A NNALES DE L ’I. H. P., SECTION B
W ANSOO T. R HEE
On the distribution of the norm for a gaussian measure
Annales de l’I. H. P., section B, tome 20, n
o3 (1984), p. 277-286
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On the distribution of the
normfor
agaussian
measureWanSoo T. RHEE Professor WanSoo T. Rhee,
Faculty of Management Sciences, The Ohio State University, Columbus, Ohio 43210, U. S. A.
Annales de l’lnstitut Henri Poin0ncaré,
Vol. 20, n° 3, 1984, p. 277-286. Probabilites et Statistiques
SUMMARY. - Let E be an infinite dimensional Banach space with
norm [ [
Then for each ~ >0,
there exists a norm N which is(1
+s)- equivalent
and a centered Gaussian measure p on E such that the distributionof N(’) for p
has an unboundeddensity
withrespect
toLebesgue
measure.
RESUME. - Soit E un espace de Banach de dimension infinie avec la
norme ~..11. Alors,
pourchaque
s >0,
il y a une norme Nqui
est( 1
+8)- equivalente à 1B . II,
et une mesuregaussienne centrée
sur E telle que la distribution deN( . )
pour ait une densite nonbornée
parrapport
a lamesure de
Lebesgue.
1. INTRODUCTION
Consider an infinite dimensional Banach space,
and
a centered Gaus- sian measure onE,
that is a Radon measure on E such that for each x* E E*the law of x* is Normal centered. For t E
R +,
let EE; 1B
t~,
and = The function has remarkable
properties.
Letgiven by 0(M) == exp A remarkable recent result
Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques - Vol. 20 0246-0203 84/03/277/10/$ 3,00/(0 Gauthier-Villars
’
of A. Ehrhard
[1 ]
asserts that ’ = is concave. It follows that foreach to
>0,
there is a constantCo
suchthat ~(t)-~F(~ ~ Co(t - u)
for t,
u >_
to. It followsthat |03C6(t)-03C6(u)| ~ Co(t - u) for t, u >_
to since 03A6 islipschutz
of constant This shows that the distributionof 11.B1 (
has a bounded
density
withrespect
toLebesgue
measure on each inter- val[to,
oo[. (M. Talagrand recently
showed that thisdensity
is conti-nuous
[8 ].)
Let us consider theproblem
whether thisdensity
is boundedon
[0,oo[,
that is whether there is C such that for0 u _
t, we haveIt has been shown
by
J. Kuelbs and T. Kurtz[3] ]
that condition(*)
holds for each
gaussian
,u when E =l2(N) provided
with the usual norm.These results have been
considerably generalized by
the author and M. Tala-grand,
who showed that it isenough
to assume the norm of E isuniformly
convex and that the modulus of uniform
convexity
is of powertype (that
is
>
~p for some p and 8 >0).
In the
opposite direction,
it has been shownindependently by
V. Pau-laskas and
by
the author and M.Talagrand
that condition(*)
fails in gene- ral[5 ].
A furtherexample by
the author and M.Talagrand
exhibits a cøoorenorming
of0.(N),
such that all the differentials of the norm remain boundedon the unit
sphere,
and still condition(*)
fails for thisrenorming [~ ].
Closely
connected to condition(*)
is theproblem
of the rate of conver-gence in the central limit theorem
(C.
L.T.).
If X is an E-valued r. v. with zeroexpectation
and moments of order2,
we say that X ispregaussian
if thereexists a
gaussian
measure on E with the same covariance asS,
that isIf are i. i. d.
copies
ofX,
the rate of convergence in the C. L. T. is often estimatedby
J. Kuelbs and T. Kutz showed that if condition
(*)
holds and theI
is three times differentiable with these differentials bounded on the unit
sphere,
thenLln
=o(n-1~6)
ifX has a third moment. F. Gotze[7] ] reduced
this bound to the best estimate under
slightly stronger
conditions.For a >
1,
a linearisomorphism
T from E to F is called an a-isomor-phism
if for x E E wehave ~x~/03B1 ~ ~ T(x) ~~03B1~x ~.
