Lecture Notes in Mathematics Edited by A. Dold and BEckmann Subsenes Adviser
Instltut de Mathematlque, Unlverslte de Strasbourg P A Meyer
1193
Geometrical and Statistical Aspects of Probability in Banach Spaces Actes des Journees SMF de Calcul des Probabllites dans les Espaces de Banach, organlsees a Strasbourg les 19 et 20 JUIn 1985
Edited by X Fernlque, B Heinkel, M. B. Marcus and P.A. Meyer
Springer-Verlag Berlin Heidelberg New York Tokyo
Editors
Xavier Fernique Bernard Heinkel Paul-Andre Meyer Institut de Recherche Mathematique Avancee 7 rue Rene Descartes 67084 Strasbourg Cedex, France Michael B. Marcus Department of Mathematics Texas A & M University College Station, Texas 77843, USA
Mathematics Subject Classification (1980): 46820, 60B05, 60B 10, 60B 12, 60F05, 60F 15, 60F 17, 60G 15,62005, 62E20 ISBN 3-540-16487-1 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-16487-1 Spnnger-Verlag New York Heidelberg Berlin Tokyo
This work IS subject to copyright. All rights are reserved, whether the whole or part of the material IS concerned, specifically those of translalion, reprinting, re-use of Illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage In data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee IS payable to "Verwertungsgesellschaft Wort", Munich
© by Springer-Verlag Berlin Heidelberg 1986 Pnnted In Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Preface le calcul des probabilites dans les espaces de Banach est a~tuellement un sujet
en plein essor auquel des rencontres internationales sont consacrees regulierement depuis une dizaine d'annees. Les 19 et 20 juin 1985, une trentaine de specialistes de ce sujet se sont reunis
a
Strasbourg sous le patronage de la Societe Mathematique
de France, pour faire le point des developpements les plus recents, notamment en matiere de fonctions aleatoires gaussiennes, de processus empiriques et de theoremes limites pour des variables aleatoires
a valeurs
dans un espace de Banach. Les
principaux exposes de ces deux journees ont ete rediges par leurs auteurs, ce qui a permis de composer ces Actes que la Societe Springer a eu l'amabilite d'accueillir dans sa collection Lecture Notes in Mathematics. Ces deux journees ont ete assombries par la disparition, le 7 juin 1985, d'Antoine Ehrhard qui etait l'un des plus brillants representants de la jeune generation de probabilistes. Nous avons ressenti cruellement son absence, celle du mathematicien bien sur, mais surtout celle de l'homme de coeur sensible et attachant qu'il etait.
Les editeurs
Table of Contents
BORELL, C.,
A brief survey of Antoine Ehrhard's scientific work.
1
DOUKHAN, P. and LEON, J.R., Invariance principles for the empirical measure of a mixing sequence and for the local time of Markov processes. GUERRE, S., Almost exchangeable sequences in
q L ,1:S:q d/2
in the first framework for
random variables with values in a compact riemannian manifold in the continuous case of the brownian motion on
* universite
Paris-Sud U.A. CNRS 743 "Statistique Appliquee" Mathematique, Bat. 425 91405
ORSAY (France)
E becomes
s > d/2-1
E
** Universidad
Central de Venezuela Facultad de Ciencias Departamentado de Matematicas Apartado Postal nO 21201 CARACAS (Venezuela).
5
1.
I NTRODUCT ION
This work is divided in two parts.
The first one is devoted to investigate a
rate of convergence in the weak invariance principle for the empirical process {~k;
of a strictly stationary strongly mixing sequence
Xn k=0,1 , ... } valued in a
L2(E,~) of uniformly bounded 1 n functions satisfying an entropy condition: X (f) = L [f(~k)-Ef(i;k)]' f E F n /ri k=1
metric space
E indexed by a compact class
F of
The typical case is obtained for a d-dimensional E with
riemannian compact manifold
F unit ball of the Sobolev space
result can be shown only if s > d/2.
Hs of the manifold (see Gine [14]) ; a In this discrete case we expose some of the
results of [10] made in collaboration with Frederic Portal.
Rates of convergence
essentially depend on the entropy condition for F . The second part of this paper studies the asymptotic behaviour of n
ZnU) =
~ Jf f(X u ) du, fEL2(~). In 0
Here
{X t : t> O}
is a continuous parameter
recurrent ergodic stationary Markov process with values in a compact riemannian manifold E or in lRd ; ~ denotes the invariant measure of the process. We first give an invariance principle in a general framework. brownian motion on
E;
We also study the case of the
we give an invariance principle and a L.I.L. uniform on the
classe F , unit ball of Hs for s > d/2-1. We also study the case of diffusions on Finally we discretize the Zn process by L f(X) f E F. O 0 CX
and
~1]
is said to be strongly mixing, with mixing coefficient
where,
v = Sup {llP (A n B) - lP (A) lP (B) I ; A E r~ , Bn;+v,e.2. O}
r;
is the
a-field generated by
{xv; s
d/2
for discrete parameter becomes
s > d/2-1
for continuous parameter.
2. INVARIANCE PRINCIPLE FOR THE EMPIRICAL MEASURE OF A MIXING SEQUENCE OF RANDOM VARIABLES. In this section we expose some results of [10] made in collaboration with Frederic Portal ; a complete version of this work will appear elsewhere with proofs. Let
(~k)k>O
a strictly stationary sequence of strongly mixing random varia-
bles with values in a polish measured space a-field of E and
~
a non negative a-finite measure.
compact subset of L2(E,~), X (f) n The class
(E,B(E) ,~)
vie
define n 1 L
/ri
k=1
[f(~k)-Ef(~k)]
where Let
B(E) r
is the Borel
a finite entropy
fEr
r is supposed uniformly bounded, the law of
~
has a bounded o density with respect to ~ ,and there is some aE]0,1/3] such that L cx~ < k=O (here {cx k} denote the mixing coeffirients of the sequence {~k})' The process Xn is C(F) valued, where (C(F), II.IIJ is the space of contiriuous functions on the 00
00
compact set
r e'lui p!Jed with uniform norm.
7
We give an estimate of Prohorov distance of Xn to the centered gaussian process Y with covariance defined for f ,9 E F by:
g"
and
Eg(t:o ) .
A reconstruction of the process
Y gives a weak invariance principle with rate
of convergence. The method is based on estimations of central limit theorem rates in Prohorov
Xn depending on the dimension of repartitions and, from another hand, on estimations of
metric convergence given in [11] for the finite repartitions of the process
the oscillations of the THEOREM 1.
Fa
L:t
I:
satisfying
k=O
a~ E
such that
{fEF-F;
=
5/6
Remark : The main interest of this result is to escape from a compact context. In [10] and [9] invariance principles for non parametric estimators are obtained with the same methods.
Kernel and projection estimators of density and regression
function are considered.
3.
F varies with the index n.
In this case the class
L.I.L.
INVARIANCE PRINCIPLE AND
FOR THE LOCAL TIME OF CONTINUOUS
PARAMETER MARKOV PROCESSES. {X t ; t > O}
We write
a continuous parameter Markov process with values in a
complete separable metric space stationary with marginal law
E
~
This process is supposed to be homogeneous and Moreover, its infinitesimal generator
unbounded non-negative linear operator, whose domain,
D(L)
is dense in
satisfies (i)
L is self-adjoint and onto.
(ii) the spectrum of
L is discrete.
