Adaptive global thresholding on the sphere
Claudio Durastantia,1

arXiv:1601.02844v2 [] 26 Jul 2016

a Fakultät

für Matematik, Ruhr Universität, Bochum

Abstract
This work is concerned with the study of the adaptivity properties of nonparametric regression estimators over the d-dimensional sphere within the global
thresholding framework. The estimators are constructed by means of a form
of spherical wavelets, the so-called needlets, which enjoy strong concentration
properties in both harmonic and real domains. The author establishes the convergence rates of the Lp -risks of these estimators, focussing on their minimax
properties and proving their optimality over a scale of nonparametric regularity
function spaces, namely, the Besov spaces.
Keywords: Global thresholding, needlets, spherical data, nonparametric
regression, U-statistics, Besov spaces, adaptivity.
2010 MSC: 62G08, 62G20, 65T60

1. Introduction
The purpose of this paper is to establish adaptivity for the Lp -risk of regression function estimators in the nonparametric setting over the d-dimensional
sphere Sd . The optimality of the Lp risk is established by means of global
thresholding techniques and spherical wavelets known as needlets.
Let (X1 , Y1 ), . . . , (Xn , Yn ) be independent pairs of random variables such
that, for each i ∈ {1, . . . , n}, Xi ∈ Sd and Yi ∈ R. The random variables
X1 , . . . , Xn are assumed to be mutually independent and uniformly distributed
locations on the sphere. It is further assumed that, for each i ∈ {1, . . . , n},
Yi = f (Xi ) + εi ,

(1)

where f : Sd 7→ R is an unknown bounded function, i.e., there exists M > 0
such that
sup |f (x)| ≤ M < ∞.
(2)
x∈Sd

Moreover, the random variables 1 , . . . , n in Eq. (1) are assumed to be mutually
independent and identically distributed with zero mean. Roughly speaking,

Email address: claudio.durastanti@gmail.com (Claudio Durastanti)
author is supported by Deutsche Forschungsgemeinschaft (DFG) - GRK 2131, “Highdimensional Phenomena in Probability — Fluctuations and Discontinuity”.
1 The

Preprint submitted to Journal of Multivariate Analysis

February 19, 2018

they can be viewed as the observational errors and in what follows, they will be
assumed to be sub-Gaussian.
In this paper, we study the properties of nonlinear global hard thresholding
estimators, in order to establish the optimal rates of convergence of Lp -risks for
functions belonging to the so-called Besov spaces.
1.1. An overview of the literature
In recent years, the issue of minimax estimation in nonparametric settings
has received considerable attention in the statistical inference literature. The
seminal contribution in this area is due to Donoho et al. [7]. In this paper, the
authors provide nonlinear wavelet estimators for density functions on R, lying
over a wide nonparametric regularity function class, which attain optimal rates
of convergence up to a logarithmic factor. Following this work, the interaction
between wavelet systems and nonparametric function estimation has led to a
considerable amount of developments, mainly in the standard Euclidean framework; see, e.g., [3, 5, 24, 26, 27, 28, 30] and the textbooks [22, 44] for further
details and discussions.
More recently, thresholding methods have been applied to broader settings.
In particular, nonparametric estimation results have been achieved on Sd by using a second generation wavelet system, namely, the spherical needlets. Needlets
were introduced by Narcowich et al. [39, 40], while their stochastic properties
dealing with various applications to spherical random fields were examined in
[2, 6, 34, 35, 36]. Needlet-like constructions were also established over more
general manifolds by Geller and Mayeli [18, 19, 20, 21], Kerkyacharian et al.
[25] and Pesenson [41] among others, and over spin fiber bundles by Geller and
Marinucci [16, 17].
In the nonparametric setting, needlets have found various applications on
directional statistics. Baldi et al. [1] established minimax rates of convergence
for the Lp -risk of nonlinear needlet density estimators within the hard local
thresholding paradigm, while analogous results concerning regression function
estimation were established by Monnier [38]. The block thresholding framework
was investigated in Durastanti [9]. Furthermore, the adaptivity of nonparametric regression estimators of spin function was studied in Durastanti et al. [10].
In this case, the regression function takes as its values algebraical curves lying
on the tangent plane for each point on S2 and the wavelets used are the so-called
spin (pure and mixed) needlets; see Geller and Marinucci [16, 17].
The asymptotic properties of other estimators for spherical data, not concerning the needlet framework, were investigated by Kim and Koo [31, 32, 33],
while needlet-like nearly-tight frames were used in Durastanti [8] to establish
the asymptotic properties of density function estimators on the circle. Finally,
in Gautier and Le Pennec [15], the adaptive estimation by needlet thresholding
was introduced in the nonparametric random coefficients binary choice model.
Regarding the applications of these methods in practical scenarios, see, e.g.,
[13, 14, 23], where they were fruitfully applied to some astrophysical problems,
concerning, for instance, high-energy cosmic rays and Gamma rays.

2

1.2. Main results
Consider the regression model given in Eq. (1) and let {ψj,k : j ≥ 0, k =
1, . . . , Kj } be the set of d-dimensional spherical needlets. Roughly speaking,
j and Kj denote the resolution level j and the cardinality of needlets at the
resolution level j, respectively. The regression function f can be rewritten in
terms of its needlet expansion. Namely, for all x ∈ Sd , one has
f (x) =

Kj
XX

βj,k ψj,k (x) ,

j≥0 k=1

where {βj,k : j ≥ 0, k = 1, . . . , Kj } is the set of needlet coefficients.
For each j ≥ 0 and k ∈ {1, . . . , Kj }, a natural unbiased estimator for βj,k is
given by the corresponding empirical needlet coefficient, viz.
n

1X
Yi ψj,k (Xi ) ;
βbj,k =
n i=1

(3)

see, e.g., Baldi et al. [1] and Härdle et al. [22]. Therefore, the global thresholding
needlet estimator of f is given, for each x ∈ Sd , by
fˆn (x) =

Jn
X
j=0

KJn

τj

X

βbj,k ψj,k (x) ,

(4)

k=1

where τj is a nonlinear threshold function comparing the given j-dependent
b j (p), built on a subsample of p < n observations, to a threshold based
statistic Θ
b j (p) is above the threshold, the whole
on the observational sample size. If Θ
j-level is kept; otherwise it is discarded.
Loosely speaking, this procedure allows one to delete the coefficients corresponding to a resolution level j whose contribution to the reconstruction of the
regression function f is not clearly distinguishable from the noise. Following
Kerkyacharian et al. [30], we consider the so-called hard thresholding framework, defined as
τj = τj (p) = 1{Θ̂j (p) ≥ B dj n−p/2 },
where p ∈ N is even. Further details regarding the statistic Θ̂j (p) will be
discussed in Section 3.4, where the choice of the threshold B dj n−p/2 will also be
motivated.
b j (p) as an unbiased statistic of
For the rest of this section, we consider Θ
p
p
|βj,1 | + · · · + |βj,Kj | . The so-called truncation bandwidth Jn , on the other
hand, is the higher frequency on which the empirical coefficients β̂j,1 , . . . , β̂j,Kj
are computed. The optimal choice of the truncation level is Jn = lnB (n1/d );
for details, see Section 3. This allows the error due to the approximation of f ,
which is an infinite sum with respect to j, to be controlled by a finite sum, such
as the estimator fˆn .

3

Our objective is to estimate the global error measure for the regression estimator fˆn . For this reason, we study the worst possible performance over a
so-called nonparametric regularity class {Fα : α ∈ A} of function spaces of the
Lp -risk, i.e.,



Rn fˆn ; Fα = sup E kfˆn − f kpLp (Sd ) .
f ∈Fα

Recall that an estimator fˆn is said to be adaptive for the Lp -risk and for the
scale of classes {Fα : α ∈ A} if, for every α ∈ A, there exists a constant cα > 0
such that



E kfˆn − f kpLp (Sd ) ≤ cα Rn fˆn ; Fα ;
see, e.g., [1, 22, 30].
For r > 0 and for p ∈ [1, r], we will establish that the regression estimator
s
, where 1 ≤ q ≤ ∞ and d/p ≤
fˆn is adaptive for the class of Besov spaces Bp,q
s < r + 1. Finally, let R ∈ (0, ∞) be the radius of the Besov ball on which f
is defined. The proper choice of r will be motivated in Section 2.1. Our main
result is described by the following theorem.
Theorem 1.1. Given r ∈ (1, ∞), let p ∈ [1, r]. Also, let fˆn be given by Eq. (4),
with Jn = lnB n1/d . Then, for 1 ≤ q ≤ ∞, d/p ≤ s < r + 1 and 0 < R < ∞,
there exists C > 0 such that


−sp
sup E kfˆn − f kpLp (Sd ) ≤ Cn 2s+d .
s (R)
f ∈Br,q

The behavior of the L∞ -risk function will be studied separately in Section 3
and the analogous result is described in Theorem 3.2. Moreover, the details
concerning the choice of r will be presented in Remark 3.1 and other properties
of Lp -risk functions, such as optimality, will be discussed in Remark 3.3.
1.3. Comparison with other results
The bound given in Eq. (12) is consistent with the results of Kerkyacharian
et al. [30], where global thresholding techniques were introduced on R. As far
as nonparametric inference over spherical datasets is concerned, our results can
be viewed as an alternative proposal to the existing nonparametric regression
methods (see, e.g., [1, 9, 10, 38]), related to the local and block thresholding
procedures.
Recall that in local thresholding paradigm, each empirical estimator βbj,k is
compared to a threshold τj,k and it is, therefore, kept or discarded if its absolute
value is above or below τj,k respectively, i.e., the threshold function is given by
1{|βbj,k | ≥ τj,k }. Typically, the threshold is chosen such that τj,k = κ (ln n/n),
where κ depends explicitly on two parameters, namely, the radius R of the Besov
ball on which the function f is defined and its supremum M ; see, e.g., Baldi
et al. [1]. An alternative and partially data-driven choice for κ is proposed by
Monnier [38], i.e., here
n
κ0 X
2
κ=
ψj,k (Xi ) .
n i=1
4

Even if this stochastic approach is proved to outperform the deterministic one,
the threshold still depends on both R and M , which control κ0 . Also according
to the results established on R (see Härdle et al. [22]), local techniques entail
nearly optimality rates for the Lp -risks over a wide variety of regularity function
s
spaces. In this case, the regression function f belongs to Bp,q
(R), where s ≥ d/r,
p ∈ {1, ∞}, q ∈ {1, ∞} and 0 < R < ∞ (cf. [1, 10, 22]). However, these adaptive
rates of convergence are achieved on the expense of having an extra logarithmic
term and of requiring explicit knowledge of the radius of the Besov balls on
which f is defined, in order to establish an optimal threshold.
As far as the block thresholding is concerned, for any fixed resolution level
this procedure collects the coefficients β̂j,1 , . . . , β̂j,Kj into ` = ` (n) blocks denoted Bj,1 , . . . , Bj,` of dimension depending on the sample size. Each block is
then compared to a threshold and then it is retained or discarded. This method
has exact convergence rate (i.e., without the logarithmic extra term), although
it requires explicit knowledge of the Besov radius R. Furthermore, the estimator
is adaptive only over a narrower subset of the scale of Besov spaces, the so-called
regular zone; see Härdle et al. [22]. The construction of blocks on Sd can also
be a difficult procedure, as it requires a precise knowledge of the pixelization
of the sphere, namely, the structure of the subregions on which the sphere is
partitioned, in order to build spherical wavelets.
On the other hand, the global techniques presented in this paper do not
require any knowledge regarding the radius of Besov ball and have exact optimal
convergence rates even over the narrowest scale of regularity function spaces.
1.4. Plan of the paper
This paper is organized as follows. Section 2 presents some preliminary
results, such as the construction of spherical needlet frames on the sphere, Besov
spaces and their properties. In Section 3, we describe the statistical methods
we apply within the global thresholding paradigm. This section also includes an
introduction to the properties of the sub-Gaussian random variables and of the
b j (p), which are key for establishing the thresholding procedure.
U -statistic Θ
Section 4 provides some numerical evidence. Finally, the proofs of all of our
results are collected in Section 5.
2. Preliminaries
This section presents details concerning the construction of needlet frames,
the definition of spherical Besov spaces and their properties. In what is to follow
the main bibliographical references are [1, 2, 7, 21, 22, 24, 37, 39, 40].
2.1. Harmonic analysis on Sd and spherical needlets


Consider the simplified notation L2 Sd = L2 Sd , dx , where dx is the
uniform Lebesgue measure over Sd . Also, let H` be the restriction to Sd of

5

the harmonic homogeneous polynomials of degree `; see, e.g., Stein and Weiss
[43]. Thus, the following decomposition holds
2

L

d

S



=

∞
M

H` .

`=0

An orthonormal basis for H` is provided by the set of spherical harmonics
{Y`,m : m = 1, . . . , g`,d } of dimension g`,d given by


d−1
` + ηd ` + 2ηd − 1
, ηd =
.
g`,d =
ηd
`
2

For any function f ∈ L2 Sd , we define the Fourier coefficients as
Z
Y`,m (x) f (x) dx,
a`,m :=
Sd

such that the kernel operator denoting the orthogonal projection over H` is
given, for all x ∈ Sd , by
P`,d f (x) =

g`,d
X

a`,m Y`,m (x) .

m=1

Also, let the measure of the surface of Sd be given by
. d + 1
(d+1)/2
ωd = 2π
Γ
.
2
The kernel associated to the projector P`,d links spherical harmonics to the
(η )
Gegenbauer polynomial of parameter ηd and order `, labelled by C` q . Indeed,
the following summation formula holds
P`,d (x1 , x2 ) =

g`,d
X

Y`,m (x1 ) Y`,m x2 =

m=1

` + ηd (ηd )
C
(hx1 , x2 i) ,
ηd ωd `

where h·, ·i is the standard scalar product on Rd+1 ; see, e.g., Marinucci and
Peccati [37].
Following Narcowich et al. [40], K` = ⊕`i=0 Hi is the linear space of homogeneous polynomials on Sd of degree smaller or equal to `; see also [1, 37, 39].
Thus, there exist a set of positive cubature points Q` ∈ Sd and a set of cubature
weights {λξ }, indexed by ξ ∈ Q` , such that, for any f ∈ K` ,
Z
X
f (x) dx =
λξ f (ξ) .
Sd

ξ∈Q`

In the following, the notation a ≈ b denotes that there exist c1 , c2 > 0 such
that c1 b ≤ a ≤ c2 b. For a fixed resolution level j and a scale parameter B, let
6


Kj = card Q[2B j+1 ] . Therefore, {ξj,k : k = 1, . . . , Kj } is the set of cubature
points associated to the resolution level j, while {λj,k : k = 1, . . . , Kj } contains
the corresponding cubature weights. These are typically chosen such that
Kj ≈ B dj

and ∀k∈{1,...,Kj } λj,k ≈ B −dj .

Define the real-valued weight (or window) function b on (0, ∞) so that


(i) b lies on a compact support B −1 , B ;
P
2
j
(ii) the partitions of unity property holds, namely,
j≥0 b (`/B ) = 1, for
` ≥ B;
(iii) b ∈ C ρ (0, ∞) for some ρ ≥ 1.
Remark 2.1. Note that ρ can be either a positive integer or equal to ∞. In
the first case, the function b(·) can be built by means of a standard B-spline approach, using linear combinations of the so-called Bernstein polynomials, while
in the other case, it is constructed by means of integration of scaled exponential functions (see also Section 4). Further details can be found in the textbook
Marinucci and Peccati [37].
For any j ≥ 0 and k ∈ {1, . . . , Kj }, spherical needlets are defined as
X  ` 
p
P`,d (x, ξj,k ).
b
ψj,k (x) = λj,k
Bj
`≥0

Spherical needlets feature some important properties descending on the structure of the window function b. Using the compactness of the frequency domain,
it follows that ψj,k is different from zero only on a finite set of frequencies `, so
that we can rewrite the spherical needlets as
X  ` 
p
P`,d (x, ξj,k ),
ψj,k (x) = λj,k
b
Bj
`∈Λj




 

where Λj = u : u ∈ B j−1 , B j+1
and [u], u ∈ R, denotes the integer part
of u. From the partitions of unity property, the spherical needlets
form a tight

frame over Sd with unitary tightness constant. For f ∈ L2 Sd ,
2
kf kL2 (Sd )

=

Kj
XX

2

|βj,k | ,

j≥0 k=1

where

Z
βj,k =

f (x) ψj,k (x) dx,

(5)

Sd

are the so-called needlet coefficients. Therefore, we can define the following
reconstruction formula (holding in the L2 -sense): for all x ∈ Sd ,
f (x) =

Kj
XX
j≥0 k=1

7

βj,k ψj,k (x) .

From the differentiability of b, we obtain the following quasi-exponential localization property; for x ∈ Sd and any η ∈ N such that η ≤ ρ, there exists cη > 0
such that
cη B jd/2
,
(6)
|ψj,k (x)| ≤
jd/2
{1 + B
d (x, ξj,k )}η
where d (·, ·) denotes the geodesic distance over Sd .
Roughly speaking, |ψj,k (x)| ≈ B jd/2 if x belongs to the pixel of area B −dj
surrounding the cubature point ξj,k ; otherwise, it is almost negligible. The
localization result yields a similar boundedness property for the Lp -norm, which
is crucial for our purposes. In particular, for any p ∈ [ 1, ∞) there exist two
constants cp , Cp > 0 such that
cp B jd( 2 − p ) ≤ kψj,k kLp (Sd ) ≤ Cp B jd( 2 − p ) ,
1

1

1

1

(7)

and there exist two constants c∞ , C∞ > 0 such that
d

c∞ B j 2 ≤ kψj,k kL∞ (Sd ) ≤ C∞ B jd/2 .
According to Lemma 2 in Baldi et al. [1], the following two inequalities hold.
For every 0 < p ≤ ∞,
Kj
X

≤ cB jd( 2 − p ) kβj,k k`p ,
1

βj,k ψj,k

k=1

1

(8)

Lp (Sd )

and for every 1 ≤ p ≤ ∞,
kβj,k k`p B jd( 2 − p ) ≤ c kf kLp (Sd ) ,
1

1

where `p denotes the space of p-summable sequences. The generalization for the
case p = ∞ is trivial.
The following lemma presents a result based on the localization property.
Lemma 2.1. For x ∈ Sd , let ψj,k (x) be given by Eq. (2.1). Then, for q ≥ 2,
ki1 6= ki2 , for i1 6= i2 = 1, . . . , q, and for any η ≥ 2, there exists Cη > 0 such
that
Z Y
q
B dj(q−1)
ψj,ki (x) dx ≤
,
η(q−1)
Sd i=1
(1 + B dj ∆)
where
∆=

min

i1 ,i2 ∈{1,...,q},i1 6=i2

d(ξj,ki1 , ξj,ki2 ).

Remark 2.2. As discussed in Geller and Pesenson [21] and Kerkyacharian et
al. [25], needlet-like wavelets can be built over more general spaces, namely, over
compact manifolds. In particular, let {M, g} be a smooth compact homogeneous
manifold of dimension d, with no boundaries. For the sake of simplicity, we
assume that there exists a Laplace–Beltrami operator on M with respect to the
8

action g, labelled by ∆M . The set {γq : q ≥ 0} contains the eigenvalues of
∆M associated to the eigenfunctions {uq : q ≥ 0}, which are orthonormal with
respect to the Lebesgue measure over M and they form an orthonormal basis in
L2 (M); see [20, 21]. Every function f ∈ L2 (M) can be described in terms of
its harmonic coefficients, given by aq = hf, uq iL2 (M) , so that, for all x ∈ M,
X
aq uq (x) .
f (x) =
q≥1

Therefore, it is possible to define a wavelet system over {M, g} describing a
tight frame over M along the same lines as in Narcowich et al. [40] for Sd ; see
also [21, 25, 41] and the references therein, such as Geller and Mayeli [19, 20].
Here we just provide the definition of the needlet (scaling) function on M, given
by
j+1
B
X  √−γq 
p
ψj,k (x) = λj,k
b
uq (x) ū (ξj,k ) ,
Bj
j−1
q=B

where in this case the set {ξj,k , λj,k } p
characterizes a suitable partition of M,
given by a ε-lattice on M, with ε = λj,k . Further details and technicalities
concerning ε-lattices can be found in Pesenson [41]. Analogously to the spherical
case, for f ∈ L2 (M) and arbitrary j ≥ 0 and k ∈ {1, . . . , Kj }, the needlet
coefficient corresponding to ψj,k is given by
B
X  √−γq 
p
= λj,k
b
aq uq (ξj,k ) .
Bj
j−1
j+1

βj,k = hf, ψj,k iL2 (Sd )

q=B

These wavelets preserve all the properties featured by needlets on the sphere: because, as shown in the following sections, the main results presented here do not
depend strictly on the underlying manifold (namely, the sphere) but rather they
can be easily extended to more general frameworks such as compact manifolds,
where the concentration properties of the wavelets and the smooth approximation
properties of Besov spaces still hold.
2.2. Besov space on the sphere
Here we will recall the definition of spherical Besov spaces and their main
approximation properties for wavelet coefficients. We refer to [1, 10, 22, 39] for
more details and further technicalities.
Suppose that one has a scale of functional classes Gt , depending on the qdimensional set of parameters t ∈ T ⊆ Rq . The approximation error Gt (f ; p)
concerning the replacement of f by an element g ∈ Gt is given by
Gt (f ; p) = inf kf − gkLp (Sd ) .
g∈Gt

s
Therefore, the Besov space Bp,q
is the space of functions such that f ∈ Lp Sd
and
X1
{ts Gt (f ; p)}q < ∞,
t
t≥0

9



which is equivalent to
X

B j {GB j (f ; p)}q < ∞.

j≥0
s
The function f belongs to the Besov space Bp,q
if and only if




Kj
X

1/p
= B −js wj ,

{|βj,k | kψj,k kLp (Sd ) }p 

(9)

k=1

where wj ∈ `q , the standard space of q-power summable infinite sequences.
Loosely speaking, the parameters s ≥ 0, 1 ≤ p ≤ ∞ and 1 ≤ q ≤ ∞ of
s
the Besov space Bp,q
can be viewed as follows: given B > 1, the parameter p
denotes the p-norm of the wavelet coefficients taken at a fixed resolution j, the
parameter q describes the weighted q-norm taken across the scale j, and the
parameter r controls the smoothness of the rate of decay across the scale j. In
view of Eq. (7), the Besov norm is defined as

q/p 1/q



Kj

X
X
p
jq{s+d(1/2−1/p)} 
B
|βj,k |
kf kBs = kf kLp (Sd ) +
p,q



j≥0
k=1
= kf kLp (Sd ) + B j{s+d(1/2−1/p)} kβj,k k`p

,
`q

for q ≥ 1. The extension to the case q = ∞ is trivial.
We conclude this section by introducing the Besov embedding, discussed in
[1, 29, 30] among others. For p < r, one has
s
s
Br,q
⊂ Bp,q

s
s−d(1/p−1/r)
and Bp,q
⊂ Br,q
,

or, equivalently,
Kj
X


p

|βj,k | ≤ 

k=1
Kj
X
k=1

Kj
X


1−p/r
r
;
|βj,k |  Kj

(10)

k=1
r

|βj,k | ≤

Kj
X

p

|βj,k | .

(11)

k=1

Proofs and further details can be found, for instance, in [1, 10].
3. Global thresholding with spherical needlets
This section provides a detailed description of the global thresholding technique applied to the nonparametric regression problem on the d-dimensional
sphere. We refer to [12, 22, 30] for an extensive description of global thresholding methods and to [1, 10] for further details on nonparametric estimation in
the spherical framework.
10

3.1. The regression model
Recall the regression model given by Eq. (1), i.e., for all i ∈ {1, . . . , n},
Yi = f (Xi ) + εi .
While {X1 , . . . , Xn } denotes the set of uniformly sampled random directions
over Sd , {Y1 , . . . , Yn } is the set of the independent observations which are related
to {X1 , . . . , Xn } through the regression function f and affected by {ε1 , . . . , εn },
which is the set of the observational errors. The independent and identically
distributed random variables ε1 , . . . , εn are such that, for all i ∈ {1, . . . , n},

E (εi ) = 0, E ε2i = σε2 < ∞,
and they are assumed to be sub-Gaussian. Further details are given in Secs
tion 3.2. Assume that f ∈ Bp,q
, d/p ≤ s < r + 1, 1 ≤ p ≤ r and 1 ≤ q ≤ ∞,
where r is fixed, and that there exists R > 0 such that kf kBp,q
≤ R. As mens
tioned in Section 1.2 and Section 2, the regression function can be expanded in
terms of needlet coefficients as
f (x) =

Kj
XX

βj,k ψj,k (x) ,

j≥0 k=1

where βj,k are given in Eq. (5).
Remark 3.1. As discussed in Section 1.3, we do not require explicit knowledge
of the Besov radius R. Although it can be difficult to determine r explicitly,
we suggest the following criterion. Consider Remark 2.1; if ρ < ∞, we choose
r = ρ (see again [30]). If ρ = ∞, we choose r = B d(Jn +1) empirically, using the
so-called vanishing moment condition on Sd , properly adapted for the needlet
framework; see, e.g., Schröder and Sweldens [42].
3.2. The observational noise
Following Durastanti et al. [10], we assume that ε1 , . . . , εn follow a subGaussian distribution; see also Buldygin and Kozachenko [4]). A random variable ε is said to be sub-Gaussian of parameter a if, for all λ ∈ R, there exists
a ≥ 0 such that
2 2
E(eλε ) ≤ ea λ /2 .
Sub-Gaussian random variables are characterized by the sub-Gaussian standard,
given by
n
o
2 2
ζ (ε) := inf a ≥ 0 : E(eλε ) ≤ ea λ /2 , λ ∈ R ,
which is finite. As proved in [4],
(
ζ (ε) = sup
λ6=0

2 ln E eλε
λ2

 )1/2
;

11

E(eλε ) ≤ exp



λ2 ζ 2 (ε)
2


.

Following Lemma 1.4 in [4], for p > 0

p
E (ε) = 0; E ε2 ≤ ζ (ε) ; E (|ε| ) ≤ 2



p
exp

p/2

ζ p (ε) .

Therefore, sub-Gaussian random variables are characterized by the same moment inequalities and concentration properties featured by null-mean Gaussian
or bounded random variables.
Remark 3.2. In order to establish the probabilistic bounds described in Sections 3.3 and 3.4, it would be sufficient for ε1 , . . . , εn to be null-mean independent random variables with finite absolute pth moment. However, we include
the notion of sub-Gaussianity in order to be consistent with the existing literature; see [10]. Furthermore, sub-Gaussianity involves a wide class of random
variables, including Gaussian and bounded random variables and, in general,
all the random variables such that their moment generating function has a an
upper bound in terms of the moment generating function of a centered Gaussian
random variable of variance a. Hence the term “sub-Gaussian.”
3.3. The estimation procedure
We note again that the method established here can be viewed as an extension of global thresholding techniques to the needlet regression function estimation. In this sense, our results are strongly related to those presented in
[1, 10, 30], as discussed in Section 1.3.
For any j ≥ 0 and k ∈ {1, . . . , Kj }, the empirical needlet estimator is given
by
n
1X
b
Yi ψj,k (Xi ) ,
βj,k =
n i=1
and it is unbiased, i.e.,
n

E(βbj,k ) =

1X
[E{f (Xi ) ψj,k (Xi )} + E(εi )E{ψj,k (Xi )}] = βj,k .
n i=1

The empirical needlet coefficients are moreover characterized by the following
stochastic property.
Proposition 3.1. Let βj,k and βbj,k be as in Eq. (5) and Eq. (3), respectively.
Thus, for p ≥ 1, there exists c̃p such that
E(|β̂j,k − βj,k |p ) ≤ c̃p n−p/2 .
Therefore, we define the global thresholding needlet regression estimator at
every x ∈ Sd by
Jn
X
X
fˆn (x) =
τj
βbj,k ψj,k (x) ;
j=0

k

see Eq. (4). Recall now the main results, stated in Section 1.
12

Theorem 1.1. Given r ∈ (1, ∞), let p ∈ [1, r]. Also, let fˆn be given by Eq. (4),
with Jn = lnB n1/d . Then, for 1 ≤ q ≤ ∞, d/p ≤ s < r + 1 and 0 < R < ∞,
there exists C > 0 such that


−sp
sup E kfˆn − f kpLp (Sd ) ≤ Cn 2s+d .
(12)
s (R)
f ∈Br,q

In the nonparametric thresholding settings, the Lp -risk is generally bounded as
follows
 

Jn



X
p

≤ 2p−1 E 
E kfˆn − f kpLp (Sd )
(βbj,k − βj,k )ψj,k

Lp (Sd )
j=0

p


X
+
βj,k ψj,k


j≥Jn
p
d
L (S )

=

S + B,

where S is the stochastic error, due to the randomness of the observations and
B is the (deterministic) bias error. The so-called truncation level Jn is chosen
so that B dJn = n.
In this case the bias term term does not affect the rate of convergence for
s ∈ (d/p, r + 1). As far as S is concerned, its asymptotic behavior is established
by means of the so-called optimal bandwidth selection, i.e., a frequency Js such
1
that B dJs = n (2s+d) ; see [12, 30]. Note that trivially, Js < Jn . The meaning
and the relevance of the optimal bandwidth selection is given in Section 5, in
the proof of Theorem 1.1. However, in the next section it will also be crucial
for the construction of the threshold function.
Consider now the case p = ∞. First, we have to modify the threshold
function given in Eq. (13) slightly, in view of the explicit dependence on p.
b∞ = Θ
b j (1) = |β̂j,1 |+· · ·+
Hence, in the selection procedure we use the statistic Θ
j
dj −1/2
|β̂j,Kj |, which will be compared to B n
. Further details on the threshold
will also be given in the next section. Under this assumption, we obtain the
following result.
Theorem 3.2. Let fˆn by given by Eq. (4). Given r ∈ (1, ∞), for any d < s <
r + 1, there exists C > 0 such that


s−d
sup E kfˆn − f kL∞ (Sd ) ≤ Cn− 2s+d .
s
f ∈Br,q

Remark 3.3. As far as optimality is concerned, Eq. (12) given in Theorem 1.1,
achieves the same optimal minimax rates provided on R by Kerkyacharian et al.
[30], where the established optimal rate of convergence was given by n−sp/(2p+1) .
Moreover, this rate is consistent with the results provided over the so-called regular zone by [1, 9, 10, 38] for local and block thresholding estimates by needlets

13

on the sphere, where the rates are nearly optimal due to the presence of a logarithmic term.
Regarding the L∞ -risk function, according to Theorem 3.2, the rate established is not optimal; see, e.g., Baldi et al. [1]. In the global thresholding
b j (p)
paradigm, a straightforward generalization of the thresholding function Θ
given by Eq. (13) is not available (see Remark 3.4). Therefore, the upper bound
for the case p = ∞ is established in a different framework, which can be reasonably assumed to cause the lack of optimality.
3.4. The threshold
The construction of the threshold function τj is strictly related to Efromovich
[12] and Kerkyacharian et al. [30], where analogous results were established in
the real case. Let
Kj
X
p
Θj (p) =
|βj,k | .
k=1
s
,
Using Eq. (9), it follows immediately that, if f ∈ Bp,q
1

1

Θj (p) ≤ CB −jp{s+d( 2 − p )} .
Consider now the optimal bandwidth selection Js . If j ≤ Js ,
1

1

B −jp{s+d( 2 − p )} ≥

B dj
.
np/2

Thus, even if j ≤ Js doesn’t imply Θj (p) > B dj /np/2 , according to [12, 30], one
p/2
has that Θj (p) ≥ B dj /n
implies j ≤ Js .

Clearly, the case Θj (p) ≤ B dj /np/2 , j ≤ Js provides no guarantee of a
better performance if compared to the linear estimate, whose error is of order
B dj /np/2 ; see Härdle et al. [22]. Thus, the natural choice is to construct a
threshold function that keeps the level j if and only if
Θj (p) ≥

B dj
.
np/2

As pointed out in [12, 30], the natural estimator |β̂j,1 |p + · · · + |β̂j,Kj |p for
Θj (p) yields an extremely large bias term and, therefore, undersmoothing effects. Hence, following the procedure as suggested in [12, 30] (see also [22]) but
properly adapted to the needlet framework (see Lemma 2.1), we propose an
alternative method as described below.
Let p ∈ N be even, Σp denoting the set of p-dimensional vectors chosen in
p
{1, . . . , n} and also let υ = (i1 , . . . , ip ) be the generic element belonging to Σp .
b j (p) by
Define the U-statistic Θ
Kj
X
X ⊗p
b j (p) = 1
Ψj,k (Xυ , ευ ) ,
Θ
n
p

k=1 υ∈Σp

14

(13)

where
Ψ⊗p
j,k (Xυ , ευ ) =

p
Y

{f (Xih ) + εih } ψj,k (Xih ) .

h=1

Given that the sets of variables {X1 , . . . , Xn } and {ε1 , . . . , εn } are independent,
it can be easily seen that
E{Θ̂j (p)} =

Kj
X

p

|βj,k | = Θj .

(14)

k=1

Remark 3.4. As mentioned in the previous section, if we consider the case
b j (p). Hence, we choose
p = ∞, we lack a straightforward extension of Θ
b∞
Θ
j =

Kj
X

|β̂j,k |.

k=1

so that the threshold function is given by


B dj
∞
τj = 1 Θj ≥ 1/2 .
n
Our purpose is to establish two probabilistic bounds related to the mth
b j (p), respectively. We have that
moment and to the mth centered moment Θ
 −m
m
n X Y
o
X
n
m
E
Ψ⊗p
E[{Θ̂j (p)} ] =
j,k` (Xυ` , ευ` ) .
p
υ1 ,...,υm ∈Σp

k1 ,...,km `=1

For any fixed configuration υ1 , . . . , υm , let the sequence c1 , . . . , cm denote the
b j (p)}.
cardinality of the E [{f (X) + }ψj,k (X)]` of size ` appearing in E{Θ
Observe that
m
X
`c` = mp.
`=1

Following Kerkyacharian et al. [30], the next results hold.
b j (p) be given by Eq. (13). Also, let p be an even integer.
Proposition 3.3. Let Θ
Then, for m ∈ N, there exists C˜1 such that
Pm
 jd ( mp
d
`=1 c` )
2 −
X
B
B −j{c1 (s− p )−d(1−γ)}
m
E[{Θ̂j (p)} ] ≤ C̃1
,
n
nmp/2
(c1 ,...,cm )∈Γm,p

where Γm,p = {c1 , . . . , cm :

Pm

`=1

`c` = mp}.

b j (p) and Θj be given by (13) and (14) respectively.
Proposition 3.4. Let Θ
Also, let p, m be even integers. Then, there exists C˜2 such that
p  jd  mh
p
X
B
E{|Θ̂j (p) − Θj (p)|m } ≤ C˜2
{Θj (p)}(1−h/p)m .
p/2
n
h=1
15

Remark 3.5. According to [30], this procedure can be easily extended to the case
of p is not being an even natural number, by means of an interpolation method.
Indeed, by fixing p1 , p2 ∈ N, both even, we can rewrite p = δp1 + (1 − δ) p2 , as
b j (p) = {Θ̂j (p1 )}δ {Θ̂j (p2 )}1−δ .
Θ
The following lemma is crucial for the application of our interpolation method.
As in [30], we consider for the sake of simplicity just the case p2 − p1 = 2, so
that p0 ≤ p ≤ p00 , p00 = p0 + 2.
Lemma 3.5. For any even m ∈ N,

E

1
Θ̂j (p0 ) − Θj (p00 )
n

m

0

≤ C˜2

p0
X

{Θj (p00 − h)}m n−mh/2 .

h=1

b ∞.
We conclude this section with a result regarding the behavior of Θ
j
b ∞ and Θ∞ be given by (13) and (14) for p = 1, reProposition 3.6. Let Θ
j
j
spectively. Then, there exists C̃∞ such that
∞ 2
dj
b∞
E(|Θ
j − Θj | ) ≤ C̃∞ B /n.

4. Simulations
In this section, we present the results of some numerical experiments performed over the unit circle S1 . In particular, we are mainly concerned with
the empirical evaluation of L2 -risks obtained by global thresholding techniques,
which are then compared to the L2 -loss functions for linear wavelet estimators.
As in any practical situation, the simulations are computed over finite samples and therefore, they can be considered as a reasonable hint. Furthermore,
they can be viewed as a preliminary study to practical applications on real
data concerning estimation of the power spectrum of the Cosmic Microwave
Background radiation; see, e.g., Faÿ et al. [14].
The needlets over the circle used here are based on a weight function b which,
analogously to Baldi et al. [1], is a properly rescaled primitive of the function
2

x 7→ e−1/(1−x ) ;
see also Marinucci and Peccati [37]. Following Theorem 1.1, we fix B = 2 and
b j (2), correspondn = 26 , 27 , 28 and Jn = 6, 7, 8, respectively. The U -statistic Θ
2
ing to the L risk considered here, results in considerable computational effort,
because it is built over 2,016, 8,128 or 32,640 possible combinations of needlet
empirical coefficients for Jn = 6, 7, 8, respectively.
By choosing a test function F and fixing the set of locations X1 , . . . , Xn , we
obtain the corresponding Y1 , . . . , Yn by adding to F (Xi ) a Gaussian noise, with
three different amplitudes, i.e., the noise standard deviation σε is chosen to be
equal to 0.25M , 0.5M or 0.75M , where M is the L∞ -norm of the test function.
Therefore, the following three numerical experiments are performed.
16

Example 4.2
Jn ↓ \σε →
6
7
8

0.25M
7.82
1.90
1.77

Global
0.50M
65.29
9.38
18.88

0.75M
108.46
67.09
54.03

0.25M
80.23
26.75
36.69

Linear
0.50M
411.38
141.41
96.82

0.75M
889.15
451.61
434.80

Table 1: Example 4.2 - Values for L2 risk
Example 4.1. According to Baldi et al. [1], we use the uniform test function
defined, for all x ∈ S1 , by
1
F1 (x) =
.
4π
In this case, for every j, k, we get βj,k = 0. The performance of our model can
be roughly evaluated by simply controlling how many resolution levels pass the
selection procedure. For all the choices of n and σ, we get τj = 0 for all j ∈
{1, . . . , Jn }. On the other hand, we consider a finite number of resolution levels
and therefore of frequencies. Thus, it is possible that higher resolution levels,
involving higher frequencies, could be selected by the thresholding procedure.
Example 4.2. In this example, we choose the function defined, for all x ∈ S1 ,
by
F2 (x) = cos (4x) .
In this case, the test function corresponds to the real part of one of the Fourier
modes over the circle (with eigenvalue 4). This choice allows us to establish
whether the thresholding procedure is able to select only the suitable corresponding resolution levels, as the amplitude of the noise increases. As expected, for
every n and for every σε , we get τj = 1 for j = 2 (containing the frequency
k = 4) and 0 otherwise. Table 1 presents the value for L2 -risks for different
values of Jn and σε , while Figure 1 illustrates the graphical results for the case
Jn = 8 and σε = 0.5M .
Example 4.3. A more general function is chosen here, which is defined, for all
x ∈ S1 , by
n
o
2
2
F3 (x) = e−(x−3π/2) + 2e−(x−2) sin (−2x) ,
depending on a larger set of Fourier modes. In this case, Table 2 gives the
values of L2 -risks corresponding to different Jn and σε , while Figure 2 presents
the graphical results for the case Jn = 8 and σε = 0.5M . Table 3 contains, for
every pair Jn , σε , the resolution levels selected by the procedure.

17

Test function + noise

−2

−1.0

0.0

0 1 2

1.0

Test function

0

1

2

3

4

5

6

0

2

3

4

5

6

Global thresholding needlet estimator

−1.0

−1.5

0.0

0.0

1.5

Linear needlet estimator

1

0

1

2

3

4

5

6

0

1

2

3

4

5

6

Figure 1: Example 4.2 — Jn = 8, σε = 0.5M

Example 4.3
Jn ↓ \σε →
6
7
8

0.25M
62.01
58.87
51.14

Global
0.50M
208.94
150.73
40.65

0.75M
625.02
277.98
NA

0.25M
227.32
150.94
321.05

Linear
0.50M
1372.64
644.31
1271.31

Table 2: Example 4.3 — Values for L2 risk

Example 4.3
Jn ↓ \σε →
6
7
8

Selected j
0.25M 0.50M 0.75M
1
1,2
1,2,3
1
1,4
1,2
1
1,2
1,3,5

Table 3: Example 4.3 - Values of the function τj

18

0.75M
2296.28
1294.59
NA

Test function + noise

−3 −1

−0.5 0.5

1

3

1.5

Test function

0

1

2

3

4

5

6

0

1

3

4

5

6

Global thresholding needlet estimator

−4

−1.0

0 2 4

0.5 1.5

Linear needlet estimator

2

0

1

2

3

4

5

6

0

1

2

3

4

5

6

Figure 2: Example 4.3 - Jn = 8, σε = 0.5M
5. Proofs
In this section, we provide proofs for the main and auxiliary results.
5.1. Proof of the main results
The proofs of Theorem 1.1 and Theorem 3.2 follow along the same lines as
the proof of Theorem 8 in Baldi et al. [1].
Proof of Theorem 1.1. Following, for instance, [1, 7, 9, 10, 22, 30] and as mentioned in Section 3.3, the Lp -risk E(kfˆn − f kpLp (Sd ) ) can be decomposed as the
sum of a stochastic and a bias term. More specifically,
 

p

Kj
Jn X

X


E(kfˆn − f kpLp (Sd ) ) ≤ 2p−1 E 
(τj βbj,k − βj,k )ψj,k



j=0 k=1
p
d
L (S )

p

Kj

X X
+
βj,k ψj,k
.


j>Jn k=1
p
d
L (S )

Using the definition of Besov spaces, we obtain for the bias term the following
inequality:
Kj
X X
j>Jn k=1

p

≤

βj,k ψj,k
Lp (Sd )

Kj
X X
j>Jn

p
ps

≤ CB −spJn ≤ Cn− 2s+d .

βj,k ψj,k

k=1

Lp (Sd )

19

Following Baldi et al. [1], the stochastic

p
Jn Kj
 XX b
(τj βj,k − βj,k )ψj,k
E
j=0 k=1

term can be split into four terms, i.e.,


p−1
 ≤ 4 (Aa + Au + U a + U a),

Lp (Sd )

where



 

Jn K j
dj
B dj
 XX b
b j (p) ≥ B
(τj βj,k − βj,k )ψj,k 1 Θ
Aa = E
1
Θ
(p)
≥
j
np/2
2np/2
j=0 k=1



 

Jn Kj
B dj
B dj
XX b
b
(τj βj,k − βj,k )ψj,k 1 Θj (p) ≥ p/2 1 Θj (p) ≤ p/2
Au = E
n
2n
j=0 k=1




p


,
Lp (Sd )



p


,
Lp (Sd )



p

Kj
Jn X
X


 

B dj
2B dj

b
b
(τj βj,k − βj,k )ψj,k 1 Θj (p) ≤ p/2 1 Θj (p) ≥ p/2
U a = E
n
n
j=0 k=1



 

Jn Kj
dj
2B dj
XX b
b j (p) ≤ B
(τj βj,k −βj,k )ψj,k 1 Θ
U u = E
1
Θ
(p)
≤
j
np/2
np/2
j=0 k=1


,
Lp (Sd )

p



.

Lp (Sd )

Following Durastanti et al. [10], the labels A and U denote the regions where
b
Θj (p) is larger and smaller than the threshold B dj n−p/2 respectively, whereas a
and u refer to the regions where the deterministic Θj (p) are above and under
a new threshold, given by 2−1 B dj n−p/2 for a and 2B dj n−p/2 for u. The decay
of Aa and U u depends on the properties of Besov spaces, while the bounds on
b j (p),
Au and U a depend on the probabilistic inequalities concerning βbj,k and Θ
given in Propositions 3.1, 3.3 and 3.4.
Let p ∈ N be even. Then, using the definition of the optimal bandwidth
selection, we have

Kj
Jn

 
X
X
B dj
p−1
p
Aa ≤ C10 (Jn + 1)
kψj,k kLp (Sd )
E |β̂j,k − βj,k |p 1 Θj (p) ≥ p/2
2n
j=0
k=1


Kj
Js
Jn X
X
X
Θj (p) 
≤ C100
B jdp/2 n−p/2 + n−p/2
B dj(p/2−1) dj −p/2


B n
j=0
j=Js k=1



Kj
Jn


X
X
p
p

≤ C1000 B Js dp/2 n−p/2 +
|βj,k | kψj,k kLp (Sd ) 


j=Js
k=1


≤ C1 B Js dp/2 n−p/2 + B −Js sp
=

sp

C1 n− 2s+d ,
20

given that
dp

sp

B Js dp/2 n−p/2 = n 2(2s+d) −p/2 = n− 2s+d .