We say that E and FAnnales de l’lnstitut Henri Poincaré - Probabilités et Statistiques
279 ON THE DISTRIBUTION OF THE NORM FOR A GAUSSIAN MEASURE
are
a-isomorphic
if there exists ana-isomorphism
between E and F. Wesay that two
N( . )
on E are(x-equivalent
if theidentity
is an a-iso-morphism
from to(E, N( . )).
THEOREM. - Let
.11)
be an infinite dimensional Banach space.Let a > 0 and
(çn)
be a sequenceconverging
to zero. Then there exists a normN( -)
on E and an E valued r. v. X such thata) N( . )
is( 1
+~-equivalent to ]] . ]] , b)
X is bounded andpregaussian,
c) if J1
is thegaussian
measure on E with the same covariance asX, J1
failscondition
(*)
for the normN( . ), d )
theinequality
holds for
infinitely
many n.2. SOME TOOLS
Let l2
be the n dimensional Hilbert space, and be the canonical basis. Let y~ be thegaussian
measureon l2
such that the dual functionalsei*
are
independent
and standardnomally
distributed. Thefollowing
obser-vations are crucial.
OBSERVATION 1. - Since the variable
(ei*)2
areequidistributed
inde-pendent
ofexpectation
1 and variance3,
the one-dimensional C. L. T. asserts that the distributionof ~ x~2 = (xi(x))2
is close toN(n,
Inparticular
i-n
1.
Notice also
that ~~
x = n.OBSERVATION 2. - Let
Yn
be a r. v. valuedin 12
such that for ie { 1, 2, 3, ... , n ~ 1,
1},
it takes the value withprobability 1/2n.
Let
(Y~)
be i. i. d. likeYn. If q
is much smallerthan n,
withprobability
closeto
1,
the r. v.Sn,q = q-1~2 y~
takes values of thetype
ieI aiei whereVol. 20, n° 3-1984.
card I = q
= so ~Sn,g~
= in this case. Sofor q fixed,
We shall also make essential use of the
following
Banach space result.THEOREM 1. - Let E be an infinite dimensional Banach space, and F be a finite dimensional
subspace
ofE, r
> 1 and n E N. Then there is an n-dimensionalsubspace
G of E ofdimension n,
that isi-isomorphic to ln2
and such that for x E
G,
y E F wei II x + y ~.
We shall need the
following
version of Dvoretzski’s theorem : Given a >1, and p ~ N,
there is a numberq(p, a)
such that any finite dimensional Banach space H of dimension >q(p, a)
contains asubspace a-isomorphic
to1~ [4 ].
Let H be a
complement
of F. Let a= -rl/4.
We canassume n >_
1 + dim F.Let
G1
be asubspace
of H that isa-isomorphic to 1’2
withq = q(2n, a).
On
G1
consider thenorm ~ .111 given
= x + yII ;
y EF }.
Dvoretzski’s theorem
gives
asubspace G2
ofGl
such that(G2, I I !!i)
isa-isomorphic
to1)~.
Let
Ti (resp. T~)
be ana-isomorphism
from]] ) (resp.
to
l2n
and let T =Tl 1).
Thequadratic
formQ(x) _ ~ ~
onl2~
can be
diagonalized
in an orthonormal We can assumethe
eigenvalues
are such that~ ~ ~ ~ . - ~ ~2~-
For i n, there exist Ui, vi withuf
+vf
=1,
+ =~n.
Let G’ be the spacegenerated by
the vectors uiei + For x EG’,
wehave ~ T(x)~2
=03BBn I x
Let G =
T1-1(G’).
For x EG,
we haveand
similarly À~/211 x ~ ~ 1 (X211
Since dim G > dimF,
it follows from[4],
lemma 2 . 8 C that there is
x0~G with ~x0~ =
1and ~x0~1
== 1. Thisshows that
~,n ~~ _ (x~. Hence !! II X 111
for x E G.3. CONSTRUCTION
Let
03B2p
be a sequence with >1, fl j3i
1 + e.By
induction over p,we construct sets
Bp of E, integers q( p),
real numbers and r. v.Zp
suchthat the
following
conditions are satisfied.( 1) B p
is convexbalanced; B1
is the unit ball ofE ; 2,
Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques
281
ON THE DISTRIBUTION OF THE NORM FOR A GAUSSIAN MEASURE
(2) Z p
isvalued
in a finite dimensional space ; for each 2 - p and the sequence(Zp)
isindependent.
