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0 : Vk E1NT,
1 X0 s K (k/L 2k)
a. s.
Let's define for every integer n and every f E B' : Z
X(n, f) :-• 2 -2n k
E I(n)
E f2 (X ) k
•
Suppose that assumptions a), b), c) of Theorem 1 are fulfilled and that the following one holds also : YE> 0 , 1TfEBI,
d)
Z n 1
exp ( _ E / X(n, f) ) < + cr .
Then : P( tu : S n(tu ) /n --' 0 weakly ) = 1.
PROOF : By Theorem 1, we know that : P( sup n A standard argument then gives [13] : E
sup n
(7)
Furthermore the one dimensional Prohorov SLLN [20 1 implies : Vf E B',
f( Sn /n) -' 0
a. s.
This property and (7) show that (Sn in , a(X 1 , ... , X n) ) is a weak sequential amart of class (B) (for the definition and the main properties of weak sequential amarts, see for instance [4] V.3 ).
37
The space B being reflexive,
the conclusion of Theorem 2 follows immediately
from a well known convergence theorem of weak sequential amarts due to Brunel and Sucheston [21.
§ 4. PROHOROV'S STRONG LAW OF LARGE NUMBERS.
The sufficient condition for the SLLN in the Prohorov setting can now be guessed easily from Theorems 1 and 2 ; the statement is as follows :
THEOREM 3 : Let (X) be a sequence of independent, centered, r. v. with values r,
in a real, separable, p-uniformly smooth ( 1
0:
1X k l sK(k/L 2 k)
a. s.
Suppose that the following hold : S
b')
2 -2nP Log n Z k E I(n)
c')
y e >0,
n
Then : S
/n
n
0
in probability.
a')
Xk 2p
-' 0 in probability
E exp (_ C / A(n) ) 0 , S 11m Lnf W[
n
n
0 .
■/171
(Actually, as detailled in [T] , this equivalence holds in the more general setting of non-separable range spaces and in the framework of empirical processes.)
Although it seems rather difficult to verify these conditions on small balls, the preceding property is intriguing since it reduces a central limit property in Banach spaces to some kind of weak convergence on the line by taking norm. This property also lies at some intermediate stage since, as we will see below, a random
45
variable X K in
E
with values in
E
satisfies the CLT iff there exists a compact set
such that
lim ink IP( --21-1 EK1 > 0 n (S
and the sequence
/fn)
n E
is stochastically bounded as soon as for some
M>0
11S lim inf IP( n-'
n <M 1171
> 0 .
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
(i)
E • Then X satisfies the CLT iff
lim t 2 F( t
a uniformly convex Banach space and X a random variable
be
E
X
>t3=o
oc)
and (ii) for each E > 0 , lim inf TP( n cx)
Is
F
< C
3>o.
/71
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 CLT which follows easily from the concentration's inequality of M. Kanter [K] . I
am
grateful to Prof. X. Fernique for useful informations on this result.
PROPOSITION 2 . Let X
be a random variable with values in a Banach space
Then X satisfies the CLT iff there is a compact
lim ink IP n
E K
> O .
set
K in
E
E.
such that
(1)
46
Further, the sequence (S
//17)
n)
n E
is stochastically bounded iff for some M > 0
IS I < M
11m 1sf IPC fl - Oo
> 0 .
(2)
Proof. The necessity of (1) and (2) implies the stochastic boundedness of X is symmetric. There exist 8 > 0
is obvious. Let us first show why (2)
CS n /N51) n and k
o
• Assume to begin
E
with
that
such that for all integers k k
o
and n S
IPC
nk
<M
Firc
>8.
By Kanter's inequality
dsnd
3
•(1
kip(
> 1,1,./Tc ))
1
8
sin and thus
!
Is
> m sA- 3
kip(
_2_ 48
JR
2
It follows that the sequence (S
ECIlx cy
0
X
satisfies the
and
be fixed
CLT iff (3)
8 . 8(e) > 0
E
such
holds.
be such that
1S 6
< c 1 >
lim inf n
Choose a finite dimensional subspace
E
map
Et
E/Ii and
T(X)11 2 1
.
H
of
E
quotient norm given by
8.E 2 . For each
n
, 11T(S n)11 -
T
such that if
Ircx) T(S)
denotes the quotient
= d(x,H) , then
11
can be written as a
martingale
!T(S)
IECHT(S n )D
= E
di
i=1
withincrementsd.,i = 1,...,n , such that, for each Efd2i 3 s lEf
IT( xi )
2
i
1
[Y] ) and thus by Chebyschev's inequality
(cf
ITC
-11 )11 _
RUT(
-2-1 )1111 > E 1 5 E -2 ECHT(X) xfil
Since
e
1 t
lim inf 13 (
cto
11T(
--11
)fJ
2
1
s ô.
48
and hence, intersecting,
lim sup TEC T( n 00 X
)d1 < 2c
AT:1
therefore satisfies the CLT
by classical arguments (cf [P2] )
A short analysis of this proof shows the central rae of the martingale trick leading to the quadratic estimate and of the integrability condition
EC
4
2
3
0 there is a
8 = 8 (c) > 0 such that for all x,y
Itx _ y one has 1 - ---2
with
in E
c,
Y = 1 and 1 x y
> 8 . According to a well-known fundamental result of G. Pisier
[P1] , every uniformly convex Banach space E is p-smooth for some p > 1 i.e. admits an equivalent norm (denoted again
)
with corresponding modulus of
smoothness
p(t) = satisfying
sup ( kix + ty
p(t)
1
-ty0 - 1 ,
+
Kt' for all t > 0
= 1 '}
and some positive finite constant K .
This p-smooth norm is uniformly Fréchet-differentiable away from the origin with derivative
D : E
(01
and F(0) , 0 , then 1 F (x) C > 0 (cf
P-1 D(x/I
E* such that if F (x) = =
P-1
for all x
X
) for x / 0
in E and, for some constant
DI-J] ),
F (x) -F (y)11
C x
y p -1
for all x,y in E .
(4)
The following lemma was the key point in the proof of the main result of EL2] . It will allow to achieve our wish in the next section.
49
LEMMA 5 . Let E
be a p-smooth Banach space for some p > 1 with norm 1
satisfying (4) . Let also (Y.)
be a finite sequence of independent bounded hen HSH11P
E-valuedrandomvariablesandlet
1E(
1 1)/
can be
written as a martingale
1 1)- ECI with increments d
1 = 1,...,n , such that, for each i
2 r 2 2p lEtF (S -Y1)(-Yi))
r Etd3 . 1
2c2 E rjj y. d2p 1 CH
1 11
where C is the constant appearing in (4) .
Before turning to the proof of Theorem 1 , let us point out that a quotient of a p-smooth Banach space E is also p-smooth, and, if norm of E , property (4) holds true for any quotient norm of
denotes the p-smooth with uniform
constant C .
Proof of Theorem 1 . We may and do assume that E is equipped with a p-smooth
1.1
for some p > 1
for which (4)
and Lemma 5 hold. By the previous remark,
these will also hold for every quotient norm with uniform constant C . We assume moreover p 0
be fixed. For each n ,define
(5)
50 X.
u. 1
= u(n)
=
-- 2--
/17_
I,t1 tilX i 1 S A 1 ' i -
n andsetUri = . Eu.;(i) and 1 1=1 number 8 = 8(e) > 0 such that Urnif IF(
n
< e
Un
Since the sequence
(1i)
combine to imply the existence of a real
>6
(S /A) n n E
does not contain an isomorphic copy of
co
[P-Z]
Theorem 5.1 of
is pregaussian, that is, there exists a Gaussian random variable the same covariance structure as
IEC
T( G )11 2 1
s
e2 P
E
and
ensures that G in
X
E with
X . The integrability of Gaussian random vectors
allows then to choose a finite dimensional subspace denotes the quotient map
(ii)
is stochastically bounded under
H of
E
such that if
T
E E/H 2 p2 ic2 P 1 )
We now apply Lemma 5 to the sum
-
(7)
T(U ) n
in
E/H ; F p will therefore denote below
the Fréchet derivative of the quotient norm of
E/H . For each
n , we have by
orthogonality,
1E01 T(un)1P - F[IT(un)P3 1 2 1 n n 2p2 E IE(F 2p (T(Un -u.1 ))(T(u i ))1 + 2 0 2 r lENT(u i )11 2P 1 1 .=.1 i=1 n 2(p-1)3 + 2 0 2 n 7E( u 1 (n)11 2P 1 S 2p2 n-1 7E (1 T(G) 2 1 E lEf U - u n il 1=1
(8 )
since by independence 2
sup 1E flx*(T(x)) I 1 lEaT(U n - ue (P-1) 1 1E(F2p (T(Un -u ))(T(u i ))1 s n-1 x* E (E/ )* H 1 (where the supremum runs over the unit ball
*
sup
)
E(Ix* (T(X))1 2 1 =
H 1 Now, by symmetry and
sup
(E/H )1- of the dual of
2 21 1E(Ix* (T(G))1 1 s laT(G))1
x* e(E/ )* I-1 1 (5) , for each
E/H ) and
i = 1,...,n ,
51
E[ n u.
(
(3E(
12 C p 1) 1 ) 1/2 (p 1 ) -
-
U
K .
n
Further, n
TECdu i (n)
2P )
13 ( 7., 11> t
n i-P
dt 2P
0 1
r( X >
n
dt 2 P
so that lim
nIE( u(n)
0
n
by
(i) and dominated convergence. These observations and (8)
lim sup IP( n 00
I
therefore imply that
T(Un )d P - 1E( T(un )
e
-2p
,2 ( p-1) 2] ( 22 p I\ TEt T(c)
s
(by (7) ) and since (6) holds, by intersection,
11m sup
n
1E (
1T un) (
1 •
For these choices, we see from (2.10) that
12 J ([0,2n j ,d) < en
even though,
by (2.8) and the integrability properties of Gaussian quadratic foins, X (s,t) g does not converge uniformly a.s..
63
Thus (1.6) can not be replaced by (1.7), with any increasing function f satisfying
lim -4
f(x) //x = 0
and (2.6), as a sufficient condition for the
CO
uniform convergence a.s. of (1.1). Note that exactly the same argument applies for the corresponding Rademacher Fourier quadratic form, i.e., with (c c 3 m n (gmgril
replacing
in (2.7). 12 . In some cases J 1/2 ([0,27J ,d) < œ is necessary and sufficient for the
uniform convergence a.s. of (1.1) and (1.2). One of these, which is trivial, is when the coefficients
(am,nI
vanish outside some one—dimensional set of indices.
We write (1.1) in the form
(2.11)
E
where m < n
akgnagnice ik(ms + nt)
( s ,t) E [0,271 2
are non—negative integers. Using Gaussian decoupling we see that this
series converges uniformly a.s. if and only if z
(2.12)
e ik(ms + nt)
k "Oic mnk
,
(s,t) E [0,2171 2
converges uniformly a.s. where (g;a 3 is an independent copy of f (gmk grilk )
is an independent symmetric sequence in k
. Since
it follows from Theorem 1.1
[8,Chapter I] that the series in (2.12) and consequently (2.11) converges uniformly a-s. if and only if J holds
ifmkgnk/
1/2
([0 211] 2 ,d) < oo . Once again exactly the same argument '
in (2.11) is replaced by (e c ) mk nk *
The finiteness of
1 J1/2([0'27]2 ,d)
marginal processes formed from X (s,t)
Theorem 2.2. Let X (s,t) the processes X (s,t ) o t oE [0,27]
also implies the continuity a.s. of the and X (s,t) . c
be as given in (1.1). Then if
j 1/2 (1° ' 2712 ' d)
e int l n [0,27] n n
sup
tE x* E B* .
where
Therefore uniform convergence a.s. of random Fourier series with coefficients in a Banach space is characterized through conditions involving
a family of
classical one-dimensional entropies. We will see in the proof of Theorem 1.2 that
(3.1) can be used to obtain a similar result for Gaussian and Rademacher Fourier quadratic forms. Using a set of one-dimensional entropy conditions instead of a single two-dimensional entropy condition to characterize uniform convergence a.s. of random Fourier quadratic forms seems to be necessary, as was shown by the class
of examples described in § 2. Theorem 3.1 sheds some light on these examples. Indeed, the quadratic form (2.7) can almost be realized as a random Fourier series with coefficients in a Hilbert space. Define a sequence
(x
n
of elements in
setting
Vj> 1 , VnEI(j)
where
(eic l
xn
=
r
denotes the canonical basis of
X(t).Egxe int , n nn
b.
/
e
) 1/2
Ni
kEI(j) k
A2 • Consider
tE[0,273 .
02
by
67
Clearly
Hx(t)11 2 = E
n
11 2 +
E < x ,x > g g
mn mn
m n
e i(ITI-11)t
(3.2)
= E
Thus (I)
11X(t)11
+ 2 E b. j
J
j
E
EI(j) m n n t E [0,27] n sup
E
b 1/2
sup
t E [0,27]
C (b iN i logN i ) for some absolute constant of the form
1/
E n E T(j)
g e inti n
2
C . Likewise the examples of §2 show that no condition
J f ([0,2171,15)
1 2 1e
1nt
e
int'12)1/2
.)
We now use Theorem 3.1 to obtain Theorem 1.1.
Proof of Theorem 1.1. We will give the proof in the case
(1.4) we will prove this theorem with
X (s,t) and X (s,t) g
X'(s,t) -Eaggie m,n m n g m g m m
m
m > n
and where the sequences
have been defined in (1.11). Indeed, Theorem 1.4 [8,Chapter I] then clearly
implies (1.12). Following the notation of the preceding proof we show (3.7) for
j = 2
and with the left hand side replaced by
EHX 1 (s,t)11
proof of Theorem 1.1 (cf. (3.3) , (3.4) and the estimate of
by (1.3). As in the I ), we have
70
E supl Xics,t) I s,t
(3.8)
E sup l E ( E la 12) m,n n m s
II
II
m
ims ) g e int n
the second term to the right of (3.8). It is plain that
= E
2 ims E A g e
sup
m m C
where El is the norm on
then follows easily from a new application a.s. . (3.7)
of Theorem 3.1 but this time in
II
2 ge ims l
g e sup E , sup I E ( E a m,n m t n in
+ E
Let us call
1/
-E
g
C
a.s.
2 TEA g rn
. Indeed ,
sup TE C*
m m
ims 2 E sup E < T,A > g e m m s m a.E
I
and by Theorem 1.4 [8,Chapter I] ,
2 E A g m
m m -
= EsuplE(Ea g) 9'e int I t n m m,n m n E sl)-P I E Ç E t n in
am,n
2 ) -1/2_, s Int
Appendix.
Proof of (1.3) and (1.4) : the argument that we give here was shown to us by Gilles Pisier. We will first prove (1.4). It follows from Theorem 2 , 17=1 that for
p > 1
(A.1)
E 11X c (s,t)11 1) -
EH E
a
m t)
4 F stands for the
mations of convergence,
vn
for any positive
uniform norm over
F
t,
and to build strong uniform
gaussian process indexed by
by some regular say
,
F with some speed
approxiof
(b n ) .
First let us recall the main known results about the subject in the classical case described above.
1.2.
THE CLASSICAL BIBLIOGRAPHY.
We only submit here a succinct bibliography in order to allow an easy comparison with our results (for a more complete bibliography see Concerning the real case
P
and are optimal
(d=1) ,
[26]) .
the results mentioned below do not depend on
:
1.2.1.
(1.1.1) is bounded by to
Dvoretsky, Kiefer
and
C exp(-2t 2 ),
where
Wolfowitz [24] (C < 4/2
C
is a universal constant according according to
[17]).
75
1.2.2. The strong invariance principle holds with
b n - log(n) ,
Komlos,
according to
In
Major and
Tusnady [37] .
In the multidimensional case (d
_> 2).
1.2.3.
(1.1.1) Kiefer
C(E) exp (-(2-E)t 2 ),
is bounded by
[34] .
In this expression
E
E > 0,
for any
cannot be removed (see
[35]
according to
[28]).
but also
1.2.4. The strong invariance principle holds with b
n
= n
1 2(2d-1) Log(n),
according
[8].
to Borisov
P
This result is not known to be optimal, besides it can be improved when
[0,11 d .
uniformly distributed on
1.2.5.
If
d
-
In this case we have
is
:
2:
The strong invariance principle holds with
b n - (Log(n))2 , /6
according to
Tusnady [50]. 1.2.6. If d > 3 :
—
The strong invariance principle holds with to
Csbrgb
and
1.2.5
1 3 2(d+1) (Log(n)) 2 , b n -n
Révész [14] .
and
1.2.6
are not known to be optimal.
Let us note that even the asymptotic distribution of (the case where in
according
d
- 2
and
P
ky n il F
is the uniform distribution on
is not well known
[0,1] 2
is studied
[12]). Now we describe the way which has already been used to extend the above results.
1.3. THE WORKS OF VAPNIK, hRVONENKIS, DUDLEY AND POLLARD. Vapnik
V
and
Cervonenkis
rally called V..-classes
-
introduce in
[51]
some classes of sets
for which they prove a strong
large numbers and an :exponential bound for
(1.1.1) •
-
which are gene-
Glivenko-Cantelli
law of
76
P.
Assouad
[40]
also
studies these classes in detail and gives many examples in
[3]
(see
for a table of examples).
P-Donsker
The functional
classes (that is to say those uniformly over which some
central limit theorem holds) were introduced and characterized for the first time by Dudley in
[20]
and were studied by Dudley himself in
[27]
Some sufficient (and sometimes necessary, see bounded) conditions for of entropy conditions
F
to be a
P-Donsker
[21]
and later by Pollard in
F
in case
[44].
is uniformly
class used in these works are some kind
:
Conditions where functions are approximated from above and below (bracketing, see
[20])
are used in case
F
is a
P-Donsker
restricted set restricted set of laws on respect to the
Lebesgue
X (P
F
is a
P-Donsker
of laws including any finite support law (the
-
belongs to some
is often absolutely continuous with
v .. Kol icnskil
measure in the applications) whereas
conditions are used in case
bility assumptions
P
class whenever
class whenever
V.L-classes
the classes of sets of this kind, see
are
P -
and Pollard's
belongs to some set under some measura-
[21]).
In our study we are interested in the latter kind of the above classes. Let us recall the already existing results in this particular direction. Whenever
F
is some V..-class and under some measurability conditions, we have:
1.3.1.
(1.1.1) Alexander in
is bounded by
[1]
and more precisely by
C(F) where
C(F,E) exp(-(2-dt 2 )
for any
in
E
]0,1],
according to
:
(i+t2)2048(D+1) exp(-2t 2 )
D stands for the integer density of
F
(from
,
in
[2] *
Assouad's
,
terminology in
[3]) .
1.3.2. f
(1.1.1)
is bounded by
4e'
D
E
\i=0
2 ))p(-2t ex 2 ) , according to Devroye in [16].
* Our result of the same kind (inequality 3.3.1°)a) in the present work) seems to have been announced earlier (in [41]) than K. Alexander's one.
77
1.3.3.
1 The strong invariance principle holds with b n -.= n
2700(D1), + according to
Dudley and Philipp in [23]. Now let us describe the scope of our work more precisely.
2, ENTROPY AND MEASURABILITY.
From now on we assume the existence of a non-negative measurable function F such that !f! < F,
for any f
in F.
v We use in this work Kolcinskii's entropy notion following Pollard [44] and the same measurability condition as Dudley in [21] . Let us define Kolcinskii's entropy notion. Let
p
and p (p) (X) F 2.1.
be in [1,+.[ . A(X) stands for the set of laws with finite support for the set of the laws making FP integrable.
DEFINITIONS. Let c
be in ]0,1[ and Q
N ( P ) (c F Q) F "
be in 4. 0 (X) .
stands for the maximal cardinality of a subset
G
of
F
for which:
(!f g!P) > EPQ( FP)
Q holds for any f,g c-net of (F,F)
in
G
with fg (such a maximal cardinality family is called an
11P) (.,F,Q) . relating to Q). We set NP ) (.,F) = sup QEA(X)
Log(N F(p)(.,F)) is called the (p)-entropy function of (F,F) . The finite or infinite quantities :
d(F) = inf fs>0 ; limsup 0
s
E N
(p) F
(c ' F) < col
e(F) =inf fs>0 ; limsup c s Log(N ( P ) (c,F)) < col F F 6-* 0 are respectively called (p)-entropy dimension and (p)-entropy exponent of (F,F).
78 Entropy computations. from that of a uniformly bounded family as
F
We can compute the entropy of
:
follows
- (T 1 (F>0) '
Let I
fE Fl ,
then
:
( ) N( F . ,' F) < N 1 P ( ' I) Q
For, given FP
so
E A(X)
in
A(X),
either
Q(F)=0
and so
qP ) (.,F,Q) = 1,
or
Q(F) > 0,
:
and then
Q(F P )
NPP ))(. ,I 1 ( 1 Q(F) n
N F( P ) (.,F,Q)
Some other properties of the
(p)-entropy
are collected in
The main examples of uniformly bounded classes with finite
[40]. (p)-entropy
dimension
or exponent are described below.
2.2.
COMPUTING A DIMENSION According to Dudley
:
THE V..-CLASSES.
[20]
on the one hand and to
Assouad [3]
on the other we
have
d (1 P (S) = pd whenever
[3]).
S
is some
[45] .
COMPUTING AN EXPONENT Let d
Dk
with real density
d
(this notion can be found in
Concerning V..-classes of functions, an analogous computation and its appli-
cations are given in
2.3.
V.C.-class
See also
:
HOLDERIAN
THE
a
be an integer and
[21]
for a converse. FUNCTIONS.
be some positive real number.
We write
8
for the greatest integer strictly less than a.
Whenever
x
belongs to
P„c1
for the differential operator
Let Let
II.11
be some norm on
k to A d , lk a kl k k ' Bx 1 1 ...x d d
and
stands for
f
and
Rd
Ad be the family of the restrictions to the unit cube of
13-differentiable functions
k 1 ”' +lc d
such that
:
Rd
of the
79
max sup
ID
!k! (1-a) Tr E)
P n-PN
, where n = N-n .
this lemma see [16] using Dudley's measurability
of
arguments in
[21] (sec. 12) . Statement of the results.
3.3. THEOREM. The Pr
following quantities
( !v n !! F > t) 1°)
a)
h)
where k 2°)
are, for
any
positive
t
and
ri , upper bounds
for
:
if
d (2) (F-u) = 2d
if
2 0 1-1,F (1) (1 + L) 3(c1-1-1-1) exp (-2 U2 (2) < 2 , e (F-u) =
,
I
)
t k+n t2 0n,F (1) exp (O 1,F (1) (0) ) exp (-2 -7) U2 6-c C( 777 ) (when c increases from 0 to 2 so does k). Suppose a) if
that !!720 F < a2 , with G < U, then d u(2) (F-u) = 2d , 2 3 ( d + fl) ex p( o n,F (1) (2)- 4(d+n) (1 + L) U 02
h) if
0 n,F (1) exp(0
2(0- 2 +(3U+t)) ) in
4 2) (F-u) = c < 2 ,
11,F (1) (.g)U
where p - 2c(4-C)
t2
(when
5 (; ) 2n r
+1-, ) e Xp
t
2(0.2 + increases from 0 to 2 so does 2p) .
2 (3u ()P+fl +t)))
84 The constants appearing in these bounds depend on
N (2) (.,F-u)
F
only through
n .
and of course on
Comments.
2.2.,
From section
F
d (2) 1
the assumption
is some V..-class with real density Thus
bound
the factor
1')
1°)
0(F,) t 2nd
1°)
in
a) improves on
1.2.3. 1°)
moreover the optimality of
d.
F
but is less sharp than
on
F - A a,d Bakhvalov proves
[0,1] d
then
(2t2)i
2.3.
then, from section in
[4]
P
that if
in the real case
;
exp (-2t 2 )
we have
:
(*)
d e (2) 1 (F) = a- .
In other
stands for the uniform distribution
:
1_
Hn H
a
d
F > C n2 1°)
Thus we cannot get any inequality of the
e (2) 1
1.2.1.
i=0
Suppose that
,
a) is discussed in the appendix where we prove that
1'H- cc
respects,
in another connection
is the collection of quadrants on R d )
( Ilm n r F > t) > 2
Pr
1.3.1. ;
a) is specified in the appendix.
d-1 lim
is typically fulfilled whenever
a) is sharper than those of
In the classical case (i.e. bound
(F)
2 . The
border line case
:
,
For any modulus of continuity in the same way as
Ad by changing
greatest integer for which
(1)(u) u -I240
It is an easy exercise, using
we can introduce a family of functions
u÷u a
q)
and defining
as the
holds.
Bakhvalov's
!v
into
A ,d
method, to show that
:
> C (Log(n))Y
d
provided Of course
that
e 1(2)
4)(u) = u 2 (log(u -1 )) Y (Ad) 4D,
)=2
and
P
is uniformly distributed on
and we cannot get bounds such as in theorem
[0,11 d .
3.3.
(*) So, there is a gap for the degree of the polynomial factor in the bound 3.3.2 ° )a) between 2(d-1) and 6(d+n) .
85
But the above result is rather rough and we want to go further in the analysis of the families Then the
around the border line.
Acp,1
(2)-entropy
A a, 1 concerning the Donsker
plays the same role for
property as the metric entropy in a Hilbert space for the Hilbert ellipsoids concer-
pregaussian
ning the
1
1 (i)
A(1),1
:
property, that is to say that the following holds
P-Donsker
is a functional
2 (2) (Log(N 1 (E,A ))) q), 1
class whenever 10
dc
1
V 1 Log(u)1 u )1 ,1
belongs to
,
[32]
about Rademacher trigonometric series.
(E n !W(e n )!)
(E n )
with
being independent of
L n>1
>
A (1),1
W(e nLog(n) n )1
W
some Wiener process provided that of G
.
So
(*) We write
A
f
(lb, 1
is not
g
, when
brownian
W(1)
pregaussian
0 < lim
that
1,
where
(E n )
we may write (We n ))
1 nLog(n)
bridge G for
N(0,1)
(ii)
is proved.
(fg 1 ) 0
-4 bound 2°)a) depends on
In order to specify in what way the constant in
F,
we indicate the following variant of 3.3.2°)a) .
3.5. PROPOSITION. If we assume that o
E
1
]0,1[
in
Ha 2PII
]0,1[ that
in
N
(2)
(E,F-u) < C (Et E) -2d for any
< a2
with
F—
depending only on
E
o
g
not exceeding
and a constant
K
E
]0,1[
in
and some
U, then there exists some depending only on
C such
that : Pr ( Hy n !! F >
From now on
L
t 2 14d ( -d o -4d t)
0
for any positive
87 (Provided that
in D n = o(--) such a choice of o Ln ' n
does exist).
Comment. { v n (f), f E F n } [38] (Lemma 2) and applying 3.6. the process i admits finite dimensional approximations whenever D n = o (Lon g(n)) and provided According to Le Cam
that Le Cam's assumption (Al) is fulfilled. This result improves on Le Cam's corollary of proposition
< 21 is needed.
for some
y
Proof of
3.6. F
Let
4, V.C.-class with entire density
be a
w > d (or w > d if d
show that, for any
(2)
in
]0,1[ ,
(2) (E,F) N1 C
C 31 e 5D
2 3D
in
w-D , K =
d
.
it is easy to
:
2w
(2D) D .
: e
for any
c
Hence, for any
1'
is "achieved"), we have
with in particular when
So from Stirling's formula we get
tant
and real density
1+(112 LogE) exp (2w) (1 + 2!LogE )w (E,F) t)
o V,F,r1W
t 2 3(d+I)
(n(1 + 7))
In the situation where
3.3.2°)a) .
is, for any positive
U
t, given by :
a _4(d+n) exp ( (0)
t2 2 U 2 (o + -- (CLLn (IL + c5)+t)))
VT' in is large this inequality may be more efficient than
88 Application to the estimation of densities : minimax risk.
K
Let
K q)
where and
M
= 11)(y'M )
(y)
kxk
is some
V.L-class
[45]
(.-x),
for any
P
Now if we assume that
Rk ,
measure on
is a
We set
K
E
ME
Rk
2
1 1 E- 2,2]
, xE R k
: 10,1[
in
C
where
w
and
depend only on
is absolutely continuous with respect to the
f
the classical kernel estimator of its density
n
K
into
that the class
of functions and so
N 1(2) (c ' K) < CE -w
(x) = h -k P n
with fixed
tp
Lebesgue
:
h
M
and
is
k.
so that
f 2 (x)dx
where
we get, setting
2 (c 2
fn -T
by choosing
:
il .[K2(x)dx. j
D -n stip!f f(x)1 : - x (x)n
t2 0 (J_Ln )
t
)
41" for any
t
in
[1 + 04
and some positive a and
E (vrj D n ) < T + 0(n œ )
T
exp
2(c 2
13 .
T2 0 ( LLn ) %KJ
for any
T
in
We choose
[1, + .[,
provided that
T = 0(/g) ,
thus
nh k >
:
E(D) n
0 ((-15) 2 )
Hence, after an integration:
T In7
)
89 f
Provided that
T-f can be evaluated so that the minimax risk associated to the uniform R k and to 9 can be controlled with the same speed of convergence as
expression distance on in
[29] ,
belongs to some subset of regular functions G , the bias
via an appropriate choice of h
.
3.8, SKETCHES OF PROOFS OF 3.3., 3.5., 3.7.
= (fT?, fEF1
First, by studying the class G
u=0
and
v=1.
s,13
and positive
We set
3.3..
Let us proof theorem
parameters such as
: N =
:
p
a,
(in
F,
instead of
[42]) .
we may assume that
All along the proof we need to introduce
]0,1[) ; r, m
; a
(in
]1,+.1); q
(in
(in
10,21)
which are all chosen in due time.
and y
,
mn
(More details are given in
and
E = --
E' ' (
1 - 1)
(1-a)
E .
1/171
We write Pr (N)
(x
(.)
The
IP n
(
r
( Iv n
- P N II > EI)
> t)
for short.
will follow, via
3.2., from
a bound for
which is at first performed conditionally on
(x
1'
...,xN ) .
chain argument. Let
(T i ) i>1 be
a positive sequence decreasing to zero.
For each integer j of
.4
instead of
and
' xN )
A bound for Pr Pr
for the probability distribution conditional on
2.4).
A projection
2 < T. — J
2
a
T.-net F. can be measurably selected (with the help
7i may be defined from
F onto
Fi so that
holds.
Then
Il(P n - P N ) ° (Id - 7 r )I
r+1 So,if(.)is na
- P N ) ° (7. j
a positive series such that
j>r+1 Pr
where
A and
B
(N) (P nN
are the
I
>
E')
1
Under each of the following assumptions some sequence versions of
brownian
defined on
0
a) if
such that
0 for every B*) * * * * x E and symmetric (i.e., T(x * ,y* ) T(y ,x ) for all x ,y E B* ) Since f is uniformly bounded we have * * * * T(x ,y ) (E E x () 2 ) 1/2 (E y (dk) 2 ) 1/2
weak second order. Furthermore,
= IS(x1S(y f) 2 2
= b[*6 2 1 H*0 2 M1 x which means that T
,
l y*
is bounded.
** B . Using a theorem of Banach (see * e.g. [1 ] ), which states that an element u of B belongs to the image of B in * ** the natural embedding of B into B , if the functional u(v), v E B , is * * is contained in B . continuous in the B-topology on B , we show that TB T
We can consider
Assume that
E
* B
as a map of
into
a) is satisfied. Since
* 1/2 S(e:f) 11 2 en =E (E(E(e* (111. )) 2 ) 1/2 e = E (Te *(e)) en n k n n n
the series
*, *, 1/2 n le n ))
E
By theorem 2.1. in
B , i.e., * * Tx (x ) =
lx* (x)1
T 2
B .
is a covariance operator of a Gaussian measure
Y(dx) .
B
Since
11x* f1 2
[2]
is convergent in
11S(x* f)11 = Tx* (x* )
the implication a) Conversely, if
b)
follows.
b) is satisfied, the series
)) 1/2 e E 11S(e* 011 e = E (Te*n(e* n n n n n n 2 is convergent in
B
as
T
is a covariance operator of
'y
'y
on
112
We recall that a Banach space does not contain
n
uniformly if it is of certain
• Banach spaces of Rademacher type 2 are of some Rademacher
Rademacher cotype r
O .
'
It is reasonable to ask whether analogous results hold for those X satisfying the CLT . The following theorems answer this query in the affirmative, showing that the study of the CLT in separable Hilbert spaces
11
N ([ 2 ] (E,F1,P) , where
is conveniently studied through the use of
*
15 the unit ball or the Hilbert dual.
In fact, in the same way that the equivalence of
X CLT
and EX}
0 , I1X1! 2 < co
characterizes, modulo an isomorphism, separable Hilbert spaces, the following results show that the same is true for the equivalence of the conditions B
1
is P-Donsker and inf N (2) (E B * P) ' s >0 [ 1
c°'
P
centered.
117
This is contained in the following results.
X be a random variable with vaiues in (B, 11 ) which need not be separable; L(X) = P Then for all p >1 the following are equivalent: (i) EIIXIIP < , and Theorem 2. Let
,
(ii)
If
inf N ( P ) (a , B 1* ,P) < oc. c>0 P is tight then the following are equivalent: 0 * Conversely, if P is a law on any then B 1 is a P-Donsker class. separable Banach space B and if Iq is P - Donsker, then for all p 0 there If
is a compact set KB such that
f
11x1I P dP < c 13 / 4
The elements of
BI , restricted to
K , form a uniformly bounded
120 equicontinueus family and hence this family is totally bounded for the sup norm on K by the Arzel- Ascoli theorem. Take f , , fm e B m < , such that Vf B i 1 ' I!f - f 11 for some
K
j .
g5. +Ill = f.3 + c/ 4
1 and a j ..-j c4 for some 0< a < $ - 1 . Since elements of Fag define measures on 11\1' , we may view Fa, $ as a subset of the dual to (C,Ir
.
We will consider the cases p =1 , p >2 and l< p< 2 in this order; much of what follows may be found in [19]. Our first proposition shows that a theorem of Dudley, which is recalled below, is far from being the "best possible".
Theorem (cf. Theorem 3.1 of [S]). Suppose that F has envelope F F c LP(A,A,P) for some p> 2 . Suppose that there exists y , 0