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Stochastic expansions in an overcomplete wavelet dictionary . Probability Theory & Related

Stochastic Expansions in an Overcomplete Wavelet Dictionary
F. Abramovich, T. Sapatinas and B. W. Silverman
Tel Aviv University, University of Kent at Canterbury, University of Bristol.

March 23, 1999

Abstract We consider random functions de?ned in terms of members of an overcomplete wavelet dictionary. The function is modelled as a sum of wavelet components at arbitrary positions and scales where the locations of the wavelet components and the magnitudes of their coef?cients are chosen with respect to a marked Poisson process model. The relationships between the parameters of the model and the parameters of those Besov spaces within which realizations will fall are investigated. The models allow functions with speci?ed regularity properties to be generated. They can potentially be used as priors in a Bayesian approach to curve estimation, extending current standard wavelet methods to be free from the dyadic positions and scales of the basis functions. Keywords: B ESOV S PACES ; C ONTINUOUS WAVELET T RANSFORM ; OVER COMPLETE WAVELET D ICTIONARIES ; P OISSON P ROCESSES.

1 Introduction
1.1 Background
Wavelets have recently been of great interest in various statistical areas such as nonparametric regression, density estimation, inverse problems, change point problems, and time series analysis. Surveys of wavelet applications in these and other related
Address for correspondence: Institute of Mathematics and Statistics, University of Kent at Canterbury, Canterbury, Kent CT2 7NF, United Kingdom. Email: T.Sapatinas@ukc.ac.uk

1

statistical areas can be found, for example, in Ogden (1997), H¨ rdle, Kerkyachara ian, Picard & Tsybakov (1998), Abramovich, Bailey & Sapatinas (1999), Antoniadis (1999), Silverman (1999) and Vidakovic (1999). An interesting development, motivated by Bayesian approaches to curve estimation, is the modelling of a function as an orthonormal wavelet expansion with random coef?cients. Abramovich, Sapatinas & Silverman (1998) considered such models in detail, and studied the Besov regularity properties of the functions produced by the models. They consider the application of the models in a Bayesian context, and also give references to related work by other authors; the results obtained have generally been very encouraging.

1.2 Abandoning dyadic constraints
Orthonormal wavelet bases have the disadvantage that the positions and the scales of the basis functions are subject to dyadic constraints. In order to avoid these constraints, this paper considers random functions de?ned by expansions in a continuous wavelet dictionary, where functions are built up from wavelet components that may have arbitrary positions and scales. The models provide a constructive method of simulating functions with varying degrees of regularity and spatial homogeneity, and our results give the explicit regularity properties of the functions thus produced, in terms of Besov spaces. Some users might wish to be able to simulate or construct functions with speci?c Besov parameters in mind. Others might wish to use the models to gain intuition about the meaning of the Besov parameters, by generating functions that lie just inside and just outside particular Besov spaces. The models lay open the possibility of building a Bayesian curve estimation approach with the advantages of standard wavelet methods, in that inhomogeneous functions can be modelled under the prior, but without the arti?cial dyadic constraints on the positions and scales of the basis functions. The improvement to standard wavelet thresholding methods obtained by moving from the discrete (decimated) wavelet transform to the non-decimated wavelet transform (see, for example, Coifman & Donoho, 1995; Nason & Silverman, 1995; Johnstone & Silverman, 1997) suggests that a Bayesian approach freed from dyadic positions and scales may result in yet better wavelet shrinkage estimators. The algorithmic details, probably involving modern Bayesian computational methods, have yet to be worked out in detail, and this is an interesting topic for further research.

1.3 Models in continuous wavelet dictionaries
For simplicity of exposition we work with functions periodic on 0; 1]. Suppose that and are the compact-support scaling function and mother wavelet respectively that correspond to an r-regular multiresolution analysis, for some integer r > 0 (see, for example, Daubechies, 1992). Take a0 = 2j0 , for some integer j0 , such that a0 is at least 2

twice the length of the support of . For indices = (a; b) with a > a0 and 0 < b < 1 we de?ne (t) = a1=2 (a(t ? b)) wrapping periodically if necessary. We model our function as the sum of a ‘coarse-scale’ function f0 and a ‘?ne-scale’ function f . The function f0 is given by

f0 (t) =

M X i=1

i

i

(t)

(1)

a0 , and some real for some ?nite set of indices (ai ; bi); i = 1; 2; : : : ; M , with ai numbers i . Here has an analogous de?nition to . The function f is generated by a stochastic mechanism and is given by f (t) =

X
2S

!

(t):

(2)

The locations of the wavelet components and the magnitudes of their coef?cients are chosen with respect to a marked Poisson process model. Speci?cally, the set of indices = (a; b) is sampled from a Poisson process S on a0; 1) 0; 1] with intensity ( ). Conditional on S , the wavelet coef?cients ! are assumed to be independent normal random variables ! j S N (0; 2( )): (3) It is assumed that the variance 2 ( ) and the intensity ( ) depend on the scale a only, and are of the form
a

2 / a? > 0.

and

a

/ a? ;

a

1;

(4)

where ,

0, with +

The stochastic wavelet expansions we consider allow intuitive notions about the functions genuinely to be modelled. The parameter controls the relative rarity of ?nescale wavelet components in the function, while the parameter controls the size of the contribution of these components when they appear. For example, if is small and is large, there will be a considerable number of ?ne-scale components but these will each have fairly low contribution, so one might expect the functions to be reasonably smooth and homogeneous. On the other hand, if is large and is small, there will be occasional large ?ne-scale effects in the functions. In the remainder of the paper, we investigate the regularity properties of the random functions generated by the proposed model. We reveal relations between the parameters and of the model and the parameters of those Besov spaces within which realizations from the model will fall.

3

2 Regularity properties of the random functions
An important tool in our argument will be the equivalence between the Besov norm of the function f on 0; 1] and the corresponding sequence norm of its orthonormal wavelet coef?cients. For details of Besov spaces see, for example, Meyer (1992, Chapter 6), H¨ rdle, Kerkyacharian, Picard & Tsybakov (1998, Chapter 9). a

array w by

hf; jk i; 0 hf; j0k i; 0

For j

j0 , de?ne wj to be the vector of orthonormal wavelet coef?cients wjk = k 2j ? 1. De?ne also the vector uj0 to have elements uj0k = k 2j0 ? 1. Let s0 = s + 1=2 ? 1=p and de?ne the norm of the

jjwjjb jjwjjb

s p;q

=

(1 X
j =j0

s p;1

= sup 2js jjwj jjp
0

n

2 jj jj
js0 q

)1=q
; :

wj q p

o

1

q < 1;

Then, for 0 < s < r, 1 1998, Theorem 2)

s 1, the Besov norm jjf jjBp;q on 0; 1] is equivalent to the sequence space norm jjuj0 jjp + jjwjjb (see, for example, Donoho & Johnstone,

j j0

p; q

s p;q

Because f0 given by (1) is a ?nite linear combination of functions , it will belong to the same Besov spaces as the scaling functions, including all those for which 0 < s < r. For these parameter values, we consider in detail the necessary and suf?cient conditions for f given by (2) to fall (with probability one) in any particular Besov space. Theorem 1 Let and be the compact-support scaling function and mother wavelet respectively that correspond to an r-regular multiresolution analysis. Consider a func2 tion f as de?ned in (2), with the conditional variances a / a? and the intensity of the Poisson process a / a? . Assume that 0, 0 1, and that + > 0: Assume also that (and hence ) are suf?ciently regular that < 2r + 2 ? 1, where 2 (0; 1) is the exponent of H¨ lder continuity of the r-th derivative of and . Then o

f
if and only if

s 2 Bp;q

almost surely

0 < s < r; 1

p; q

1

s + 1=2 ? =p ? =2 < 0 if 1 p < 1 s + 1=2 ? =2 < 0 if p = 1:

Proof. De?ne p = 1.

= s + 1=2 ?

=p ? =2 if

1

p<

1 and = s + 1=2 ? =2 if
= (2j ; 2?j k). For resolution
p<

We ?rst prove the suf?cient part. Consider the case 1 thonormal wavelet coef?cients wjk = hf; jk i and set jk 4

1. Consider the or-

and spatial indices j and k with j have wjk where K ( ; 0 ) = h

j0 and k

=

X
2S

= 0; 1; : : : ; 2j ? 1 respectively, we then
jk

K( ;

)!