(3)
If is thegaussian
measure with the same covariance asZ p,
then(4)
If vp is andNp
the is thegaussian
gauge measure ofBp,
we with have the for samer - p,
covariance asXp = I ~ Z~,
(5)
If(X)i
are i. i. d.copies
ofXp,
forr _ p,
we have~p ~~tP)’p’
" ’
.
We
proceed
to the firststep
of the construction. We choose such thatq(1) 1/6, and 03B41
= follows from observations 1 and 2 that there exists n such that10n-l/2 5i
and thatThere exists d with
n 1 ~2
> d >n 1 ~2/2
such thatand
automatically
we haveThere is 1 a 2 such that
From Dvoretzski’s
theorem,
there is asubspace G
of E and an a-isomor-phism
Tfrom l2
to G. Letb =1/(8d)
andZl
=bT(Yn).
Vol. 20, n° 3-1984.
(2)
followsfrom )) 2bn 1 ~2 __ 1/2.
We check
(3).
Since = we haveso
(3)
holds.Let al
=1/8.
We check(4).
Since~1/b >_ 10,
we haveand hence
(4)
holds. To check(5),
we note thatso
(5)
follows from(12). Finally (6)
holdsby
construction. The firststep
iscompleted.
Let us now assume that the
first p steps
have beencompleted.
Thereexist two numbers 1 a
[3p
and b > 0 such that forr _ p
we haveWe can assume
b _ 2~P~~.
Let c =bjq(p).
Letq(p
+1)
belarge enough
that
=
~q( p + ~ ~l(p
+1).
From observations 1 and2,
there exists nwith
5p+~ ~ 2on -1~2
andAnnales de l’lnstitut Henri Poincaré - Probabilités et Statistiques
283 ON THE DISTRIBUTION OF THE NORM FOR A GAUSSIAN MEASURE
Let d with d and
Let T with
T3
=(a
+1)/2.
Let F be a finite dimensional space of E in which
Xp
is valued. We useTheorem 1 for
(E, Np).
So there is a finite dimensional space G of E andi-isomorphism
Tfrom 11
to G such that for x EG,
y E F we haveWe define
Bp+ 1
as the closed convex hull of the setFor
xeG, y E F, II [ T -1 (x) [ [ _ ~ 2, II (a - 1)/2,
we haveN p(x) 23,
so
Np(x
+y) c Np(x)
+ a.In
particular Bp
crBp+ 1
c so(1)
holds since a/3p.
Moreover for xeE we have
NpM.
We now propose the
following
fact.Fact. - For
x E G, II = i2, y G F, 11 (cc - 1)/2,
we haveNp+ 1(x
+y) =
1.We
already
know that +y)
1. There is a linearfunctional 03C61
on G such
that 03C61(x)
= 1 1when [ T-1(x’) [ [ -- z 2,
so thereis a linear
functional ~ 2
on F + G such that =1, ~2(~) ~
1 whenever1,
x’ E Gand 4>2
= 0 on F. In Ifthen i, so
’[z,
so~2(x’+y’) __
1. Thisshows that
Np( ~2) __
1. So4>z
can beextended
on Eby
with 1.Since +
y’) --
1 for x’ EG, II z2
andy’
E F andsince 4>
1on
Bp,
the definition of shows thatNp+ 1( ~) _
1. As~(x + y)
=1,
the fact isproved.
Remark. This fact motivated the choice of
Bp +
1.We set
Zp+ 1 (cv)
=(c/d )T(Y(co)).
Now(2)
follows fromAlso, (3)
follows fromVol. 20, n° 3-1984.
We now check
(4).
We first show that forr _ p,
we haveWe notice that vp + 1 is a measure on F +
G,
that identifies to vp @ r~ p + 1. LetA= ~ xEG; II
xII 16c ~.