(iii) 0 is a single eigenvalue of L associated to the constant eigenfunction 1 . ? L-(~)
Under those hypotheses, the Hilbert space 11.11
(resp. (.,.)) its norm (resp. scalar product)
an orthonormal basis of
L2(~)
is separable; we write
and
{em; m = 0,1,2, ... } is
such that
00
Lf = L A (f,e)e m=O m m m Thus the spectrum of
L is
for
fED(L), here
{Am; m ~ O}
Ao = 0
0
with norm (resp. scalar product)
1/2 ),s (f,e )2 1 m=1 m m J w
lifll s =
[L
are formally described by Seeley [20] for r E [ 2 Gs / 2 and the domain of Ls / . We also write H_ s it is an Hil bert space with the norm 11.II_ s
IITII_ s = Sup {IT(f)1 ; We write
for
< w}
AS (f,e )2
is the range of
the dual space of
H s
defined by :
11·ll s (resp. (.,.)s)
Note that
> O}
l-
with range
is the identity operator on
t
(Rosenblatt [21]) .
a thus : Gf = L: Am m=1
{X+c
fEH s '
Ilfll s=1} =
the unit closed ball of
C~1 A~s
(T(e m))2f/2
H . s
The aim of this work is the study of asymptotic behaviour of the functionnal
In
Z (f) = -1
n
In
L2 (fJ)
It is defined on an
H-s
f(X ) du
0
u
because
valued random variable for 00
(*)
L:
m=1
A-(1+s) m 00
Indeed,
EIIZnll: s
= E
L:
m=1
s
a
verifying:
CIn r
n
00
-s Am
J0
2 e
(X ) m u dU)
11
r
00
A- s (1 - n~) (p u em' em) du m=1 m 0 -A u n ~) e m du 2 I: A-ms r (1 E II zn II 2-s n Jo m=1 -nA 2A-(2+s) (1-e m) I: E II zn II 2-s = 2 I: A-(1+s) n m=1 m m=1 m E II Zn II 2-s
2
I:
00
00
00
This calculus
is analogous to those of Baxter and Brosamler ([4] , theorem
(4.3)) .
We now define a gaussian random vector
Z on
H
-s
For this we assume the
following technical hypothesis: (H)
\ There is a uniform random variable on the interval
~ probability spRce (0,A, We consider an i.i.d.
~)
(~ ) '0
sequence
with variance 1 and we define, for Z(f) =
{X t ; t
that
m m/
~
O}
[0,1] defined on the same and independent of it.
of gaussian centered random variables
s satisfying (*)
-1/2 (f, e ) t: m ' f E Bs Am m m=1 2 I: A-(1+s) thus ZEH -s a. s. m=1 m
/2
00
I:
00
E II ZII :s
Note that
An abstract construction of Z can be made using the operator T = 2 G1+s 2 ~ A~(1+S) m=1 hypothesis (*), satisfying:
whose trace is
I
H_ s
1}
-
converges in distribution to
(iii) and (*) •
Z
in
C(B s ) ,
12
Proof.
Under the former hypotheses, Battacharya shows convergence of finite repar-
titions ([3], remark 2.1.1.).
{Z n ; n -> 1} result from flattly concentrated property using De Acosta's method [1] . Let
FmcH -s
The tightness of the sequence
be the m-dimensional space defined by ;
F = {T E H ; T e = 0 for m -s k FE the m
We note
k = 0 and
E-vicinity of Fm in 00
M+ m
Note that
will
= {T E H-s
; T e
0
=0
k > m}
H-s and -s
L A k=m+1 k
M~ cF~ and E Z~ (e k) < 2/A k ' so Bienayme- Tchebi~ev inequal ity
implies: lP (Z n
t.
FE) > lP (Z n m-
t.
lP (Z n
t.
lP (Z n
t.
EE) > 1 - lP ( L A-s Zn2 m\k=m+1 k A- s E FE) > 1 - E-2 L mk=m+1 k
ME) m 00
(e) k
~
E2)
00
Z2 (e ) n k
00
L
k=m+1 Thus the sequence
Zn is flattly concentrated
COROLLARY 5. The sequence of real random variables tion to
IIZII: s '
X2
infinite sum of weighted
theorem 4 follows. 2
II Zn II -s
converges in distribu-
random variables.
: Using the direct construction of Z, note that: liZ II 2-s
B~~~r~
(t:m)m>O
2
00
-(1+s) 2 t:m
L Am
m=1
being an i.i.d. sequence of normal random variables.
In view to investigate iterated logarithm behaviour of construction of the brownian process
Zt
Levy's construction ([16], 1.5, p. 19).
with base
M, law of
Zn' we now make a direct Z
We use the
Let
2 {Xn,k ; n = 0,1, ..• , k odd, k = 1, ... ,2 n-1} the Haar basis of L [0,1] defined by 'V 2- n/2 for k i 2nt i k+1, = 0 else, and Z k an i.i.d. array of reali'V n, N t 'V r X k(u) du . zations of Z defined on (~,A, lP), we write ZN(t) = L L Z n=O k n,k Jo n, This serie converges normally a.s. Note, for this, that:
13
From another hand Fernique ([12], theorem 1.3.2.) shows that there is an 2 / a 2) < such th a t : A = E exp (11z11 -s
00
Th us
•
a >0 lP (EN ~ a 2-(N+1)/2 V I,n(22N)) _< A 2- N. k_ 'V
Borel-Cantelli lemma implies then the continuity of the limit quence
of the se-
The limiting covariance is computed as
ZN(t) 'V
'V
UAV
00
E (Z(u),U)_s = (Z(V),V)_s = L n=O
L
k odd O O} is the brownian process with basis M;
it belongs to C (R+,H -s ) under condition (*). PROPOSITION 6.
The a.s.
compact subset
/2 B1
~rQQf.
cluster set of
H_ s (B 1
of
We see, like Bolthausen ([5])
Z(t) (2 t £n[£n t])-1/2 for {fEL 2(fJ); IIfl11 ~ 1}) under that this set is
t ~
00
is the
condition
(*).
/f (B s ) = /2 B1 using the
results of Kuelbs and Lepage ([18]) . 3.1. Brownian motion on a compact riemannian manifold. The space {X t ; t > O} LEMMA
7.
Remark
E is here a d-dimensional compact riemannian manifold and
is the brownian motion on
(i).
For
(ii).
Vo, S
S > d/2, > 0,
For s = d/2
------
For s > d/2-1 Proof.
where
E.
fJ
is the uniform measure on
E.
is a bounded r.v.
s > d/2 - 2/(2+0)=> E II Z1 11 :;0
0 verifying Iii).
is the Laplace operator of the Riemann The eigenvalues of L , (Am)m>O satisfy Am 'V c m2/d ([ 19]) for m.-+ 00,
Note first that
manifold
IIZ111:s
E and
L=-
c > 0 is some constant.
(1"
where
(I,
From another hand, Gine ([14]) shows:
14
VS
(;)
> d/2
II Z1 11
00
:S
II Z1 11 :s (i ;)
3C L
m=1
~
h, k
116xll:s~C
, VX E E
A- s m
[fo em
2 (X )
u dU]
1 1 2 2 1 + L A-ms r em (X u) du = r 116 x II -s du < C J J u m=1 o o 00
fr1 = E[ r ~ A-ms \J em Jm= 1 o
E Ilz111:;
Let
>0
(X)
du
)2 11+6/2 J
> 0 satisfying h+k = s and p = (2+6)/6, q E II z II 2+ 6< E A B 1 -s -
inequality implies
A=
1 [~ A~hp (\Jr em (X) u o m=1
B
1 A-kp (Jr e (X) dU)\2 m=1 m o m u
where
\2 6/2 du \ 1 ) J
because
d/2, and
~
EB
Z(k) 1
(2+6)/2, Holder
L
m=1
zi 1 )
[r
= Z1'
Moreover, Baxter and Brosamler ([4])
show that this sequence is
¢-mixing (with ¢n ~ c an ,0 < a < 1). Thus the following theorem 8 will result from lemma 7 and : THEOREM A.
(Dehling, Philipp [s]).
Let
{x
V
, v> 1} -
a strictly stationary sequence
of random variables with values in a separable Hilbert space tation and having a strongly mixing with
(2+6) CJ.
n
-order fini te moment
=
(0 < 6 < 1). for
together with a brownian motion
{Z\t) ; t > O}
n~
= 0 (It £n ( £n t)
a.s.
centered at expec-
If the sequence is
it can be reconstructed
with covariance
probability space such that :
II L x V - X( t) II v d/2-1 ,
Let
S
compact in
H
and its a.s. cluster s",t for
-s
0 E0
There is some
o
n
is
-'1
=
and
lR
f E Hs which is another conseXv = z;v) (f). Those random variable
E Iz;k) (f)1 2+0 ~ Ilfll ;+0 E Ilz;k) II
with same law that
(Z;k) (f))k>1
L Uk - W(t) a = 0 (It £n(£n t) ) a.s., k 0 under assumption Am i.
k->oo
k
3R
>0
o < Inf
{V(x)
> R} < Sup
Ixl
ii.
VX E lR
IDa V(x) I iii.
3C
COROLLARY 9. assumptions
s
> d/2
+
If
Am
dim -
>0 ~
Ixl
< Ca
+
m
; Ixl
> R}
R => x.V(x) ~ C Ix 1m V
is equivalent to Lebesgue measure,
m> 0
for some
and 13 =
2 ¢
-1
2
= ¢-1
6¢
4
17¢ E L (~) n L1oc (lR
Zn
implies that the sequence
Conditions
(1
-
{V(x) Ixl-
which
d
satisfies
),
then
converges in distribution to
Am are satisfied by homogeneous polynomials
Z
in
V with degree
m•
V(x) = c Ixl 2 for some constant
A multidimensional Ornstein-Uhlenbeck (i.e. c
> 0)
satisfies hypotheses of Corollary 9
if
s
> d-1
.
We do not get here an iterated logarithm law (L.I.L.) because the lemma of the former section is no longer valid.
To avoid this problem and obtain a uniform L.I.L.
we now reduce the class of functions used. From here we suppose ¢ bounded, thus L2 (lR d ) cL 2(~); the Sobolev space of L2 (lR d ) constructed with tensor products of Hermite functions is denoted by
* Hand
The process
s
Z1
is a bounded random variable with values in
Here and
h is the m-th normalized Hermite function. With the help of the estimate m II hmII = 0 (m-1 I 12) of Szego ([23]) we see, summing by parts on spheres of 00
that: liZ 11*2 < C d 1- s 1 -s-
~
d- s -7/6 •
p
p=1 The strongly mixing property of the
logous method than for theorem 8,
{X t
t
> O} process implies, by an ana-
18
> d - 1/6 ,
S
For
THEOREM 10.
there is some
In
lim [(Zn(f) (w) n-+oo
II· II
The Use of the norm
Proof : Note that
J IfI2+8d~
d/2 and if {X t ; t ~ O} is a diffusion on a compact riemannian d-dimensional manifold this condition is realized. Suppose here that
The discretization of the process
Z
n
is, for 6
>0
Z (f)".l n,6
In
For 6
> 0 fixed, we see that [8] implies Zn,6 converge in distribution to a
gaussian process
Z6
such that -v 1/2
if
k(v)
1 e (-+-) 1-e -v
19
With the help of an i.i.d. sequence of normal realizations explicit constructions of Z and Z6 00 Z= L E:m em m m=1 00 Thus EIIZ 6-ZII_2s = 6 L Ik(A m6) m=1
4
(~)m>1
we give
00 Z6 = IS. L kO'm 6 ) E: em m m=1 An6 2 A- s 2..1 A-s < 6 L {2 A (T)} Am6· m m m=1
;;r.
00
For a diffusion on a compact riemannian manifold: 2 0, if s>d/2 EIIZ-ZI1 6 -s Then the Prohorov and Levy's distances of those gaussian random variables on Hare 0(6 0/3) for 6 -> 0, 0 = s - d/2+1 -s From the other hand, a precise analysis of the results of [11] shows that Dudley distance
d3 can be estimate :
d3 (lP Z ' lP Z,) = 0 (6- 11 / 4 n- 1/ 4 ) for n->oo , 6-> 0 if s > d/2. n,6 Ll Thus the discretized process Zn,6(n) converge in distribution to Z for 6 (n) = 0 (n- 1/ 11 ) when n->00 If, moreover 6(n) = 0 (n -1/(40+11)) then ' lP ) - 0 (n- 0/(40+11)) d (lP 3 Zn,6(n) ZFor great values of
s this speed is approximately
n- 1/ 4
20
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ALMOST EXCHANGEABLE SEQUENCES q L ,1';q 2 , every weakly null sequence in
q L :
- either is isomorphic to the unit vector basis of
of
t
2
- or has a subsequence which is almost equivalent to the unit vector basis q t •
Dacunha-Castelle For
D. Aldous
[4J. i-symmetric subspaces of
1,; q < 2 [lJ. (Case
[10J.
Every infinite dimensional subspace of is isomorphic to
for some
More precisely: If there exists (Yn)nE E
p E [l,qJ
on
are means of Orlicz spaces.
q = 1).
J.L. Krivine and B. Maurey
tP
q L
q L , 1,; q < +
00
,
has a subspace which
p
(Xn)nE fl
is a weakly null sequence in
such that for all
e> 0
q L , l,;q 2 , every weakly null sequence of
q L
has an almost symmetric
subsequence. If of
q 1,; q< 2 , every sequence in L , which is equivalent to the unit vector basis 2 t has an almost symmetric subsequence.
The case
q
1
The case
q E 2E
is due to H.P. Rosenthal. is proved in
The proof of the general case
[ 8J • [6J
uses the theory of stability
[10J
: in stable
spaces, there is a natural way to find almost symmetric sequences. First, recall a
24
few definitions : A Banach
x
space
x
in
and two ultrafilters
lim Ilx + y II = lim m,'!' n m m,'!' The~
defined by
0
V x E X
For
(O!
,rn
V x E
0(X)
E R
2
X
lim n,1/
(xn)nE E
lim n,'/)
*
0
ST(X) = lim
is defined by
+
n
y
1/
and
on
'!'
and
Ii •
m
is a function from
X to
R+
such that
Ilx + x 11 n
n,11 where
and
Ilx
, we define the ~
0!0
(x ) c.,,, n n,- .1." , we have :
is stable if given two bounded sequences
0!0*
ST
by :
lim m,'!'
(xn)nE 11
by
71 and
and
and
V •
Ibll = 0(0) = lim Il x II •
Let
n
n,V The spreading model R ON)
r 3J
def ined by
(x ) nEE n
and
'I'
is the comp let ion of
under the norm k
Ii I: i=l
.e .11
o!. ~
lim n.
~
~
«xn)nE E
is supposed to have no
convergent subsequences).
In a stable space, every spreading model is
i-symmetric. The proof of theorem 1
uses a sufficient condition (S.C.) in stable spaces for a sequence
(xn)nE E
to
have a subsequence which is almost equivalent to the fundamental sequence of its spreading model : Let
(S.C.)
0
(xn)nE ~
be the type defined by
on a stable space
X • If
*
O! k0 and IITII,; 1} K (0) = [T /3: k E:N , 3: O! l' .. • ,O! k E Rk suc h th a t T --O! 10 ~,. ... 1 is relatively compact for the uniform convergence on bounded sets of X , then
(xn)nE 11 by
has a subsequence which is almost equivalent to the spreading model defined
(xn)nE:N
and thus almost symmetric.
This condition was used by J.L. Krivine and B. Maurey in the case of written in
[6J. Its proof uses Ascoli's theorem.
Let now
(Xn)nE ~
q
L
be a weakly null sequence in
equivalent to the unit vector basis of
.e
2
that
(Xn)nE 11
type
0
verifies (S.C.)
defined by
can show that
0
2
L (0 X [0,+ oo[
(Xn)nE 11
00
q > 2
which is this is the
[9J). It is shown in
[6J
suppose for simplicity that 1~ q< 2 and that the q is symmetric ri.e.: V xE L , 0(X)=0(-X)J • One
n
is entirely determined by 0 11 and a function Tf belonging to dt dP @ ----=t1) which is the weak limit in that space of t
Xn)nE E
1 ~ q < +
(even in the case
only case to consider because of Kadec-Pelczynski's result
(1- cos t
.eP-types and is
q
' by the formula:
q V X E L , +00
where
K
q
I
'0
(1 -
25
ux = 1 - cos t
X
and
is the inner product in
Moreover, this representation has the following property q (] ~ (] unifonnly on bounded sets of L
n
(]
u
n-+ CO
n
in
II(] 1\ n
) Ilcrll
n
-+
+00
To prove theorem 1, it is thus sufficient to show that if 1"
belongs to Lq(dP
@
t
«(])
K
1 d11) • As q
then
1"
=O'n(] n 1
*
1,
n Ct
k
cr n
(u n)nE N
has a subsequence which converges in 2 is equivalent to Q , one can show that:
(Xn)nE N
such that :
whe re
lim
€ (w, t) = 0 a •e • 0 On the other hand, we know that k n t~
0' ,(]
n
1"
1- U
n
n
=
(1 -
u ~ )
[6J
i=l We deduce from t\ese two facts that n 2 - I: (O'~t) A(w) k 1" n i=l ~ U n 2 1 - e + I: (O'~) ~ i=l where
S
n
=
[10' ~ I,
Sup
~
n
n
I: (0',) ~ i=l S
2
(w, S t) n
l,;i,;k}.
Taking a subsequence of k
€
n
(Xn)nE N
'
we can suppose
_ 0 ' >0
TI-+CO
___
~>
0 •
n
2
Then
converges a.e. to
~
1- e -at A(w)
1"
gence theorem
(U n)
2
L (cP 0
d11) • This implies that q q is relatively compact for the unifonn convergence on bounded sets of L nE IN
converges also in
and by Lebesgue's dominated conver-
t
K «(]) 1 prove s theo rem 1 by (S. C.) in tha t ca se.
and
This theorem does not give any answer to problems 1 or 2 for weakly null sequenq ces of L 1,; q < 2 which are not equivalent to the unit vector basis of £2. The following result gives a negative answer to problem 2 in that case : THEOREM 2. Let (i)
1,; q < p< 2
(Xn)nE lli
There exists a bounded sequence in
is equivalent to the unit vector basis of
such that
26
has no almost exchangeable subsequence after any change of
(Xn)nE f>l
(ii)
density. If
(Xn)nE f>l K «(J) 1 q for the uniform convergence on bounded sets of L • (iii)
(J
is the type defined by
Remark: Property (iii)
gives a hope that
(Xn)nE ~
is not relatively compact
has no almost symmetric subse-
quence but this question is still open. In fact, the two natural ways to find almost symmetric subsequences in
Lp-spaces (namely the theory of probability with almost
exchangeable subsequences and the stability of those spaces) do not work for this (Xn)nE ~
sequence
of proof
Sketch
[7J.
-I Let where
U(w, t)
l-e
X(u,w)
1
+00 (l-costu) X(u,w) 0
if
Sk (w) fN
du u 1'+1
u E [O,l/NJ u E [N2k-l,N2k+1J
if
is a fixed positive constant and
(Sk)k E 11
a sequence of i,i,d. random varia-
' k E~) = 1) = P( U', P (t) = sup( 1/2 ( Ilx+tYII + IIx-tyll) - 1 , Ilxll = IIYII = 1 ) ,
p:
30
satisfies: p (t)
"c
tP
C being a positive constant. It is well known that the norm II II is differentiable away from the origin
let's de-
note by D the derivative of II II.
If now one as sociates to D the following fonction F p l 'l xt-O F(x) :: Ilxll - D(xl Ilxll) and:
B
~
B'
F( 0):: :)
one can check that F has the following two properties [ 19J ( i)
IIF(x)II
B
,
::
Ilxlltl
~ C
> 0: 'l (x. y) E B 2 . Ilx+yllP - Ilxil P I" p F(x)(y) I + C liYllP P P These two properties are crucial for reducing the infinite dimensional SLLN to a (ii)
I
scalar one. Other geometrical properties of a p-uniformly smooth space that we will use are its reflexivity [5 J and the fact that it is of type p [15
J.
Usually the SLLN problem in (B. II II) is stated in the following way "Let (X ) be a sequence of independent. centered. B-valued r. v. and denote by k Sn ::: Xl +... +X the associated partial sums; under what hypotheses does n (S
In) converge a. s. to 0 ?
n
(1)"
This point of view is very restrictive because two other asymptotic behaviours of the sequence (S
p( sup II S n
n
n
In) are worth of interest:
Inll
0: 486 K' (2p
'In;;'2, 'lj EI(n),
where of course C
p
+ 2C p ) " 1/8 and :
n Ilx.II"K'2 /Logn a.s., J denotes the constant involved in the fundamental inequality
recalled in Section 1. ii )
I:
exp ( _4K ,2 / lI.(n))
0 :
I: p( II T II >t) n:2: 1 n
Zx ) s; p(
I: Q'k > x )
In a first step we will bound p( I: Q'k > x ).
+ p(
k
I: 13
k
> x ) .
I 3'k_l)
,
I 3'k - 1
) •
33
In order to do this we notice for beginning that ( 2 : ~ , 3'. ) . n l";k,,j J I"J"Z is a martingale to which we plan to apply the following comparison result of Dubins and Freedman [3 J
:
LEMMA 1 : Let (Sk ' 3'k) be a real valued martingale and denote by (Y k ) its increments. 1£ one defines for every k := 1, .•• , n : Vk
=:
E ( Y
~
I 3'k-l)
V a>O. '1b>O,
- with 3' 0 := ( (/J , 0)
P(3:j:=I, ... ,n:(Y
I
- then:
+... +Y ); 0, such that: sup sup n n l"k"Z
(E liT _zk I18 (p-l)) 1 /4
"
L.
34
Hence: E I: E
(a~
I d k _1 )
"
3 4
8p2 lI.a /
L
..,
ac"p
+
K'
2
a/Log n
and so by assumption cl , one has for n large enough: E I: E (a 2 d ) s: a 3 / 4 k k-l
I
Applying now Lemma 1 for a ::: a - I /2 and b ::: 1, one obtains: p( I: a
>x)" p( I: a
k
k
+ p(
I: a
1 2 " a /
+
> x, I: a
> x, I: a
k
1 (a /4 /(
1 2 a- / I: E (
:2:
k
k
a~
< a-I /2 I: E (
x-l/ ) "
I d k _1 ) + 1 a~ I d k _ 1 ) + 1
)
1 4
2 a /
From the inequalities: a
"p(
sup 1 "k" 2 n
II
z.
I:
1"j" k
II> y/3) "2 p( II
I: ZJ'
II >y/3 )
J
an application of Hoffmann-J~rgensen'sLemma ( [14 J Lemma 4.4 ) gives that for every y
p(
~
a
k
:2: 81 : > x)" 2 (2)3/4 p
1 2 / ( II I: Zk
I> y/9)
"4 (2)3/4 P( II I: Zk II > y/27)
"
32 p
2
( II I: Zk II > y/81 ).
Now. fix Y:2: sup ( 486. 243 c), such that: 'l n E f'-I ,
32 P( II I: Zk II > y /81 ) "
1/2
- such a choice is of course possible by hypothesis a) -
; y being fixed, we put:
x:::y/243. For such a couple (x, y) one has:
p( II T
liP - E
II T
liP
> 2x) "
(6)
2 p( I: 13 k > x )
In the next step of the proof we will bound the right-hand side of (6) by using another martingale result, also due to Dubins and Freedman [3J
:
LEMMA 2 : Let (Sk ' d k ) l"k"n be a real valued martingale and denote by (Y k ) increments. Suppose that: 'lk:::l, •.. , n Then:
lykl"l
i!2
a.s . .