(15)

Similarly, for the region Au we obtain
Au ≤ C20 (Au1 + Au2 ) ,
where
Au1 =

Kj
Js X
X

p

kψj,k kLp (Sd ) E(|β̂j,k − βj,k |p ),

j=0 k=1

Au2 =

Kj
Jn X
X
j=Js




d
p
b j (p) ≥ B j
kψj,k kLp (Sd ) E |β̂j,k − βj,k |p 1 Θ
.
n−p/2
k=1

Using Eq.s (7) and (15), it is easy to see that
sp

Au1 ≤ C2 n− 2s+d .
Regarding Au2 , using Hölder inequality with 1/α0 + 1/α = 1, the generalized
Markov inequality with even m ∈ N and Proposition 3.3, we obtain
Au2

≤

C

Kj
Jn X
X

p
kψj,k kLp (Sd )

j=Js k=1
Kj
Jn X
X




B dj
p
b
E |β̂j,k − βj,k | 1 Θj (p) ≥ p/2
n

1/α
n 
o1/α0  
B dj
pα0
b
B
E |β̂j,k − βj,k |
≤ C
Pr Θj (p) ≥ p/2
n
j=Js k=1
hn
oi

1/α
b j (p)m
Jn
E Θ
X
 
B jdp/2 n−p/2 
≤ C
dj /np/2 m
B
j=Js
1/α

Pm
 jd ( mp
Jn
`=1 c` )
j dp
2 −
X
X
2
d
B
B

≤ C
B −j{c1 (s− p )} B dj{(1−m)−γ}
n
np/2
j=Js

jd(p/2−1)

(c1 ,...,cm )∈Γm,p

≤ Au2,1 + Au2,2 ,
(+)

where Au2,1 and Au2,2 are defined by splitting Γm,p into two subsets, Γm,p and
(−)
Γm,p . These subsets are defined as
(
)
m
mp X
(+)
Γm,p := c1 , . . . , cm :
−
cm ≥ 0 ;
2
i=i
)
(
m
mp X
(−)
−
cm ≤ 0 .
Γm,p := c1 , . . . , cm :
2
i=i

21

Note that 1 − γ ∈ [0, 1]. Hence we choose m, α so that m > 1 + αp/2. It can be
easily verified that
≤

Au2,1

Jn
X

C0

Bj

dp
2

n−p/2

h

j=Js

≤
≤

C 00 B Js

X

B dj{(1−m)−γ}

i1/α

(+)

(c1 ,...,cm )∈Γm,p

dp
2

n−p/2





X




(c1 ,...,cm )∈Γm,p

B dJs (1−m)

1/α





(+)

sp

C 000 n− 2s+d

On the other hand,
≤ C0

Au2,2

X

B

c (s−d/p)
−Js 1 α

≤ C

≤ C

00





B dJs
n

p/2

B dJs
n

p/2

B jdp/2 n−p/2

X

B

−Js

c1
α

(−)
(c1 ,...,cm )∈Γm,p

n−

X


Pm
 B jd ( mp
`=1 c` )/α
2 −
n



j=Js

(−)
(c1 ,...,cm )∈Γm,p

00

Jn
X

(

B

1−m−γ
α



Pm
 Js d ( mp
`=1 c` )/α
2 −
1−m−γ
(s− dp ) B
B α dJs
n

)

c1 s− d
p
α(2s+d)

{n (2s+d)α (
2s

mp Pm
`=1
2 −

c` )

}n

d(1−m−γ)
α(2s+d)

(−)

(c1 ,...,cm )∈Γm,p

≤ C 000 n

sp
− 2s+d

,

and therefore,

sp

Au ≤ C2 n− 2s+d .
Consider now U a. We have that

p
Kj
Js
X
X
0 
βj,k ψj,k
U a ≤ C3 
j=0

k=1

Lp (Sd )

+

j=Js

=

 
 

dj
B dj
b j (p) ≤ B
E 1 Θ
1
Θ
≥
2
j
np/2
np/2

Kj
Jn
X
X
k=1

p

βj,k ψj,k





Lp (Sd )

U a1 + U a2 .

It is easy to see that
−sp

U a2 ≤ C300 B −Js ps = C300 n− 2s+d .
On the other hand, using the generalized Markov inequality, Proposition 3.4

22

dj




with m = p and Eq. (15), we have that
U a1

Js
X



 

dj
B dj
b j (p) ≤ B
B dj(p/2−1) E Θj 1 Θ
1
Θ
≥
2
j
np/2
np/2
j=0

=

≤ C

≤ C

≤ C

Js
X



n
B dj
1 o
B dj(p/2−1) Θj Pr |Θ̂j (p) − Θj | ≥ Θj 1 Θj ≥ 2 p/2
2
n
j=0

Js
X


 

B dj
m
1
Θ
≥
2
B dj(p/2−1) Θ1−m
E
|
Θ̂
(p)
−
Θ
|
j
j
j
j
np/2
j=0

Js
X

B

dj(p/2−1)

j=0

≤ C

Js
X

Θj

p 
X

Θ−1
j

`=1

B dj
np/2

 m`
p

dj

B dj(p/2−1)

j=0

B
np/2

−sp

≤ Cn 2s+d ,
and therefore,
−sp

U a ≤ C3 n− 2s+d .
Finally, in view of Eq. (7) and Eq. (15), we have that


 X

Kj
Jn X
Js
dj
X
B
p
p
U u ≤ C40 
|βj,k | kψj,k kLp (Sd ) 
B jd(p/2−1) Θj (p) 1 Θj ≤ 2 p/2 +
n
j=0
j=Js k=1


Jn
Js
X
X
B −jsp 
≤ C400 
B jdp/2 n−p/2 +
j=0

≤ C4 n

−sp
2s+d

j=Js

.

We now need to extend these results to any p ∈ (1, r) using the interpolation method described in Remark 3.5. The two terms that have to be studied
separately are Au and U a, in particular Au2 and U a1 , since they involve the
probabilistic inequalities described in Propositions 3.3 and 3.4, holding only for
even p ∈ N. According to [30], the generalization in the case of Au2 is obtained
by bounding
E

hn
om i
hn
om iδ hn
om i1−δ
Θ̂j (p)
≤ CE Θ̂j (p1 )
E Θ̂j (p2 )

23

Indeed,
Au2

≤ C

Jn
X





B

j

dp1
2

n

p1
2

−

j=Js

×B


(c1 ,...,cm )∈Γm,p1



−jc1 s− pd
1


×



X

X



B jd
n

( mp2 1 −Pm
`=1 c` )

1/α #δ h
dp2
p2
B j 2 n− 2

B

dj{(1−m)−γ}

jd

1/α 1−δ
( mp2 2 −Pm


`=1 c` )
−jc s− pd

2 B dj{(1−m)−γ} 
.
B 1


B
n

(c1 ,...,cm )∈Γm,p2

The result above follows from Eq.s (10) and (11), so that
s
Bp,q

s
⊂Bps1 ,q ; Bp,q

⊂



1
s−d p
− p1
2
Bp2 ,q
.

Straightforward calculations lead to the claimed result. On the other hand, in
order to study U a1 , we apply Lemma 3.5 to obtain
U a1 ≤C

Js
X

B

dj(p/2−1)

j=0
Js
X



 

B dj
B dj
b
E Θj 1 Θj (p) ≤ p/2 1 Θj (p) ≥ 2 p/2
n
n



 

B dj
B dj
B
E Θj 1 Θ̂j (p0 ) ≤ p0 1 Θj (p) ≥ 2 p/2
≤C
n
n2
j=0
(
) 
#

B dj
B dj
1 Θ̂j (p00 ) ≤ p0 1 Θj (p) ≥ 2 p/2
0
n
n2



J
s
X
Θj (p00 )
0
0
dj(p/2−1)
B
Pr Θ̂j (p0 ) − Θj (p0 ) ≥
≤C
2
j=0


Θj (p00 )
1
+ Pr
Θ̂j (p0 ) − Θj (p00 ) ≥
,
n
2

o 
n
dj
dj
because Θj (p) ≥ 2 nBp/2 ⊂ Θj (p00 ) ≥ 2 Bp0 . Finally, by applying Markov
dj(p/2−1)

n

24

0
2

inequality with m > p, m ∈ N even, we have that
U a1 ≤C

Js
X

0

B

p0
X

dj(p/2−1)

j=0

m

{Θj (p00 − h)} n−

h=1

(
×

{Θj (p00

1−m

− h)}

Θj (p00 )

1

≥2

B dj
n

≤C

Js
X

B


dj 1− pp0 + mh
p0
0

{Θj (p00 )}

p−mh
p00

h=1
 (

1

0

Θj (p00 )

≤C

p00
2

≥2

0

B dj(p/2−1)

j=0

p0
X

B dj

h=1

p00
2

n

n−

B dj
n

Js
X

)

0

p0
X

dj(p/2−1)

j=0

×B

mh
2

mh
2

)

p00
2

! p−mh
0
p0

− mh
2

n

B



dj 1− pp0 + mh
0
p
0

0

sp
− 2s+d

≤Cn

Proof of Theorem 3.2. Similarly to the previous proof, note that
 


Kj 
Jn X




X


E kfˆn − f kL∞ (Sd ) ≤C E 
τj βbj,k − βj,k ψj,k



j=0 k=1
∞
d
L (S )


Kj

X X
.
(βj,k ) ψj,k
+


j>Jn k=1
∞
d
L

s
If f ∈ B∞,∞
, |βj,k | ≤ M B −j (
(8) with p = ∞, we get

s+ d
2

Kj
X X
j>Jn k=1

) for any k = 1, . . . , K , then by Eq.s (7) and
j

≤

βj,k ψj,k
L∞ (Sd )

(S )

Kj
X X
j>Jn

≤C

βj,k ψj,k

k=1

X

L∞ (Sd )

sup

j>Jn k=1,...,Kj

≤C

X

|βj,k | kψj,k kL∞ (Sd )


s
s
B −js = O n− d = O n− 2s+d .

j>Jn

As far as the other term is concerned, following the same procedure as described
in the proof of Theorem 1.1, see also [1, 10], we obtain


Kj 
Jn X

X


E
τj βbj,k − βj,k ψj,k
 ≤ C (Aa + Au + U a + U u) ,
j=0 k=1

L∞ (Sd )

25

where




 


Kj 

B dj
B dj
 X
∞
∞
b
b
τj βj,k − βj,k ψj,k 1 Θj ≥ 1 1 Θj ≥
Aa =
E
1
n2
2n 2
j=1
k=1


 

Kj 
J
n

X  X
B dj
B dj
∞
b∞
≥
τj βbj,k − βj,k ψj,k 1 Θ
Au =
E
1
Θ
<
1
1
j
j
n2
2n 2
j=1
k=1



 

Kj
Jn
dj
dj
X
X
B
2B


b∞
βj,k ψj,k 1 Θ
≤ 1 1 Θ∞
Ua=
E

1
j
j ≥
2
2
n
n
j=1
k=1
∞
d
L (S )






Kj
J
n
dj
X  X
B dj

b ∞ < B 1 1 Θ∞
βj,k ψj,k 1 Θ
Uu=
E

1
j <
j
2
2
n
2n
j=1
k=1
Jn
X



L∞ (Sd )




L∞ (Sd )

L∞ (Sd )

1

dj
2
Now Θ∞
j ≥ B /n implies j ≤ Js (see Section 3.4) and in view of Eq.s (7) and
(8) with p = ∞, we get
!
Js
X
d
j
2
B E
Aa ≤C
sup |β̂j,k − βj,k |
k=1,...,Kj

j=1

≤CB

d
2 Js

1

(Js + 1) n− 2

s

=O n− 2s+d

Consider now Au. Using Js , we split this term into Au = Au1 + Au2 , as in the
proof of Theorem 1.1. Trivially, we get

s
Au1 = O n− 2s+d .
On the other hand, using Eq. (8) and Proposition 3.6, we get
Au ≤

Jn
X

! 12
B

d
2j

Jn
X

|β̂j,k − βj,k |

sup

d

1


Pr

k=1,...,Kj

j=0

≤

E

2

B dj
∞
b∞
Θ
−
Θ
≥
1
j
j
2n 2

 12

d

B 2 j (j + 1) n− 2 B − 2 j

j=0




s
1
=O Jn n− 2 = o n− 2s+d
1

dj
2
As far as U a is concerned, again Θ∞
j ≥ B /n implies j ≤ Js (see Section 3.4),

26

so that
Ua ≤

Js
X

d

Bj 2

j=1
Js
X

Kj
X

B dj
∞
b∞
Θ
≥ 1
j − Θj
n2


Pr

βj,k ψj,k

k=1



L∞ (Sd )


B dj
∞
b∞
≥
Θ
−
Θ
1
j
j
n2
j=1

 dj −2 
Js
X
2
B
jd
∞
∞
b
2
B M
≤
E Θj − Θj
1
n2
j=1
≤

d

B j 2 M Pr



1

≤ Js n− 2 ,
where we used Eq. (8) and Proposition 3.6. Finally, we have that


 

Kj
J
n
X  X
B dj
B dj
∞
b∞
<
βj,k ψj,k 1 Θ
E
Uu ≤
1
Θ
<
1
1
j
j
n2
2n 2
j=1
k=1




L∞ (Sd )

≤ U u1 + U u2 ,
where
U u1 ≤

Kj
Js
X
X
j=1

≤

Js
X

βj,k ψj,k

k=1

B

3d
2 j



B dj
∞
1 Θj <
1
2n 2
L∞ (Sd )

 3d

1
1
n− 2 = O B 2 Js n− 2

j=1

and
X

U u2 =

d

B 2j

sup

|βj,k | .

k=1,...,Kj

j>Js

Note that
B

3d
2 Js

1

d−s

n− 2 = n 2s+d ,

as claimed.
5.2. Proofs of the auxiliary results
The proof of Lemma 2.1 can be viewed as a generalization of the proof of
Lemma 5.1 in Durastanti et al. [11].
Proof of Lemma 2.1. Using the needlets localization property given in Eq. (6),
we have that
Z Y
Z Y
q
q
1
jdq
ψj,ki (x) dx ≤ Cη B
η.
jq
{1
+
B
d
(x, ξj,ki )}
Sd i=1
Sd i=1
27


Let S1 = x ∈ Sd : d (x, ξj,k1 ) ≥ ∆/2 , so that Sd ⊆ S1 ∪ S1 . Therefore,
Z

q
Y

1
η ≤
jq
Sd i=1 {1 + B d (x, ξj,ki )}

q
Y

Z

S1 i=1

1
η
B jq d (x, ξj,ki )}

{1 +

q
Y

Z

1
η.
jd d (x, ξ
{1
+
B
j,ki )}
S1 i=1

+

From the definition of S1 and following Lemma 5.1 in Durastanti, Marinucci
and Peccati [11], it follows that
q
Y

1
2η(q−1)
≤
η
η(d−1)
jd
S1 i=1 {1 + B d (x, ξj,ki )}
(1 + B jd ∆)

Z

2η(q−1)

≤
(1 +

≤

(2π)
(2π)

d−1

η(q−1)

2

d−1

≤Cη0

(2π)

{1 +

η
B jd d (x, ξj,k1 )}

B

1

Sd

Z

η(d−1)

(1 + B jd ∆)

{1 +

η
B jd d (x, ξj,k1 )}

Z

2η(q−1)

(1 + B jd ∆)

1

S1

η(d−1)
B jd ∆)
d−1

=

Z

π

Z

η(d−1)

∞

0

2η(q−1) B −dj

(1 + B jd ∆)

η(d−1)

dx

sin ϑ
η dϑ
(1 + B jd ϑ)

0

−dj

dx

y
η dy
(1 + y)

.

On the other hand,
q
Y

1
2η
η
η ≤
jd
(1+B jd ∆)
S1 i=1 {1 + B d (x, ξj,ki )}

Z

Z

q−1
Y

S1 i=1

dx


1 + B jd d x, ξj,ki+1

η.

Let S2 = {x ∈ S1 : d (x, ξj,k2 ) ≥ ∆/2}. Then,
S1 ⊆ S2 ∪ S2

and

Sd ⊆ S2 ∪ S2 ∪ S1 .

As far as S2 is concerned, we apply the same chain of inequalities as those
−η
used for S1 . The integral over S2 can be bound by the factor 2η 1 + B jd ∆
multiplied by the integral of the product of q − 2 localization bounds of the
needlets.
By re-iterating
the procedure, we obtain, a set ofS nested Sg =


q
x ∈ Sg−1 , d x, ξj,kg ≥ ∆/2 , g = 1, . . . , q so that Sd ⊆ Sq ∪ g=1 Sg , which
yields the claimed result.
The proof of Proposition 3.1 is a simple modification of Proposition 6 in Durastanti et al. [10] concerning complex random spin needlet coefficients. Many
technical details are omitted for the sake of brevity.
Proof of Proposition 3.1. For p ≤ 2 we apply the classical convexity inequality
such that for a set of independent centered random variables {Zi } with finite
28

p-th absolute moment,

E

n
X

p!

Zi

 
p/2
2
n


X
.
≤ E
Zi 


i=1

i=1

For p > 2, we apply the Rosenthal inequality (see for instance Härdle et al.
[22]), that is, there exists a constant cp > 0 such that

( n
)p/2 
p!
n
n
X
X
X

p

≤ cp 
E
Zi
E (|Zi | ) +
E Zi2
i=1

i=1

i=1

On the other hand, since B dj ≤ n, we have that
p

p

E {|(f (X) + ε) ψj,k (X) − βj,k | } ≤2p−1 [E {|f (X) ψj,k (X) − βj,k | }
p

+E {|εψj,k (X)| }]
p

p

≤c0p {M p + E (|ε| )} kψj,k kLp (Sd )
p/2−1
≤c00p B jd(p/2−1) ≤ c000
p n

Hence,


p

E |β̂j,k − βj,k |




≤c̃p

np/2−1
+ n−p/2
np−1



= c̃p n−p/2

The proof of Proposition 3.3 can be considered as the counterpart in the needlet
framework of the proof of Lemma 2 in Kerkyacharian et al. [30].
Remark 5.1. Any element c` can be decomposed as the sum of integers ci1 ,...,i` ;` ,
where the `-dimensional vector {i1 , . . . , i` } ⊂ {1, . . . , m} specifies the spherical
needlets involved in each configuration (of size `) given by
h
i
`
E {f (X) + ε} ψj,ki1 (X) . . . , ψj,ki` (X) .
The notation [i1 , . . . , i` ] denotes the set of all the possible combinations of {i1 , . . . , i` }
such that
X
ci1 ,...,i` ;` = c` .
[i1 ,...,i` ]