;

(5)

;

0

i:

In the particular case where 0 = jk , we have K ( ; jk ) = K0 (2?j a; 2j b ? k ). Let L ; U ] be the support of the mother wavelet . Then K0 (u; v ) = h ; uv i 6= 0 only if L ? U =u v U ? L =u: (7) In what follows we use C to denote a generic positive constant, not necessarily the same each time it is used. We have, from Daubechies (1992, p. 48),

We now explore some properties of the reproducing kernel K that we use in subR 2 sequent calculations. Firstly, since = 1 for all , we always have K 2( ; 0 ) 1: Now de?ne K0 (u; v ) = h ; uv i: Let = (a; b) and 0 = (a0; b0). Simple calculus shows that K ( ; 0 ) = K0 (a=a0; a0 (b ? b0 )): (6)

jK0(u; v)j

Cu?(r+ +1=2); uniformly in u

1: = 1:

(8)

For u < 1, apply the symmetry of K to show that K0 (u; v ) hence jK0(u; v)j Cu(r+ +1=2); uniformly in u

K0 (1=u; ?uv ) and
(9)

Suppose that < 0. It follows from (5) that, conditionally on S , the distribution of wjk is normal with mean zero and variance
jk

2 (S ) =

X
2S

K 2( ;

jk

)a? ;

(10)

where, as usual, = (a; b): The unconditional distribution of wjk will have ?nite 2 variance if the expectation of jk (S ) over S is ?nite. If the sum in (10) is in?nite for a particular S , then, conditionally on S , the sum de?ning wjk cannot converge, because it will not converge in distribution. More generally, for p > 0, the pth absolute moment of wjk will be given by

E jwjk j
where
p

p

=

p ES

( X
2S

K 2( ;

jk

)a?

)p=2

;

(11)

is the pth absolute moment of the standard normal distribution. 0 Let Tjk be the set a0; 1) (2?j k ? 1=2; 2?j k + 1=2): By the de?nition of a0 and 0 j0 , we may restrict attention in the sum (10) to S \ Tjk since the support of any with a > a0 will be of length at most 1 , and so the terms excluded by restricting the 2 5

0 sum will all be zero. Now let Sj be a Poisson process on the half-plane f(u; v ) : u > 0; ?1 < v < 1g of intensity 2?j u? . De?ne the set Tj = 2?j ; 1) ?2j?1; 2j?1 ]: Consider the transformation of = (a; b) given by (u; v ) = (2?j a; 2j b ? k ). Applied 0 0 to the process S \ Tjk this gives a process with the same distribution as Sj \ Tj : In addition, for each , we have from (6) that K ( ; jk ) = K0 (u; v ): It follows that 8 9p=2 < X = 2 E jwjk jp = p 2?j p=2 E K0 (u; v )u? :(u;v)2S \Tj ; j 8 9p=2 < X = ?j p=2 E 2(u; v )u? (12) p2 :(u;v)2S K0 ; : j
0 0

To obtain a bound on the expectation in (12), de?ne the random sum

Zj

=

X

(u;v)2Sj
0

2 K0 (u; v )u? :

(13)

2 The bounds (8) and (9) imply that K0 (u; v )u? is bounded by Cu2r+2 +1? for 0 < ?2r?2 ?1? for u 1; hence it is uniformly bounded for all u and v . u 1, and by Cu p=2 We now apply Corollary 1 in the Appendix to investigate the behaviour of EZj . To verify the ?niteness of the ?rst integral in the corollary, it will be suf?cient to have ?niteness of the integral

Z 1Z 1
0

?1

2 K0 (u; v )u? ? dv du:

(14)

The bounds (7) on the support of the integrand, and those stated above on its order of magnitude, allow (14) to be dominated by Z1 Z1 2r+2 +1? ? (1 + 1=u)du + C C u u?2r?2 ?1? ? (1 + 1=u)du: 0 1 The assumptions of the theorem guarantees the ?niteness of the ?rst integral, while the second integral is clearly ?nite. Now verify the ?niteness of the second integral in the corollary. By similar arguments to those just used,

Z 1Z 1
0 C

?1

Z1

fK02(u; v)u? gp=2u? dv du
u(2r+2 +1? )p=2? ?1du + C

Z1

0 1 The assumptions of the theorem about the regularity of the wavelets imply easily that these integrals are both ?nite. It now follows from Corollary 1 in the Appendix that, for each ?xed p, p=2 EZj = C 2?j + o(2?j ) as j ! 1: (15)
6

u?(2r+2 +1+ )p=2? du:

Now de?ne wj to be the vector with elements wjk for k = 0; :::; 2j (15) into (11) gives E kwj kp C 2?j ( p=2+ ?1) for all j . p By Jensen’s inequality and (16), we have

? 1. Substituting
(16)

E jjwjjbs 1 p;
which is ?nite if

=

1 X
j =j0

2

js0

E jjwj jjp

1 X
j =j0

2 (E jj jj
js0

wj p p

)1=p

C

1 X
j =j0

2j

;

To complete the proof we use similar methods to show that the norm jjuj0 jjp is ?nite almost surely. For ?xed k , let T 0 be the range of indices = (a; b) with a > a0 = 2j0 for which the support of overlaps that of j0 k for some ?xed k . Then

< 0. Thus, if

< 0, jjwjjbs 1 is ?nite almost surely. p;

uj0 k

=

X

2S \T

W( ;
0

j0 k

)!

;

(17)

where L ; U ] and L ; U ] are the supports of the scaling function and the mother wavelet respectively. From Daubechies (1992, p. 48) we have again

where W ( ; j0 k ) = h ; j0 k i = W0 (2?j0 a; 2j0 (b ? 2?j0 k )) and W0 (u; v ) = h ; uv i. Note that for 2 S \ T 0, u 1. One can easily verify that W0 (u; v ) = h ; uv i 6= 0 only if L ? U =u v U ? L =u; (18)

jW0(u; v)j

Cu?(r+ +1=2); uniformly in u

1:

(19)

Exploiting (18), (19) and the fact that u 1, the same techniques used for wavelet coef?cients !jk will imply E jjuj0 jjp cj0 < 1 for any ?xed j0 and, hence, jjuj0 jjp is p always ?nite almost surely. By the equivalence of the Besov space and sequence norms, we conclude that if s < 0, jjf jjBp;1 is ?nite almost surely, and therefore, by the embedding properties of Besov spaces (see, for example, H¨ rdle, Kerkyacharian, Picard & Tsybakov, 1998, a s almost surely for all 1 Corollary 9.2, p. 124), f 2 Bp;q q 1, completing the proof for this case. Consider now the case p = 1. For any positive and c, Markov’s inequality implies that P (jwjk j c) 2e? c E (e wjk ): (20) To evaluate the expectation, we use the standard expression for the moment generating function of a normal distribution and Campbell’s theorem (see, for example, Kingman, 1993, p. 28) applied to the random sum (10) to obtain

log E (e wjk ) = log E fE (e wjk jS )g = log E expf
7

2 2(S )=2g

= =

Z

0 Tjk Z ?j 2 Tj

expf

2 K 2( ;

jk

)a? =2g ? 1]a? d

expf

2 K 2 (u; v )2?1? j u? 0

g ? 1]u? du dv

2 by the usual change of variable. Now let M = supfK0 (u; v )u? g, which was shown earlier to be ?nite. Extending the integral to the whole of the half-plane u > 0, we have, by the convexity of the exponential function,

log E (e

wjk

)

C 2? C 2?

j j

exp(2?1?

j

2M )

Z 1Z 1
0

exp(M 22?1? j ):

?1

2 K0 (u; v )u? ? dv du

Suppose 2j =2 c > 1. To obtain a bound for P (jwjk j > c), choose 2 = M ?1 21+ and substitute into (20) to obtain, for positive constants C1 and C2 ,

j

log(2j =2c),

log P (jwjk j > c) log 2 + C12( =2? )j c ? C22 j=2c logf2j =2cg:
Both s0 = s + 1=2 and are positive by the hypotheses of the theorem. Choose such that 0 < < and set c = 2?(s + )j . Then 2j =2 c = 2( ? )j > 1: We now have
0

q

log P (2(s + )j jwjk j > 1)
0

log 2 + C12( ? ? )j ? C22( ? )j log 2( ? )j ?2( ? )j ;

p

for suf?ciently large j . Since wj is of length 2j , it follows that, for suf?ciently large j ,

P (2s j jjwj jj1 > 2? j ) < 2j exp(?2( ? )j ):
0

This very rapidly decreasing bound on the tail probabilities implies that, with probability one, the sequence 2s j jjwj jj1 is bounded by a multiple of 2? j and hence its sum is convergent almost surely. The same arguments used in the case of ?nite p for scaling coef?cients show that jjuj0 jj1 is ?nite almost surely.
0

We now prove the necessary part. Noting that the function K0 (u; v ) is continuous and that K0 (1; 0) = 1, choose c0 with 0 < c0 < 1 such that K0 (u; v ) > 1=2 for all (u; v) with 1 u 1 + c0 and 0 v c0. For j j0 and k = 0; 1; : : : ; 2j ? 1, de?ne the nonoverlapping rectangles Ijk in the range of indices as Ijk = 2j ; 2j (1 + c0 )] 2?j k; 2?j (k + c0 )].

s By the equivalence of norms, we conclude that f 2 B1;1 almost surely and, therefore, by the embedding properties of Besov spaces (see, for example, H¨ rdle, Kerkya s acharian, Picard & Tsybakov, 1998, Corollary 9.2, p. 124), f 2 B1;q almost surely for all 1 q 1, completing the proof for this case, and hence we have the suf?ciency.

8

Using (4), the expected number of wavelet components falling within Ijk is then ? ?j Ijk a db da = c12 for some c1 > 0, and hence the probability that there is one or more wavelet components in Ijk is at least c2 2? j for some c2 > 0. Now de?ne

R

0 wjk

=

X

2S \Ijk

K( ;

jk

)!

0 and observe that the wjk are independent because the Ijk are disjoint. It follows from (6) that K ( ; jk ) > 1=2 for in Ijk . Note also that from (3) and (4), Var(! j 2 S \ Ijk ) 4c3 2?j for some c3 , so that Var(K ( ; jk )! j 2 S \ Ijk ) c3 2?j . For j j0 and k = 0; 1; : : : ; 2j ? 1, now de?ne independent random variables w0
jk

to have the mixture distribution

0 wjk

jN

(0; j2) + (1 ? j ) (0);

where j = c2 2? j and j2 = c32?j . For the orthonormal wavelet coef?cients wjk = 0 hf; jk i, it is obvious that jwjk j is stochastically larger than jwjk j which in turn is 0 stochastically larger than jwjk j. Hence, stochastically jjwjjbs jjw0jjbsp;q jjw0jjbsp;q p;q for any 0 < s < r, 1 p; q 1. The results of Abramovich, Sapatinas & Silverman (1998) for independent or0 thonormal wavelet coef?cients wjk show that almost surely ?niteness of jjw0 jjbs imp;q plies < 0, completing the proof of the necessity, and hence we have the theorem.

3 Concluding remarks
Theorem 1 places an upper bound restriction on the value of . In the case > 1, the intensity a / a? is integrable over the range of for which has support intersecting 0; 1]. Therefore, the number of relevant terms in the stochastic expansion of f is ?nite almost surely. With probability one, f will belong to the same Besov spaces as the mother wavelet , namely those for which 0 < s < r, 1 p 1, 1 q 1. The key conclusion of Theorem 1 we have proved is that, under suitable condis tions, the function f falls in Bp;q if + (2=p) exceeds 2s + 1. Since the ?ne-scale content of model functions depends both on the intensity of ?ne-scale components and on their size, it is not surprising that the smoothness as measured by the parameter s should depend on both parameters. The parameter p can be seen as discouraging inhomogeneity, in that the larger the value of p the more emphasis is placed on the parameter . For large , no matter how many ?ne-scale components there are, they each make a relatively low contribution. On the other hand, if p is small, then there is a trade-off where large weights on ?ne-scale components (small ) can be tolerated if the corresponding components are relatively rare (large ). 9

The constraints placed on s in the statistical literature, for example in the optimality results of Donoho & Johnstone (1998), are often stronger than those we have assumed. Typical conditions are max (0; 1=p ? 1=2) < s < r or 1=p < s < r. These constraints ensure that the Besov spaces are function spaces rather than spaces of more general distributions (see, for example, Meyer, 1992, Chapter 6).

Acknowledgements
We are delighted to thank Boris Tsirelson for fruitful discussions. The ?nancial support of the Engineering and Physical Sciences Research Council, the Israel Academy of Science, and the Royal Society are gratefully acknowledged. During periods of the work TS was a Research Associate in the School of Mathematics, University of Bristol, supported by EPSRC grant GR/K70236, and BWS was a Fellow at the Center for Advanced Study in the Behavioral Sciences, Stanford, partly supported by NSF grant SBR-9601236. Helpful comments of Peter Hall and three referees are gratefully acknowledged.

Appendix: Moments of sums of thinned Poisson processes
In this appendix we prove a lemma and corollary used in the proof of Theorem 1. They are of interest in their own right. Lemma 1 Let be a measure on and let measure " , where " > 0. Assume that

R

S" be a Poisson Process on R with intensity

Z1

?1

min (1; jxj) (dx) < 1 =

and

De?ne Y"

P

cl =

Z1
?1

jxjl (dx) < 1

for some l > 0:

(21)

X 2S"

X: Then E jY"jl = " cl + o(") as " ! 0:
(22)

Proof. Applying Campbell’s Theorem (Kingman, 1993, p. 28), condition (21) shows that the sum de?ning Y" is absolutely convergent with probability one. For any > 0, de?ne B = n ? ; ]. It follows from (21) that (B ) < 1; de?ne F ( ) = (B ). Now choose < 1 to depend on " in such a way that ! 0 and "F ( ) ! 0 as " ! 0. The dependence of on " will not be expressed explicitly. Now de?ne

R

Y"(1) =

X T X 2S" B

X

and

Y"(2) =

X

X 2S"

T? ; ]

X:

10

Consider, ?rst, the asymptotic behaviour of Y" . The number of X in S" ("F ( )) random variable and so

(1)

TB

is a Poisson

E jY (1)jl
"

j "j F ( )j E j X X jl; = exp(?"F ( )) j ! i i=1 j =1

1 X

(23)

where X1 ; X2; : : : are independent and identically distributed random variables on B with (1) distribution =F ( ). Let cl = B jxjl (dx). For j 2, we consider a bound for the expectation in (23). For l 1, we immediately see that

R

E (j

j X i=1

Xij

l

jc(1) : E jXij = jE jXij = F (l ) i=1
l l l

j X

(24)

For l > 1, using Jensen’s inequality, we have

Ej

j X i=1

Xi j

j lc(1) : j E jXij = F (l ) i=1
l?1 l

j X

(25)

Hence, in either (24) or (25), we have from (23), separating the terms for j

= 1 and j > 1,
(26)

E jY"(1)jl = " c(1) exp(?"F ( )) + R(1); " l
where

R(1)
"

"j F ( )j j (l^1)c(1) l exp(?"F ( )) j ! F( ) j =1 1 k k (l?1)+ X = "2 c(1)F ( ) exp(?"F ( )) " Fk(! ) (k + 2) 1) : l (k + k=0

1 X

(27)

As " ! 0, the sum in (27) is a Poisson expectation that converges to 2(l?1)+ , since "F ( (1) It follows that R" = o(") and hence, from (26), that, as " ! 0:

) ! 0.
(28)

"?1E jY"(1)jl ! cl;
using the facts that cl

(1) ! c and "F ( l

) ! 0.

Now consider the asymptotic behaviour of Y" . For l to the Poisson process S" ? ; ], we have

T

(2)

1, by Campbell’s theorem applied

0 X E jY"(2)jl E @ T
X 2S"

?;]

1l jX jA E

X
X 2S"

T?;]

jX j = "
l

Z
?

jxjl (dx):

(29)

Therefore, we have from (29), as ! 0, that E jY" jl = o("). Since jY" jl it follows from (28) that E jY"jl = " cl + o(") as " ! 0:

(2)

jY"(1)jl + jY"(2)jl;

11

> 1. De?ne Z" = X 2S" T ? ; ] jX j. For ? x , we ?1 (e ? 1)jxj, and so, using (21), I = R (ejxj ? 1) (dx) < 1: It then have 0 ejxj ? 1 ? follows (using equation (3.17) of Kingman, 1993), that E exp(Z" ) = exp("I ): Since l > 1, there exists a constant kl such that z l kl (ez ? 1) for all z 0. We then have
Finally, consider the case l

P

EZ"l kl(E exp(Z") ? 1) = kl(exp("I ) ? 1) = o(") as " ! 0;
since I

(30)

! 0 as ! 0. Therefore, from (30), it follows at once that E jY"(2)jl = o(") as " ! 0: Using Minkowski’s inequality applied to the norm jjX jjl = (E jX jl)1=l we have: (E jY"(1)jl )1=l ? (E jY"(2)jl )1=l (E jY"jl)1=l (E jY"(1)jl )1=l + (E jY"(2)jl )1=l
(2) (1)

and hence, since "?1 E jY" jl ! 0, the limiting values of "?1 E jY" jl and "?1 E jY" jl are the same. It follows from (28) that "?1 E jY"jl ! cl , which gives (22), completing the proof of the lemma. Corollary 1 Let be a measure on a set , and let g be a real-valued function on . Let S" be a Poisson Process on with intensity measure " , where " > 0. Assume that the induced , and assume that measure (g ?1(A)) is non-atomic for every measurable set A

Z

min(1; jg (x)j) (dx) < 1 =

and

De?ne Y"

P

cl =

Z

R

jg(x)jl (dx) < 1

for some l > 0:

X 2S"

g(X ). Then E jY"jl = " ul + o(") as " ! 0:

Proof. De?ne a measure on by g (A) = (g ?1(A)) and let Z = g (X ). Then, appealing to the Mapping theorem for Poisson processes (see, for example, Kingman, 1993, p. 18), Z is a Poisson process on with intensity measure " g and Y" = X 2S" Z . The proof of the corollary is completed by applying Lemma 1 to Z and g .

R

R

P

References
[1] Abramovich, F., Bailey, T.C. & Sapatinas, T. (1999). Wavelet analysis and its statistical applications. J. R. Statist. Soc. D, 48, 000–000. [2] Abramovich, F., Sapatinas, T. & Silverman, B.W. (1998). Wavelet thresholding via a Bayesian approach. J. R. Statist. Soc. B, 60, 725–749. [3] Antoniadis, A. (1999). Wavelets in statistics: a review (with discussion). J. It. Statist. Soc., 000-000.

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[4] Coifman, R.R. & Donoho, D.L. (1995). Translation-invariant de-noising. In Wavelets and Statistics, Lecture Notes in Statistics 103, Antoniadis, A. and Oppenheim, G. (Eds.), pp. 125-150, New York: Springer-Verlag. [5] Daubechies, I. (1992). Ten Lectures on Wavelets. Philadelphia: SIAM. [6] Donoho, D.L. & Johnstone, I.M (1998). Minimax estimation via wavelet shrinkage. Ann. Statist., 26, 879-921. [7] Johnstone, I.M. & Silverman, B.W. (1997). Wavelet threshold estimators for data with correlated noise. J. R. Statist. Soc. B, 59, 319-351. [8] H¨ rdle, W., Kerkyacharian, G., Picard, D. & Tsybakov, A. (1998). Wavelets, Apa proximation, and Statistical Applications. Lecture Notes in Statistics 129, New York: Springer-Verlag. [9] Kingman, J.F.C. (1993). Poisson Processes. Oxford: Oxford University Press. [10] Meyer, Y. (1992). Wavelets and Operators. Cambridge: Cambridge University Press. [11] Nason, G.P. & Silverman, B.W. (1995). The stationary wavelet transform and some statistical applications. In Wavelets and Statistics, Lecture Notes in Statistics 103, Antoniadis, A. and Oppenheim, G. (Eds.), pp. 281-300, New York: SpringerVerlag. [12] Ogden, T.R. (1997). Essential Wavelets for Statistical Applications and Data Analysis. Boston: Birkh¨ user. a [13] Silverman, B.W. (1999). Wavelets in statistics: beyond the standard assumptions. Phil. Trans. Roy. Soc. Lond. A, 357, 000–000. [14] Vidakovic, B. (1999). Statistical Modeling by Wavelets. New York: John Wiley & Sons.

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