'l u E R . 'l 'A > 0 , 'l v > 0
'A(u+ Y E ch (
+ ... +Y) n " ch ('Au/v) v + VI + .. ,+V n 1
where: 'l k::: 1, .... n, e(x) ::: exp x - I
V
k
::: E (
Y~
I d k _1)
exp (v e('A/v) ) ,
and:
- x
For applying this lemma to our situation, we first notice that by the same computations as for the r. v. a
k
one has:
35
I 13k I
I F(T-Zk)(Zk)I B I
,;:p E (
k
+ P E (
I F(T-Zk)(Zk)
If for every k = 1 •...• 2
Y
= (yI-p Log n
k
n
I
Bk
l3'k) + C
I I 3 xyI-p Log n n k
Remembering now that p E
J 1.2 J
and also the assumption i) made on K I. one ob-
tains : an ';:P( (I/v+V + .•. +V ) I 2n
LY
k
> 16Logn/(v+4Logn)).
Applying now Lemma 2 for A = v = 4 Log n. one gets: A an ,;: exp 4Log n I ch(8Log n) ,;: 2 n . So for the integers n -4 u ,;: 4 n
:l:
nO(x) and such that Log n ,;: K 1
2
I 11 • one has:
n
Now we go to the second case. Second case: Log n > K'
2
I 11 •
It is easy to see that the relation (6) remains true - for the same couples (x. y) -
and that the only thing to do is to apply Lemma 2 in a different manner.
If one defines: If k -1, n ••• ,2.
Yk --
one has: n Ifk=I .... ,2 .
a. s.
Furthermore: V + .•. +V I 2n So u
n
s;
K,2(8p2+8C2(K,2 ILog n)) I 11 2 (2p+2C)2 p
p
,;: 4 K ,2 I 11
can be bounded by :
01
v+v + ... +V ) L 13k > xK ' yI-p I (2p+2C ) (IIv+ 4K,2) I p 2n 2 Now we apply Lemma 2 for A = v = 4K 1 I 11 and we obtain: 2 p(
36 u
n
,,2 exp (4K,2
111.) I
ch (8K,2
111.)
,,4 exp _ (4K,2
I
11.) •
Putting together the results obtained in the two cases, one gets 'In :2: N(x), u " 4 ( n- 4 + exp _ (4K,2 I 11.) ). n
Hence: I:
u
n :2: 1
0,
I:
exp (-
k
I
E
11
2p
i\.(n) )
.... o in
0 a. s.
REMARK: It is easy to see that b ' ) holds for instance if : n - 2p L n I: II X 11 2 p .... 0 in probability. k 2 1 ~"n PROOF: By a classical argument [17
J
it is sufficient to consider the symmetrical
case. By symmetrization ( see [10 J proof of Lemma 1 ) b') implies: b ll )
lim 2-2np Log n n .... + ~
E II X
I:
k E I(n)
k
fP
:: O.
The ideas of the proof of Theorem 3 are the same as those used for proving Theorem 1 ; so we will keep the same notations as in the proof of Theorem 1 and we will only detail what needs to be detailed. First one notices [1 EIITnllp ....
J
that from a') it follows that: E II Sn
In
liP .... 0 , and also:
o.
So the the conclusion of Theorem 3 will be true if we show: Vx >
I:
0,
n :2: 1
p( II T
n
liP - E II T
n
liP> ?,x)
< + dl •
Without loss of generality it suffices to check (8) only for x E
(8 )
J 0,
243- P [ • Fix
such an x. As previously, we denote by u we will bound u
n
n
the general term of the series involved in (8) ;
by considering again two classes of indices n, slightly different
38
from those taken in the proof of Theorem 1. Let's suppose - and this of course isn't a loss of generality - that: n 'In:2:2, 'ljEI(n), Ilx.II,,2 /Logn a.s •• J For simplicity we put 8 :: exp ( -8(2p+2C ) Ix) p
We consider fir st the following situation: I\. Log n" 8.
First case:
It is easy to see that if one puts y :: 243 x
l/p
, relation (6) holds for n large enough.
Now we will again apply Lemma 2 ; for this we introduce the following sequence ( Yk ) :
'l k :: 1, • • •. 2
n
One has: 'l k :: 1, .. "
2
n
I "
Yk
1 a. s.
and:
VI
+... + V 2
where:
d
"
n
n
....
(2(Log n)
2
I\. + 2Log n d
n
),
0. by condition bll) .
Therefore:
V n :
Vl+•.• +V n
,,48Logn.
2
Applying now Lemma 2 with v:: 4 8 Log nand A :: 4 v (2p+2C ) p 3/2 u s: ( 2/ch2Log n ) exp 4 81 12 Log n ,,4 nn
It remains to consider the complementary situation:
A Log n > 8 .
Second case:
If one choos es now:
Y
:: ( 8
k
I
I\. (2p+2C) P
) Sk
one has: 'l k :: 1. ... , 2
n
,
I Yk I
,,1
a. s.
and: 'In:2:n'(x)
VI +... +V
n
"48
2
I
1\..
2
Applying finally Lemma 2 for v :: 4 8 2 u
n
,,(2
I
I
ch(x387/8/21\.)) exp (4x482/Al
I\. and
A:: v x
2
one gets 3 7 ,,4 exp (_ x 8 18 141\.)
Collecting the partial results we obtain: 'l x E JO, 243- P C, II N(x) EU'J, II 'Y (x) > 3/2 'l n:
I II),
I II)
such that:
I
:3: K > 0 : Vk E N,
Xk
II " K
a. s.
(k/L k) 2
Suppose that the following hold: 1) n- P I: 1/ X liP .... 0 in probability. k 1 "k" n 2)
'l
>0
E
I:
exp (-
E
I
S
n
/ n
.... 0 a. s.
As an obvious corollary of Theorem 4 one has the following result which can be compared with the well known law of large numbers in type p spaces of HoffmannJ~rgensen and Pisier [15J
COROLLAR Y 4 : Let (X ) be a sequence of independent, centered, r. v. with values k in a real, separable. p-uniformly smooth ( 1 < P " 2 ) Banach space (B,
I II).
Suppos e that: a)
Sn / n .... 0 in probability. I:
b)
k- 2p E
k;" 1 c)
3: j ;" 1 :
I:
I
Xk
2P 11
( !(n) ) j
< + 0
(Sn/jll)n E ~
and the sequence
Ilsnll lim inf IP [ - -
< M}
>
is stochastically bounded as soon as for some
M> 0
0 •
v£
n-oo
M. Talagrand (oral communication) raised the question whether the equivalence he proved holds without the strong second moment assumption which is not necessary in general for the
CLT • In this note, we answer this question in a positive way
in uniformlY convex spaces. Precisely, we will establish the following result
THEOREM 1 • Let with values in
E
E. Then 2
(1)
lim t t-ao
(ii)
for each
be
a uniformly convex Banach space and X satisfies the
JP[ Ilxll > t}
CLT
X a random variable
iff
0
and E
> 0 ,
Ils)1 lim inf I P [ n-'"
vn
o .
This result will follow easily from a new quadratic estimate of sums of independent random variables in uniformly convex spaces obtained in
[L2] •
Preliminary results. We begin this section by a characterization of the which follows easily from the concentration's inequality of M. Kanter
CLT
[K] • I am
grateful to Prof. X. Fernique for useful informations on this result.
PROPOS IT ION 2 Then
X
. Let
satisfies the
lim inf IP[ n- OO
S n
vn
X
CLT
E K }
be a random variable with values in a Banach space iff there is a compact set
>
0
K in
E
E such t fl-'lt
(1)
.
46
Further, the sequence IlsJ lim inf JP[-
(Sn / [ii)n EC:IN
0
o •
>
hi (1)
Proof. The necessity of
(2)
and
(S / Ill) :IN. Assume to begin wit h that nnE
implies the stochastic boundedness of X
is symmetric.
There exist
6 > 0
(2)
is obvious. Let us first show why
and
k
such that for all integers
o
k
~
k
0
n
and
Ilsnkll
IP[--
6 •
>
MJk })"""2
By Kanter's inequality Ilsnll
3
( 1 + kJP[ - -
2
1
~
6
hi
and thus
-
It follows that the sequence JE[llxII
CI }
copy of of that
Choose a finite dimensional subspace
of
H
E such that if
T
denotes the quotient
E.... E/ and 11.11 the quotient norm given by Ilr(x)11 = d(x,H) , then H 2 JE[IIT(X)11 } :s; 6.£2 • For each n, IIT(S )11 - JE[IIT(S )II} can be written as a
map
n
n
martingale n
IIT(S n )11 -
JE[l/T(S n )II}
with increments
d. , i ~
=
2: d. i= 1 ~
= 1, ••• ,n , such that, for each
i
[yJ) and thus by Chebyschev's inequality
(cf
S
F [IIIT(
Since
~ )11 -
;n
S
E[IIT( ~ )II} I
>
£
Jll
IITII:s; 1 , S
lim 1nf n ....
OO
JP[ IIT( ~)/1 In
6 ,
}
:s;
6 •
48 and hence, intersecting, S
lim sup JE[IIT(
< 2
n )II}
£
n .... oo
X therefore satisfies the
CLT
by classical arguments (cf
[P2]).
A short analysis of this proof shows the central role of the martingale trick 2 leading to the quadraUc estimate and of the integrability condition JE[llxI1 }
0
E
such that for all
CLT .
is uniformly convex if for each x,y
in
E with
Ilxll
= Ilyll
= 1
£
> 0 and
there is Ilx-yll:2: £
1 - II~II > 6 • According to a well-known fundamental result of G. Fisier
one has
2
[P1] , every uniformly convex Banach space admits an equivalent norm (denoted again
E
is p-smooth for some
p > 1
i.e.
11.11) with corresponding modulus of
smoothness
pet) satisfying
sup [ Hllx + tyll + Ilx - tyll) - 1, p(t),s; Kt P
for all
t > 0
1 }
Ilxll
and some positive finite constant
K.
This p-smooth norm is uniformly Frechet-differentiable away from the origin with derivative F (0)
and C
>
P
0
(cf
D : E - [a}
=
0 ,
then
....
E*
IIF (x)11 p
such that i f
F (x) p
= Ilx11 P- 1 for all
x
p 1 Ilxll - D(x/llxll)
=
in
for
x
I
0
and, for some constant
E
[H-J] ),
IIF (x) - F (y)11 P P
,;;
p 1 C II x _ y II -
for all
x,y
in
E.
The following lemma was the key point in the proof of the main result of It will allow to aChieve our wish in the next section.
(4)
[L2].
49
LEMMA 5 • Let satisfying
E
(4)
p > 1
be a p-smooth Banach space for some
Let also
(Y)i"n
E-valued random variables and let
with norm
11.11
be a finite sequence of independent bounded
S
written as a martingale
with increments
where
C
d
i
i
= l, ... ,n
such that, for each
is the constant appearing in
i
(4) •
Before turning to the proof of Thearem 1 , let us point out that a quotient E
norm of
holds true for any quotient norm of
E, property
constant
(4)
is also p-smooth, and, if
11.11
of a p-smooth Banach space
11.11
with uniform
C.
Proof of Theorem 1 • We may and do assume that
11.11
denotes the p-smooth
for some
p
> 1 for which
(4)
E
is equipped with a p-smooth
and Lemma 5 hold. By the previous remark,
these will also hold for every quotient norm with uniform constant moreover
p
C. We assume
g g e m
Ilx(t)11
tb.}
and
J
2
i(m -n)t
m n
I: g g cos((m -n)t) m n J m,n E I(j)
2 I: b.
J
m
Thus
n
geinti t E [O,2n] n J n n
E
E
sup tE[O,2n]
I b:/2
C (b.N .1ogN . J J J
J
f
int
I
)1/2
C. Likewise the examples of § 2 show that no condition
for some absolute constant of the form
J
g e nEI(j) n I:
([O,2TI],6)
- PN with the help of a Paul Levy's 'V 'V type inequality, we may study P -P P -P where Pn stands for the instead of n N n randomized empirical measure. Choosing some sequence of - measurably selected
- nets relating to
PN whose mesh decreases to zero and controlling the errors committed by passing from a net to another via some one dimensional exponential bounds, we can evaluate, conditionally 'V
to
PN , the quantity
82
Randomization
Setting from [1.n]
N=nm (m into
[1,N]
is an integer), let w be some random one-to-one mapping whose distribution is uniform (the "sample w is drawn without
replacement"). The inequalities in the next two lemmas are fundamental for what follows 3.1- LEMMA. For any
in
I;
N
lR N , we set SN
L
UN = (max (1;.)) - (min (I;i)); 1 d (or w -> d if d is "achieved"), we have r~ (2) (E, F)
< K1+(1/2I Lo gEI)
1
for any
exp (2w) (1 + 21 LOgE I) w E-2w 3 D K = 2D! (2D)
E in ] 0, H , with in particular when w=D ,
So from Stirling's formula we get N;2) (E,F) tant
C . 1
~ C~ e 5D 23D E- 4D for any E in
Hence, for any
E in
] 0,
1.] 12
]0,1 [ we have
and some universal cons-
NF) (E,n ~ C~ (2e)5D E- 4D thus, applying 3.5. to the class We propose below
G yields On
another variant of
3.6. inequal ity 3.3.2°)a), providing an
alternative proof of a classical result about the estimation of densities. 3.7. PROPOSITION.
-----------
If we assume that some positive bound of
2 d0 )(F-U) = 2d
t)
(n(1 +
is, for any positive
:~))3(d+T1I;
i
with
3.3.2°)a) •
In - -
for
C such that an upper
t, given by :
n (O)_4(d+ ) exp (- 2
\i
+ ..t(CLLn (..!:!.. + o)+t)))
;;; In the situation where
uv llfll oo IK 2 (X)dX.
1l > h- k > C2 , we get, setting Ln -
and some positive a
f
•
Jhk
89
Provided that
f
belongs to some subset of regular functions 8 , the bias
f-f can be evaluated so that the minimax risk associated to the uniform distance on R k and to 8 can be controlled with the same speed of convergence as
expression
in [29] , via an appropriate choice of
h .
3.8. SKETCHES OF PROOFS OF 3.3., 3.5., 3.7. (More details are given in [42]) . First, by studying the class G u=o
and
Let us proof theorem 3.3 ..
v~1.
parameters such as and positive
a, and
S,I3
We set :
I.l
( = -t
Pr(N)r.)
All along the proof we need to introduce
(in ]0,1 [) ; r, m (in }j) ; a (in ]1,+oo[); q (in ]0,2])
In
We write
(' =
and
Pr (1lvnll > t)
'V
Pr ( II Pn - PNII > E')
(1 -
1)
m
(1-a)
(
.
for the probability distribution conditional on
and 11·11 instead of 11·11 F
A bound for
instead of F , we may assume that
which are all chosen in due time.
y
N = mn
(x 1 , .. • ,x N)
f-u fE F} {iJ'
~
for short.
will follow, via 3.2., from a bound for
which is at first performed conditionally on
(x 1 ,··· ,x N)
The chain argument.
Let
be a positive sequence decreasing to zero.
For each integer of 2.4).
j
a ,.-net J
A projection '1 j 2 2 holds.
PN( ( 11 j f- f) )
i
F.
J
can be measurably selected (with the help
F.
F onto
may be defined from
J
so that
'j
Then 'V
'V
II (p n - PN) So, if
0
(Id -
'1
r )11 i
L j~r+1
II (p n - PN)
is a positive series such that
(nj)
L
j>r+1
o
(,1. - 11· 1) II JJ
n·J < I.l
'V
Pr(N) ( IIPn-PNII where
A and A B
B are the
Nr IIPr L j~r+1
(N)
> (') < A + B
(x 1 , ... ,xN)-measurable va ri ab1es 'V
(I(Pn-P N) a 11rl ? (N) 'V Nj IIPr (I(Pn-P N)
> (hI) (')11 0
,
(,1r 1f j_1)I > nj c:' ) II
we get
90
A is the principal part of the above bound and
B is the sum of the error
terms. Inequalities 1°) or 2°) of Lemma 3.1. are needed to control
A according to
whether case 1°) or 2°) is investigated. B , giving
Bound 3°) in Lemma 3.1. is used to control B
r+1 J the control of the tail of series 3.8.1. is performed via the following
Choosing a > 2),
r
=2
(so
+
elementary lemma 3.8.2. Lemma.
Let lji : [r, +00[
->-
lR.
Provided that
~I
is an increasing convex function, the
following inequality holds: exp (- lji(j)) ~ ~ exp (-lji(r)) d stands for the right-derivative of lji L
j~r+ 1
where
ljid
We choose S
= 1 under assumption a) and S
(~)
under assumption b) •
Proof of theorem 3.3. in case 1°). We choose
a
=
t- 2 ,
m = [t 2 J and
T. J
= Jl j-(a+S) and apply 3.1.1°), then In
A < 2 Nr e 10 exp (_2t 2 (1-2~)) Under assumption a).
Considering the type of inequality we are dealing N. ~ C t 2d }(a+1ld (instead of N. < C' t 2d ' }(a+1ld' J
J -
We c hoose,
"~ -- t- 2
an d a
=
with we may assume that for any
Max (2 , 1 + 3n 4d) ' so
A ~ 0n,F(1) (1+t 2 )3(d+ n ) exp (-2 t 2 ) and
d' > d) •
pIIIN_a.s.
91
P!liN -a.s. whenever
t 2 > 7+4d(a+l)
Now the above estimates are deterministic, so using
Lemma 3.2., theorem 3.3. is proved in situation 1°)a) • With the idea of proving proposition 3.5. note that, setting a method gives, under the hypothesi sin 3.5. , that
with
~
2 , the above
Pr( llvnll> t) is bounaed by:
K (E~1 t)4d (2+t 2 )12d exp (_ 2t 2) 1 K1 depending only on C, whenever t 2 > 7+12d.
Under assumption b).
We may suppose that We set
I.l =
N.
< exp
(C t1; jc;(a+S))
J -
t -2y , where 1;(1+2y(~))
~
n and for y ~ y( 1;) - 2 S > 1 to ho 1d, whe re
tion when a
~
+00
(Namely:
2( 1-y( 1;))
~
2( 1-y) , then we choose y( 1;)
k).
a 1arge enough
is the solution of the above equa-
So:
2 k A ~ 0n,F(1) exp (On,F(1) t +n ) exp (-2 t ) and B
whenever
n, F( 1)
0
eXfJ
(-2 t 2 )
t 2 ~ ~ + 5 + C t1; 22S+2 •
So theorem 3.3. is proved in case 1°). Proof of tneorem 3.3. in case 2°). We set
Q
~!
o
The variable
and choo'se m = s) i
s2
2 exp (- 2i) for any positive s.
The choice of parameters being the same as in the proof of 3.3.2°) (except 1·2
J
2 0 --2-
J.-2(a+s)) ,
P(1' )
d 4.1. 1. an d 4 .1 .2. are prove.
S'1nce 4 .1.1. 1S . a 1 so an
oscillation control, the almost sure regularity of Gp follows from Borel-Cantelli. The general case.
Since
(F,Pn) r
is separable
, the familiar extension principle may be used to
construct a regular version of brownian bridge on on a countable dense subset of F. this version.
l' from a regular version defined
Inequal ities 4.1.1. and 4.1.2. still hold for
95
Comment.
The optimality of bound 4.1.2. is discussed in the appendix. the polynomial factors are different in 3.3.1.2°)a)
The degrees of
and in 4.1.2. ; the reason is
that bound 3.1.3°) is less efficient than bound 4.2.
5. WEAK INVARIANCE PRINCIPLES WITH SPEEDS OF CONVERGENCE P(F 2+o)
t) given by a)
d~ 2)
If
7d
OF (1) n
(F) = 2d
t Sd
(0.1
(
exp \.-
a n)<Sn P are defined hereunder (we reca 11 that FE L2+6 ( P) and that
of a brownian bridge relating to where
(an)
and
(Sn)
P, G(n) p
e~2) and d~2) are defined in Section 2) : a)
If
for any
d(2)(F)
=
F
2d
2t+B) i A+B+C where
A" Pr ( Ilvnll G a
> t)
and
C" Pr ( !IGpll G a
Theorem 5.2. is used to control k ~ N~2) (~,F)
noticing that
> t)
B (with II·!! F(o)
.
i
I!·!! 2 i Ik I!·!! F(o))
according to remark 2.6 . .
Moreover C is evaluated with the help of theorem 4.1., so the calculations are
co~pleted
via an appropriate choice of
t
and
o.
6. STRONG INVARIANCE PRINCIPLES WITH SPEEDS OF CONVERGENCE.
The method to deduce strong approximations from the preceding weak invariance principles is the one used in [43] to prove theorem 2 : the weak estimates are used locally, giving strong approximations with the help of maximal inequalities and via Borel-Cantelli lemma. Maximal inequalities.
As was noticed in [23], the proofs of the following inequalities may be deduced from the one given in [10] and in [32] . Notation.
We set
X.
J
6x. - P for any integer J
j
100
6.1.
~£~~~ (Ottaviani's inequality).
k L X. .
We set Sk" j
c ~ max k ra) < Pr ( IIS n II
F
> a)
More precisely, for symmetrical variables, the following sharper inequality is available 6.2.
~£~~~ (Paul Levy's inequality).
Let
(Yi)1 a) k a)
S = L y. k j=1 J
Strong approximations for the empirical brownian bridge. 6.3. THEOREM. -------
Under each of the following assumptions some sequence versions of brownian bridges relating to defined on a)
if
~
(Yj)j~1
P that are continuous on
of independent (F,pp)'
may be
such that : d2 )(F) = 2d < F
1
Iii
00
,
n
II L
j ~1
(x·-~·)II F J
a. s.
J
for any a < 2( 1Y(O,d) +y ( o-;crn (2) a' ) if, more precisely NF (c,F) < Cc- 2d (1+Ls- 1 )d for any c in ] 0 , 1[ , n 1 (X .-y.) II = O(n-y(o,d)/(2( 1+y(o,d))) ((Ln)(1/2)+d + (Ln)(5/4)+(d/2)))a.s. In II j~ 1 J J F
b)
if
e?) (F) = I; < 2 , 1 II n (X.-Y.)II
,Iii
j~1
J
J
F
"O(Ln-(6/ 2 ))
for any 6 < 6(1;) . Where
y(.,.)
and
s(.)
aredefinedin5.5.
a. s.
101
For a proof of 6.3., see [42]. Comments.
When passing from weak invariance principles to strong ones, the speeds of convergence are transformed as follows within our framework:
incase a).
in case b) Transformation (ii) appears in theorem 6.1. (under 6.3 .) from [23], it is not the case for transformation (i) in the same theot'em (under 6.4.). On the contrary transformation (i) is present in finite dimensional principles and appears to be optimal in that case: more precisely, the rate of weak convergence towards the gaussian distribution for 3-integrable variables is ran' 1S rang1ng . a bou t n- 1/ 6 (see gl'ng about n- 1/ 2 when th e ra t e 0 f s rong t convergence [39] for the upper bound and [9] for the lower bound) lin the real case Application to V.C.-classes.
Applying theorem 6.3. with 6 =1
in the case where
v
F is a V.C. -class wi th
real dens ity d, we get a speed of convergence towards the brownian bridge that is 1 O(n-O:) for any This improves on 1.3.3. but is less sharp that 1.2.4. 0: < 18+20d in the classical case of quadrants in lR d
Following an idea from Dudley in [21] (sec. 11), the study of the general empirical processes theoretically allows one to deduce some results about random walks in general Banach spaces. metric space (5,K)
As an application of this principle
and the space
ped with the uniform norm
11.11
C(5) •
00
equipped with the Lipschitz-norm:
Let
let us consider a compact
of real continuous functions on
5,
equip-
X be the space of Lipschitz-functions on
5
102
II· II L : x
->-
II x II + tfS sup 00
We write
N(E,S,K)
ds,t) > E for any
sft
Ix(t)-x(s) I (s t) K,
for the maximal cardinality of a subset in
R of S such that
R.