Proof of Proposition 3.3. Note that
E

hn

om i n−m
b
Θj (p)
=
p

X

E

υ1 ,...,υm ∈Σp

29


 X


m
Y

k1 ,...,km `=1

Ψ⊗p
j,k` (Xυ` , ευ` )





(16)

where

 X
E


m
Y

k1 ,...,km `=1



Ψ⊗p
(X
,
ε
)
υ
υ
`
`
j,k`


=

"

m
X Y
Y

E {f (X) + ε}

k1 ,...,km `=1[i1 ,...,i` ]

=

m
X Y

(

`
Y

E f (X)

k1 ,...,km h=1

#ci1 ,...,i` ;`
ψj,kih (X)

h=1
)ch;1

ψj,kh (X)

h=1

"

m
Y
Y

`
Y

`

`

E {f (X) + ε}

`=2[i1 ,...,i` ]

`
Y

#ci1 ,...,i` ;`
ψj,kih (X)

.

h=1

Using Eq. (2) and the independence of the noise ε, for any ` ≥ 2 we have that
"
Y

`

E {f (X) + ε}

#ci1 ,...,i` ;`

`
Y

(

≤ CM,p,` E

ψj,kih (X)

h=1

[i1 ,...,i` ]

`
Y

)ci1 ,...,i` ,`
ψj,kh (X)

,

h=1



where CM,p,` = 2p−1 M ` + E ε` . In view of Lemma 2.1, we obtain
m
Y

(
Y

E

`=2 [i1 ,...,i` ]

`
Y

)ci1 ,...,i` ,`
≤C 0

ψj,kh (X)

h=1

m
Y

`ci

Y

,...,ı` ;`

1
kψj,k kL` (S
d)

`=2 [i1 ,...,i` ]

1{k = ki1 = . . . = ki` }
≤C

0

m
Y

`

P

[i

,...,i` ]

kψj,k kL` (Sd1)

ci1 ,...,ı` ;`

∆ (k, `)

`=2

≤C 00 B
=C 00 B

P
m
jd
`=2

`c`
2

P
m
jd
`=1

`c`
2

−

Pm

−

Pm

`=2

`=1

c`



c`



∆ (k, m)
B jd

c1
2

∆ (k, m) ,

Q

Note that [i1 ,...,i` ] 1{k = ki1 = . . . = ki` } implies that al least ` kh indexes are
equal. Thus, using Eq. (10), we obtain
X

m
Y

ch,1

E {f (X) ψj,kh (X)}

=

X

k1 ,...,km h=1

Pm

βj,kh=1

ch,1

k
(

≤ CB −j

Pm

h=1

ch,1 {s+d(

1
1
2−p

)} B

= CB −jc1 {s+d( 2 − p )} B jd(1−γ)
1

30

1

jd 1−

(

min p,

Pm
h=1 ch;1
p

)

)

where γ = min (p, c1 ) /p. Hence,
"
Y

E {f (X) + ε}

`

`
Y

#ci1 ,...,i` ;`
≤C 000 B jd(

ψj,kih (X)

mp Pm
`=1
2 −

c` )

h=1

[i1 ,...,i` ]

B −jc1 (s− p ) B jd(1−γ)
d

Finally, for any fixed configuration c1 , . . . , cm , the number of possible combinations is bounded by
Pm

 
 
n
n − cm
n − `=2 c`
,
C?
...
c1
cm
cm−1
where C ? denotes the possible choices of {ci1 ,...,i` ;` } for any ` and does not
depend on n. In view of the Stirling approximation, np ≈ np , the number
Pm
of possible combinations is bounded by Cn− `=1 c` . Using the aforementioned
results, Eq. (16) is bounded by
E

hn

om i
b j (p)
Θ
≤ C̃1

X



(c1 ,...,cm )∈Γm,p


d
Pm
B jd ( mp
B −j {c1 (s− p )−d(1−γ)}
`=1 c` )
2 −
,
mp
n
n 2

as claimed.
The proof of Proposition 3.4 can be viewed as the counterpart of the proof
of Lemma 3 in Kerkyacharian et al. [30] in the needlet framework.
Proof of Proposition 3.4. Following Kerkyacharian et al. [30], note that
p
Y

xi − β p =

i=1

p
X

h
Y

X

β p−h

(xti − β) .

1≤t1 <...,<th ≤p i=1

h=1

Let
Ψ̃j,k (X, ε) := {f (x) + ε} ψj,k (x) − βj,k .
so that

n
o
E Ψ̃j,k (X, ε) = 0.

(17)

We therefore obtain
b j (p) − Θj (p) =
Θ

 −1 X
Kj
p
X X
n
p−h
βj,k
p
k=1 υ∈Σp h=1

X

Ψ̃⊗h
j,k (Xι , ει ) ,

ι⊂υ,ι∈Σh

and, reversing the order of integration, we have that
b j (p) − Θj (p) =
Θ

Kj p
X
X
k=1 h=1

n−h
p−h

n
p



31

p−h
βj,k

X
ι∈Σh

Ψ̃⊗h
j,k (Xι , ει ) ,

Hence, we can rewrite
E

n

p
m o
X
b j (p) − Θj (p)
Θ
≤pm−1

(

h=1

 )m

n−h
p−h

n
p

(
X

E

ι1 ,...,ιm ∈Σh

m
Y

X

m
Y

p−h

βj,k`

k1 ,...,km `=1

)
Ψ̃⊗h
j,k`

(Xι` , ει` ) .

`=1

Similar to the proof of Proposition 3.3, we fix a configuration of indexes ι1 , . . . , ιm ∈ Σh ,
corresponding to the set of coefficients {c1 , c . . . , ch }. Because, in this case, the
considered U-statistic is degenerate, we discard all the combinations with c1 6= 0,
in view of (17). On the other hand, following Lemma 2.1, we have that
n
o
Ph
X
Ph
`
⊗h
`=2 ( 2 −1)c` }
`=1 c` B dj {
E Ψ̃⊗h
j,k1 (X1 , ε1 ) . . . Ψ̃j,kh (Xh , εh ) ≤ Cn
ι1 ,...,ιm ∈Σh
Ph

= Cn

`=1

c`

B dj (

mh
2 −

Pm

`=2

c` )

.
(18)

Pm
P
Furthermore, mh = `=2 `c` > 2 `=2 c` implies that the exponent in the last
term of (18) is positive, so that
n
o
X
mh
⊗h
⊗h
E Ψ̃j,k
(X
,
ε
)
.
.
.
Ψ̃
(X
,
ε
)
≤ Cn 2 ,
1
1
h
h
j,k
1
h
ι1 ,...,ιm ∈Σh

because B dj ≤ n. Finally, using the Stirling approximation and Eq. (10) we
have that
!m
p
om i
hn
m(p−h)
X
X
mh
n
p−h
0
b j (p) − Θj (p)
≤C
|βj,k |
n 2
E Θ
nmp
h=1
k

!1− hp m
p


X
X
h
mh
p
≤C 00
n− 2
B jd p
|βj,k |


h=1

≤C̃2

k

p 
X
h=1

 mh
jd
p

B
np/2

1− h m
{Θj (p)}( p ) ,

as claimed.
The proof of Lemma 3.5 is the counterpart of the proof of Lemma 6 in
Kerkyacharian et al. [30] in the needlet framework.
Proof of Lemma 3.5. Following Kerkyacharian et al. [30] and the results ob-

32

tained in the previous proof, we have that

 −1 X
p00
Kj

X
X
0
n
p0 −h
Θ̂j (p0 ) − Θj (p0 ) =
βj,k

p0
k=1 υ∈Σp0 h=1

+

+

1
n
1
n

1
2

⊗(h−1)

X

Ψ̃j,k

X

Ψ̃⊗h
j,k (Xι , ει )

ι⊂υ,ι∈Σh

(Xι , ει )

ι⊂υ,ι∈Σh−1

X
ι⊂υ,ι∈Σh−2

with the convention
X
⊗(0)
Ψ̃j,k (Xι , ει ) = 2;



⊗(h−2)
Ψ̃j,k
(Xι , ει ) ,


X

⊗(h)

Ψ̃j,k (Xι , ει ) = 0 for h < 0.

ι⊂υ,ι∈Σh

ι⊂υ,ι∈Σ0

Reversing the order of integration and applying an analogous procedure to the
one used in the proof of Proposition 3.4, we achieve the claimed result.
Proposition 3.6 is proved by using the general properties of the needlets.
Proof of Proposition 3.6. It is easy to see that

2

Kj n
Kj n
2 1 X
i 1X
X h
X
2
b∞
E Θ
= 2
E {ψj,k (Xi ) Yi } +
E{ψj,k (Xi ) Yi }
j
n
n
i=1
i=1
k=1

k=1

d

≤


2
B j
M 2 + σε2 kψj,k kL2 (Sd ) + Θ∞
,
j
n

as claimed.
Acknowledgement. The author wishes to thank M. Konstantinou, A. Ortiz,
A. Renzi and N. Turchi for precious discussions and hints. Furthermore, the
author wishes to acknowledge the Associate Editor, the referees and the Editorin-Chief for the insightful remarks and suggestions which led to a substantial
improvement of this work.
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