For ZEl2, II
zII 2n1~2,
we haveII (c/d)T(z) II
16c.It follows from
(17)
and the fact that r~p+ ~ that ~p+1{A) ?
1-b.Let
For x E
A,
sinceNp+ 1{x) _
we haveNp+ 16c _
16b. For y EB,
since
Np(Y)
we haveSo,
forxEA, YEB,
wehave ~ 2014 6~
+y)
~.It follows from
(13)
thatLet ap+ 1 =
c/,r2.
To finish theproof
that(4)
holds at rank p +1,
itremains to show if
then vp+
i(H) > 2~q(p+
1)’ LetFor
since
(d/c)5p+i ~ d a p + 1 >--
10. Inparticular, ~p+i(C) ~ 1/3
from(20).
Let D
y ~ ~ c((x -1)/3 }.
Wec/6,
and we canassume
--2 4/3.
We then havec(oc2014l)/3 ~
It follows that for x E
C,
we haveNp + i(x),
so x +yEH.
Hence
from ( 15),
so(4)
holds.We now check
(5).
We first show thatfor r
p, we haveAnnales de l’lnstitut Henri Poincaré - Probabilités et Statistiques
285
ON THE DISTRIBUTION OF THE NORM FOR A GAUSSIAN MEASURE
We have seen
that!! 8 b/q( p),
so sinceXp + 1= X~ + Zp + 1,
weget
so the result follows from
(14).
To check(5),
it remains to show thatWe first note that
for x E G, y e F,
we have +y) >_
since+
Ày)
is a convex function of £ that isequal
to for small ~.Hence
and the result follows from
(19)
and the definition of Z. The construction iscomplete
since(6)
holdsby
construction.4. PROOF OF THE THEOREM
It follows from
(2)
that one can define a bounded r. v. Xby X(03C9)=
For
each 1,
letUI
be a Gaussian r. v. with the same covariance asZi,
andsuch that the sequence
(Uz)
isindependent.
It follows from(3)
that theseries
(UI)
is summable in Its sum V isGaussian,
and has the samecovariance as
X,
so X ispregaussian. Let
be the distribution of V.Let
N(x)
= limN p(x)
and8p
=fli.
From condition(1)
weget
N(x) 0pN(x).
It follows from(4)
that forq _ p, r p
weget
Since v03C1 is the distribution of
Vp
=Ul,
weget by letting
p - 00 i i pVol. 20, n° 3-1984.
and
letting ~ 2014~
oogives
It follows from
(6)
that condition(*)
fails for ~. It follows from(5)
that forLetting
p - 00gives
In
particular,
which
completes
theproof
of the theorem.ACKNOWLEDGEMENT
The author thanks Professor W. Davis of the mathematics
department
of the Ohio State
University
forintroducing
her to the lemma 2 . 8 . c of[4 ].
[1] A. EHRHARD, Symetrisation dans l’espace de Gauss, to appear in Math. Scand.
[2] F. GOTZE, Asymptotic Expansions for Bivariate Von Mises Functional, W. S ahrschein- lichkeitheorie verw. Gebiete, t. 50, 1979, p. 333-355.
[3] J. KUELBS and T. KURTZ, Berry-Esseen estimates in Hilbert space and an application
to the law of iterated logarithm. Ann. Prob., t. 2, 1974, p. 387-407.
[4] J. LINDENSTRAUSS and L. TZAFRIRI, Classical Banach spaces, vol. 1, 1977. Springer Verlag.
[5] W. RHEE and M. TALAGRAND, Uniform bounds in the Central Limit Theorem for Banach space valued Dependent Random Variables, to appear in J. Multivariate
Analysis.
[6] W. RHEE and M. TALAGRAND, Bad rates of convergence for the central limit theorem in Hilbert Space, to appear in Ann. Prob.
[7] W. RHEE and M. TALAGRAND, Uniform Convexity and the distribution of the norm for
a Gaussian measure, to appear in W. S ahrscheinlichkeitheorie verw.
[8] M. TALAGRAND, Sur l’intégrabilité des vecteurs gaussiens, to appear in W. S ahrschein- lichkeitheorie verw.
(Manuscrit reçu le 9 janvier 1984)
Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques