand that Xy-+ X uniformly
over
each Ukin viewof(5).
Let Qy(yeF) be any familyof probabilitieson (S, f). It laterwill be further
givingmass one to thepoints E S, let
specified.Letting3, denotetheprobability
Pi=s =
(7)
bs x HlyerMj,s,y
(1 ? j < oo,s eS) denotethe productprobability([4] page 166) on (Q, X) whose
componentsare respectively:
bs definedon (S, f), and
= QY
Pj,s,y
= bs
if
0 ? k(y)<j,
if
j < k(y)< oo,
definedon (Sy. Y.). Clearly Pui,X = bs Pi'sXY-I = pjsy; moreover,since F
is countable,Mj s(Uk) = 1 forj1? k. Next, defineprobabilities,j(1 < j < oo) on
(Q, X) by
Hj= sEs P{S}',s.
(8)
Clearly 1X-1 = p
tjXy- = Qy
=P
if
0 ? k(y)<j
if
j<k(y)<
oo,
= I ifj? k.
and puj(Uk)
Finally,let (wk)1
(9)
Wk _
(10)
k<0,
0
be any sequenceof numbersWksatisfying
Z
Ej<k
Z Wk = 1
Ek
Clk
=
Wk<
1(1 k < oo)
andput
El<j_ k Wi(O< k _ o).
Note w0 = 0, w,, = 1. Definetheprobabilityp on (Ql,X) by
p=
(I1)
Ejwjpj.
Clearly iX- 1 = p
( 12)
(13)
1elXY -
)k(y) P + (I -t)k(y))Qy
/(Uk) >?
and
< k < oo).
k(l
withrespect
Since limk-o Ck = 1, (13) impliesthat Xy-+ X almost uniformly
in
of (12), to
in
it
view
this
suffices,
setting
to p. To comp1-tethe proof
special
of Pyto P impliestheexistenceof k(y)'ssatisfying
showthattheweak convergence
(5) and probabilitiesQysatisfying
(14)
Py=
0k(y) P +
-(l
Wk(y))QY
(14) ifand onlyif
forall yCF. Now ifk(y) = oo,thereexistsa Qysatisfying
(1 5)
P1. = P,
287
ALMOST UNIFORMLY CONVERGENT RANDOM VARIABLES
and thenany Q, willdo. On theotherhand,if0 < k(y) < oo,we see,aftersetting
(16)
qk=s,y= Py{s} +(w)k!(l -ck))(Py{s}
(17)
Mk.y=
min ES
P{S})
qk,s,y
that thereexistsa probabilityQysatisfying
(14) if any only if
Es Es qk(y),s,y = 1, and thenone musttake
(18)
Ink(y),y?
0
and
Qy = EseSqk(y),s,y6s.
> 0. Thus itsuffices
We notethatEs Esqk,s,y = 1 forall k(O < k < oo) and thatmo0y
to showthat(5) is satisfied
and (15) holdsfork(y)= oo ifwe put
k(y)=sup {j _ 0: mj, > 0}.
(19)
Now sincePy-+ P, we have
(20)
Py{s}
-+
P{s}
foreach se S(Ifs1beinga continuousboundedfunctionon the discretespace S).
Hence limyerqksy= P{s} foreach se S, 1 ? k < oo; this,togetherwiththe fact
thatqk,s,y > 0 if P{s} = 0, impliesthatqk,,,y is ultimately
foreach
nonnegative
k (1 ? k < oo). Thus since S is finite,thereexistsforeach k an indexykeIT such
that mk,y_ 0 for all y _ y(k). Since mk,y ? 0 impliesk(y)? k, (5) is satisfied.
Next,ifk(y)= oo, we have (recallZs s qk,s,y = 1)
(21)
0?
Py{s}?(+
k/(1-
k) )(Py{s}-P{S})
< 1
for each s e S and arbitrarilylarge k; since Wk/(l - Wk) -+ oc, it follows that
Py{s} = P{s} for each seS, i.e., that (15) holds. This completesthe proof of
Theorem1 forF countableand S finite.
2 in the general case. Let P, Py(yeF), ?, Y'0, and
3. Proof of Theorenm
e
be as in Theorem2. Let W(P) = {C e : P(boundaryof C) = 0} be the
y(y-F)
class of P-continuity
sets.We recall ([5] page 50) thatW(P) is an algebraand that
foreach s e S, theopen ball
(22)
{t: d(t,s) < r} E W(P)
forall but at mostcountablymanyvalues of r. The followinglemmashowsthat
theanalogueof(20) holdsforsetsCe-(P) (conferT1.1 of [8]):
LEMMA
1. In thepresentcontext,C e(P)
implies
limye rPy(C) = P(C) = limye r PY(C).
PROOF. Let F be a closed subsetof S. Since the continuousboundedfunctions
fn:s - max((1 - nd(s,F)), 0) decrease to the indicatorfunctionof F, the weak
of Py to P impliesthatlimsupyPY(F) _ limsupyPY(fn)= P(fn)I P(F).
convergence
The dual relationforopen sets is seen to hold by takingcomplements;thus for
anyC S/'we have
(23)
P(C) ? liminfy
PY(C) ? limsupyPY(C) < P(C),
288
MICHAEL J. WICHURA
where C (resp. C) denotesthe interior(resp. closure) of C. When Ce%'(P), the
extrememembersof(23) are equal. 0
We shall need a sequenceof "finiteapproximations"to S. For this,choose and
fixanytwo numericalsequences(Ak)l I k< and (8k)1 <k< . suchthat
(24)
Ak >
(25)
,
Ak = O
limk,
Ek Sk <
Ek > ?,
00-
Lettingd(C) = sup{d(y, z): y, z e C} denotethe diameterof a subsetC of S, we
thenhave
LEMMA2. In thepresentcontext,thereexistpositiveintegersnk(l _ k < oo) and
disjointsubsetsCm. . ., mk(O < mj nj, 1 j ? k) ofS suchthat
(26)
(27)
Cm1,... mk-1
CmI,
EO<mkk<nk
1<j<kmax1
maxO<j<nj,
(28)
=
*
-,,nk
_mk_nkd(Cml,
ZO<mj_nj,l 1j<kP(Cml
*
_1,1k
,mk)
Ak
-k
nmk- l,O)
n7Y?(O _ mj _ nj, 1 _ j < k).
PROOF.Let E be a separablesubsetof S suchthatP(E) = 1 and let {sn, n _ 1} be
a countabledensesubsetofE. In viewof(22), thereexistsforeach n ? 1 an open ball
in S, call itEn,centeredat Sn withradiusgreaterthan'A1 butlessthanA1,suchthat
En6W(P). Since the union of these balls covers E and hence has P-probability
one, there exists a positive integern1 such that P(Un n, En) ? 1-81. Setting
(29)
CmI = EmI -E<m<m
CMl,
c-
* * *, mk
(P)
? m1 ? n1),
Cm(1
CO =
=
S-Un<nlEn
max,?<m?nid(Cml)< A1,P(CO)
S-E
CO,C
81C,
1m?nm?nCm1Iweget
** ,Cn,1e(P)nft0
The proofis completedbyinductionon k. 0
Let Hk(1 ? k < oo) be the finitepartition of S whose members are the Cml,...mk,
and put Ho = {S}. Choose and fixnumbersWksatisfying
(9) and defineokkby(10).
<
For 0 k < oo, CeHik,and y eF, set(confer(16), (17), and (19) )
S=EOZmi?niCmi,
qk,C,y =
(30)
Py(C) + (Py(C)
-
P(C) )(k/(l
)
-
mk,y = minEenk qk,E,y
k(y)=sup {j
0: m., _ 0}.
of Pyto P implies(see theargument
In viewof(29) and Lemma 1,theconvergence
following
(20)) that
limyErk(y)= oo
(31)
For y suchthat0 < k(y)< oo,put(confer(18) )
Qy = EC E nk(y)qk(y),C,yPy(
(32)
I
C),
on (S, sly) obtainedfromPybyconditioning
wherePy( C) denotestheprobability
on theoccurrenceof theeventC. It is easy to see thatQyis itselfa probabilityon
(Sya,sy)and that(confer(14))
(33)
(-)k(y)
(ZC e
rk(y)
Py(|
C)P(C))
+(1
-
Ok(y))Qy
=
Py
289
ALMOST UNIFORMLY CONVERGENT RANDOM VARIABLES
Now for each yeF, let Sy be a copy of S, and let (Q,)-(Sx
fl, rSY,
the productof themeasurablespaces (S, 9) and (SY,.y)(y E F).
s E S, and let(confer(7))
Let Ck,S denotetheelementofHk containing
y X fly e FS1y)be
Vj'S= bs X flyvj,s,y
be the productprobabilityon (Q, d) whose componentsare respectively:b
definedon (S, 9), and
(34)
= PYO Ck(y)s)
if
0 ? k(y)<i,
j < k(y)< oo,
= bs
if
k(y)= oo,
= QY
VJ,S,Y
if
definedon (SY,sy). For eachj, the mappings -+ vj,s(A) is a randomvariableon
base (sinceforeach
A E a? is a cylindersetwitha finite-dimensional
(S, S9) whenever
manyC in Hk(y) belongsto ), and hence([4] page 74)
yeF, each of thefinitely
thismappingis a randomvariableforeach A E d/. Thus ([4] page 76) we maydefine
a probabilityVj on (Q,4) by the formula(confer(8)) vj = fsvjsP(ds). Finally
v on (Q, d) by
(confer(11) ), definetheprobability
v=
(35)
<-< < wjvj.
Once again,letthecoordinatemappingsX and Xy(ye F) be definedon Q by (4).
We have
LEMMA 3.
In thepresentcontext,
(37)
(y eF)
X is sl'-9' measurable,Xyis '-.4y measurable
vX-' = P
(38)
vX,- I = PY(yeF).
(36)
For (38),
PROOF. Relations(36) and (37) followdirectlyfromthe definitions.
observethat
vX
' (=
=
k(y)(ECenk(,)
P
Py(
IC)P(C))+(1
-Ok(y))Qy
if
if
to a?y
restricted
0 < k(y)<
k(y)= ci.
00,
In viewof (33), (38) holds when0 < k(y)< oo. It remainsto show that(38) holds
hereis similarto, butmorecomplicatedthan,thatat
whenk(y)= oo; theargument
(21). Put
(39)
Dk =
EO0mj?nj,1?j<kEl?mk<nk
CmI,
,mk(k _
1);
D = liminfkDk
and observethat(28) and (25) imply
(40)
P(D) = 1.
Let Wk be the sub-algebraof Y? made up of sums of membersof Hk and put
tWk; since(in viewof(26) )
W = Uk
290
MICHAEL J. WICHURA
Witselfis a sub-algebraof SO. Let u<K>nD (resp.synD,fnD) be thetraceon
D ([4] page 19) of C<@> (resp..?y, f), and let YD denotetheBorelc-algebraofD.
In viewof (24), (27), and (39), each open subsetof D is a union,necessarily
countable, ofsetsoftheformCnD withC Ec ; itfollowsthat YD C cr<KrD> = C<KC>nD.
Since Dk belongsto t7k, we have D E c<W>,and since([5] page 5) YD = fnD, we
have
(42)
De
ryrDcrD=
/DCCU<W>.