We may apply our results through the following choices: F=II.II L ·
F={oS,sES}and Then
(X, 1/.11
) 00
is a Suslin space (but is not Polish in general), so
fills (M) . Moreover, for any distribution Q( ( Os - 0t )2)
so
N~2)(.,F) ~ N(.,S,d . Besides
11.11
Therefore, considering a sequence tributed
00
= II.II
2 Q(F ) ,
.
F
(Xj)j~1
of independent and identically dis-
C(S)-valued random variables such that: IX1(s)-X1(t)I~Mds,t)
with
p~2)(X) we have:
Q in
~ K2(s , t)
F ful-
E(M 2+0)
t)
~
1 - hd(O,t)
therefore (ii) is proved. Comment. 'V
Theorem A.1. was proved by ourself (see [40] and [41]) but also by E. Cabana in [11] * In another connection, inequality A.1. (ii) ensures that some polynomial factor t 2h (d)
w,"th
h(d»d-1 _
b ds 331°)) canno tb e remove d", noun .. a an d412 ..•.
The calculations yielding 3.3.1°)a) are slightly modified here, where the entropy condition a) is replaced with a more explicit one. A.4. Theorem. If we assume that a'l
F is
[0,1]-valuedandthat -2 NF)(E,F) ~ K1+1/ Lo g ( E 1 (1+LOQ(E- 2 ))d E- 2d
then, an upper bound for
Pr (
Ilv n II
F > t) is, for any
for any t
in
E in
]O,n
[1,+00[, given by
4H(t) exp(13) exp(-2t 2 ) + 4H 2 (t) exp(-(t 2-5)(Lt)2) where
Proof of A.4. Lt 2 + 1 , then In the proof of 3.3. 1°)a) we choose a = ----2 LLt A ~ 2H(t) exp(13) exp(-2t 2 ) P!liN -a.s. B ~ 2H 2(t) exp(-(t 2-5)(Lt)2)
whenever
*
t
2
Thanks to
~
P!liN -a.s.
6+4d , yielding A.4. via lemma 3.2.
M. Wchebor and J. Leon for communicating this reference to us.
107
Comment.
Assumption a'l is typically fulfilled whenever case of
d
F
•
is a V.C.-class.
In that
may be the real density of F (if it is "achieved") or the integer density
F (see the proof of 3.6.).
REFERENCES
V-i.Me.~on,
[7]
ALEXANVER, K. (7982). Ph. V.
[2]
ALEXANVER, K. P~obab~y ~ne.qua£LtZ~ nO~ e.m~c.al p~oc.~~~ and a taw On Ue.M.te.d togMdhm. Anna£~ 06 Ptwbabildy (7984), vol. 72 ,N° 4,7047 -7 06 7.
[3]
ASSOUAV. P. Ve.lUUe. e.t
[4]
BAKHVALOV, N.S. On appMuma;te. c.a.£c.ula.t 0 such that the open interval (a- 0 , a+o) is disj 0 int from the union of closed intervals m _ + .U [f.(xO),f.(xO)J J=1 J J
an
fEBl*
Moreover, by the Hahn-Banach theorem, there exists
such that
there exists
1< j.:::.m
This leads to a contradiction since
a
=
such that
+
fj (x) .:::. f(x) .:::. f j (x)
VX
E
R •
Thus, we have shown that m
For all
I
1
f:(X)
-
fj (X)
J
a.s.
there exists a constant P ( (f +. - f.) (X))
J
J
Integrating this with respect to N[Pj
(s,B~,P)
completing
C . = C (p, m)
such
a.s. P
and using the definition of
shoHs that
(ii)~(i)
To show (i)=)(ii) it suffices to consider the single bracket defined by
fi (x) = -llxll
and
f~ (x)
= Ilxll
and to take
f
s = 2 Ilxll PdP (x) .
If P is a tight measure then We may show the stronger imnlication (iii)=9(iv); our proof is inspired by the proof of Proposition 6.1. 7 of [7J. Let p> 1 be fixed and note that for all s > 0 there is a compact Set K~ B such that
fBIK
IlxIIPdP<sP/4
The elements of
Bi ' restricted to
K, form a uniformly bounded
120
equicontinuous family ano hence this family is totally bounded ~or the sup norm II ilK on K by the ArzeIa-Ascoli theorem. ~ B* ' m < "" , such that Iff ~ B}* l Ilf - f. II J
for some g.
J+m
Let
J
=f.+E/4
On
J
N~Pj
(E
f , ... , f m l
< E/4
K
Then for every
Thus,
Take
g.=f.-E/4 J
K,
fE B* 1
B~,P)
J
on
K , g. (x) = - II xii
gj+m(X) = Ilxll if
J
for
Ilf - f j Ii K < E/4
< 2m < 00 , proving
x ¢K
,
then
(iii)~ (iv)
~or
for
x ¢K ,
all
j = 1, ... ,m
g. Z Suppose that there exists y , o < '( < 1 - Z/ P an d M < such that 00
for
E
small enough.
is a P-Donsker class.
Z< p < 4
For all
Proposi tion-l.
F
Then
there exist
Donsker classes
P-
F
with
-y
FF
€
LP(A,A,P)
(E,F,P) > ZE
and
where
is any number leSS
y
than liZ. Fix
Proof. p/(p-Z) above.
and
Z
0 and choose , 2a + Z + , a (p Z) < 1 . Let S
Since S > pa + 1 Theorem Z.4 of [5J, F
t a.,rp-:J
J
J
'-or
boun~
Also, by 2a + 2 implies
F F € LP
it is easily verified that is a P-Donsker class since
( 1) N . [ J (E,F,P)
are greater than or equal to
F [ J ( E, , P) j:5.- j o =jo(E) , the n N (1) if ja-S ~ E and thus We may take
> 2 j o. i
-0
Now
- c-1/(S-a) -
a.p. > J J -
E
for all
if' and only
E
Let
y
= l/(B-a)=
small enough it is clear that y l/(a+,+Z) By taking a and , Q.E.D. may assume any value less than l/Z . The next proposition shows that suIts. N (1)
[ J
In particular, if
(E , F , P)
~
ZE
.y
for
Z
0
but by Theorem 2.4 of [5J, F
Now l'f then
and find
(p-2)/(p-1-6)
0 +0
and letting
K ,
We See that
Q.E.D.
, giving the desired result.
We conclude this section by exploring the relationship between P-Donsker classes and
p > 2
and
(E, F, P),
NtPj
in this order.
1
1,
We consider the caSeS
We first show that
(p) 2 N[ J'P> ,
will not in general furnish sharp results for the P- Donsker property. Proposition 3. with
If and
q> 2
and
p < q/2 + 1 , then there are classes
F
2 E -, , ,> q , such that
mayor
N(q) (E,F, P) [ J
,...
F
may not be a P-Donsker class, We first show that there are P-Donsker classes
Proof.
P < q/2 + 1 large.
, such that
Let
q> 2
S = 2a + 2 + 6
N (q)
[ 1
(E , F , P)
p < q/2 + 1
and
and show that if
arbitrarily large for
0
> 2 E -'
, >q
for
Fa S
then
arbitrarily small.
with
FF
~
arbitrarily
N[qj
(E,F, P)
becomes
To See that this is
j ~jO(E)
q for all note that if the individual terms a~p. > E J Jjo Now a~n,>Eq if and only if then ( E, F , P) > 2
jqa-S~Eq
if and only
actually
50,
J J -
N[qj (E,F, P) that
F
~
jf
j::::. E-q/(S-qa) o
2 E -' ; moreOver
is P-Donsker and
On the other hand, i f F
LP
8 > 0 , a=2/q-2) ,
be fixed, let
F:~
F
,= q/s too
FF ~ LP , since N tqj
may not be P-Donsker, eVen if
(E, F, P) F
If as
,=q/(S-qa) 8 +0
then
Finally, We note
S > pa + 1 < 2 E -,
for
is of the form
, > q > 2 , then F a, S .
Indeed,
,
123
let Q