In viewof (41) and theadditivity
of P and Py,theconditionk(y)= oo implies(see
(30) and (21)) that for each Ce-W the inequalities0 < Py(C) + (Py(C) - P(C))
(wk/(l - Wk)) ? 1 hold forarbitrarily
largevaluesofk; sincelimk,
O k/(l - wk) = 00
it followsthatPy and P coincideover W,henceovero<W>, and hence,in viewof
(42), over . r)D. But then,in viewof (40), we have PY(J) = P(D) = 1, so thatP
and Pycoincideover.y Thiscompletestheproofofthelemma. ]
Now putA,, = 0 and set(confer(6) and (24))
(43)
Uk=
nY:k(y)>k{d(Xy, X)
?k(y)}
The Ukneednotbelongto ./ in general,althoughtheywillifF is countableand S
is separable(so thatd(Xy,X) is d1-measurable
(confer[5] page 6)). For any subset
QOof Q, let v*(Qf)= inf{v(A): Q. a A e .} denote the outer probabilityof QO
underv.
LEMMA4. In thepresentcontext,
overeach
XY-*X uniformly
(44)
Uk
Iimk, V*(Uk)= I
(45)
PROOF. We get (44) from(24), (31), and (43). For (45) put Ek = infm,kDm(1 <
k < oo),whereDm is definedby (39). Suppose that UkcA Ed/. Then thereexists
([4] page 81) a countablesubset1A of f suchthatA dependsonlyon X and theX,
withy [A; it followsthat
er
ny
e
rA;
k(y)> k {d(Xy,
X) < Ak(y)}C A
(theseton theleftneed notbelongto d/). Thus forj < k we have (confer(34) and
(27))
vj(A) = fsvj,s(An {X = s})P(ds)
>
_V__S(nY6rA;k(y)?k{d(XY,S) ? Ak}(y)P(ds)
= P(Ek).
By (35), v(A) > Zj<kWjVj(A) >? OkP(Ek); it followsthatv*(Uk)> O)kP(Ek).But (28)
impliesP(Ek) _ 1 -LmEkEm; (45) nowfollowsfrom(9) and (25).
We notethatthe Uk increasewithk. In viewof Lemmas3 and 4, to completethe
to establish
proofofTheorem2 itsuffices
ALMOST UNIFORMLY CONVERGENT RANDOM VARIABLES
291
LEMMA5. Let (Q, .4,v) be anyprobability
space and let (Uk)k>1 be an increasing
sequenceof subsetsof Q of outerprobabilitiesv*(Uk). Let X be the a-algebra
generatedbysl and theUk(l < k < oo). Thenv maybeprolongedto a probability
It
on(Q, .) suchthat
(46)
JU(Uk)
foreachk.
PROOF. Put
Bk*E sl
Bk
=
Uk-
Uk-
such that v(Bk *)
=
(1
=
V*(Uk)
? k < oo), put Boo= (supkUk)c, and choose
v*(Bk)(1
<
k ? oo). According to [4] page 43, %
coincideswiththe class of sets of the formEkAkBk, whereAkeC-;
formula
(47)
I(Yk
Ak
Bk)
=
Ek fAkfk
moreoverthe
dv
definesa probabilityju on (Q, 9), whose restriction
to (Q, ) is v, providedthat
each fk is a nonnegative,d-measurable random variable vanishingoff of
Bk*(l < k? oo) and thatEkfk = 1.
Let fk be the indicator function of Bk*
(47). Then
(48)
,u(Uk)
=
hi(Zj_kBj)
U
= YjkJffjdv
<kBj *(
=
<k ? oo) and define[t by
v(UjikBj*).
Since U j kBj* is an dW-measurable
setcontainingUk,we have
(49)
v(Uj <k Bj*) > v*(Uk).
On the otherhand, suppose UkcA E X, so thatBjc A forj < k. Then each Bj*,
and hencealso U .<kB*, is containedin A up to a v-equivalence.It followsthat
v(A) ? v(Ui<kBi*) andthat
(50)
V*(Uk) > V(Uj_kBj*)
Together(48), (49), and (50) imply(46). E]
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