arXiv:1702.08787v2 [math.PR] 18 May 2017

Compound Poisson approximation to estimate the Lévy density
Céline Duval

∗

and Ester Mariucci

†

Abstract
We construct an estimator of the Lévy density, with respect to the Lebesgue measure,
of a pure jump Lévy process from high frequency observations: we observe one trajectory
of the Lévy process over [0, T ] at the sampling rate ∆, where ∆ → 0 as T → ∞. The main
novelty of our result is that we directly estimate the Lévy density in cases where the process
may present infinite activity. Moreover, we study the risk of the estimator with respect
to Lp loss functions, 1 ≤ p < ∞, whereas existing results only focus on p ∈ {2, ∞}. The
main idea behind the estimation procedure that we propose is to use that “every infinitely
divisible distribution is the limit of a sequence of compound Poisson distributions” (see
e.g. Corollary 8.8 in Sato (1999)); we can thus profit from the abundance of techniques
developed for the estimation of the Lévy density of a compound Poisson process in the
high frequency setting. We consider linear wavelet estimators and the performance of our
procedure is studied in term of Lp loss functions, p ≥ 1, over Besov balls. The resulting
rates are minimax-optimal for a large class of Lévy processes, including subordinators.
The results are illustrated on several examples.

Keywords. Density estimation, Infinite variation, Pure jump Lévy processes.
AMS Classification. 60E07, 60G51, 62G07, 62M99.

1

Introduction

Over the past decade, there has been a growing interest for Lévy processes. They are a
fundamental building block in stochastic modeling of phenomena whose evolution in time
exhibits sudden changes in value. Many of these models have been suggested and extensively
studied in the area of mathematical finance (see e.g. [7] which explains the necessity of
considering jumps when modeling asset returns). They play a central role in many other
fields of science: in physics, for the study of turbulence, laser cooling and in quantum theory;
in engineering for the study of networks, queues and dams; in economics for continuous timeseries models, in actuarial science for the calculation of insurance and re-insurance risk (see
e.g. [1, 4, 5, 32]).
∗
Sorbonne Paris Cité, Université Paris Descartes, MAP5, UMR CNRS 8145.
E-mail:
celine.duval@parisdescartes.fr
†
Institut für Mathematik, Humboldt-Universität zu Berlin. E-mail: mariucce@math.hu-berlin.de. This
research was supported by the Federal Ministry for Education and Research through the Sponsorship provided
by the Alexander von Humboldt Foundation.

1

From a mathematical point of view the jump dynamics of a Lévy process X is dictated
by its Lévy density. If it is continuous, its value at a point x0 determines how frequent jumps
of size close to x0 are to occur per unit time. Thus, to understand the jump behavior of X,
it is of crucial importance to estimate its Lévy density.
A problem that is now well understood is the estimation of the Lévy density of a compound
Poisson process, that is, a pure jump Lévy process with a finite Lévy measure. There is a
vast literature on the nonparametric estimation for compound Poisson processes both from
high frequency and low frequency observations (see among others, [6, 8, 10, 14, 15], and [21]
for the multidimensional setting).
Building an estimator of the Lévy density for a Lévy process X with infinite Lévy measure
is a more demanding task; for instance, for any time interval [0, t], the process X will almost
certainly jump infinitely many times. In particular, its Lévy density, which we mean to
estimate, is unbounded in any neighborhood of the origin. This implies that the techniques
used for compound Poisson processes do not generalize immediately. The essential problem
is that the knowledge that an increment Xt+∆ − Xt is larger than some ε > 0 does not give
much insight on the size of the largest jump that has occurred between t and t + ∆.
Many results are nevertheless already present in the literature concerning the estimation
of the Lévy density from discrete
data without the finiteness hypothesis, i.e., if we denote by
R
f the Lévy density, when R f (x)dx = ∞. In that case, the main difficulty comes from the
presence of small jumps and from the fact that the Lévy density blows up in a neighborhood
of the origin. A number of different techniques has been employed to address this problem:
• To limit the estimation of f on a compact set away from 0;
• To study a functional of the Lévy density, such as xf (x) or x2 f (x).

The analysis systematically relies on spectral approaches, based on the use of the LévyKhintchine formula (see (5) hereafter), that allows estimates for L2 and L∞ loss functions,
but does not generalize easily to Lp for p ∈
/ {2, ∞}. A non-exhaustive list of works related to
this topic includes: [2, 9, 11, 12, 16, 18, 19, 20, 23, 24, 25, 29, 35]; a review is also available
in the textbook [3]. Projection estimators and their pointwise convergence have also been
investigated in [17] and more recently in [27], where the maximal deviation of the estimator
is examined. Two other works that, although not focused on constructing an estimator of
f , are of interest for the study of Lévy processes with infinity activity in either low or high
frequency are [30, 31]. Finally, from a theoretical point of view, one could use the asymptotic
equivalence result in [28] to construct an estimator of the Lévy density f using an estimator
of a functional of the drift in a Gaussian white noise model. However, any estimator resulting
from this procedure would have the strong disadvantage of being randomized and, above all,
would require the knowledge of the behavior of the Lévy density in a neighborhood of the
origin.
The difference in purpose between the present work and the ones listed above is that we
aim to build an estimator fb of f , without a smoothing treatment at the origin, and to study
the following risk:

Z
p
b
|f (x) − f (x)| dx ,
E
A(ε)

2

where, ∀ε > 0, A(ε) is a set non containing 0: A(ε) is included in R \ (−ε, ε) and such that
a(ε) := minx∈A(ε) |x| ≥ ε where, possibly, a(ε) → 0 as ε → 0. To the knowledge of the authors
the present work is the first attempt to build and study an estimator of the Lévy density
on a set that gets near the critical value 0 and whose risk is measured for Lp loss functions,
1 ≤ p < ∞.
More precisely, let X be a pure jump Lévy process with Lévy measure ν (see (3) for a
precise definition) and suppose we observe

Xi∆ − X(i−1)∆ , i = 1, . . . , n with ∆ → 0 as n → ∞.
(1)

Our goal is to estimate the density of the Lévy measure ν with respect to the Lebesgue
measure, f (x) := ν(dx)
dx , from the observations (1) on the set A(ε) as ε → 0. In this paper, the
Lévy measure ν may have infinite variation, i.e.
Z
Z
2
|x|ν(dx) = ∞.
(x ∧ 1)ν(dx) < ∞ but possibly
ν:
|x|≤1

R

The starting point of this investigation is to look for a translation from a probabilistic to
a statistical setting of Corollary 8.8 in [34]: “Every infinitely divisible distribution is the limit
of a sequence of compound Poisson distributions”. We are also motivated by the fact that a
compound Poisson approximation has been successfully applied to approximate general pure
jump Lévy processes, both theoretically and for applications. For example, using a sequence
of compound Poisson processes is a standard way to simulate the trajectories of a pure jump
Lévy process (see e.g. Chapter 6, Section 3 in [13]).
We briefly describe here the strategy of estimation for f on A(ε). Given the observations
(1), we choose ε ∈ (0, 1] (when ν(R) < ∞ the choice ε = 0 is also allowed) and we focus only
on the observations such that |Xi∆ − X(i−1)∆ | > ε. Let us denote by n(ε) the random number
of observations satisfying this constraint. In the sequel, we will sometimes use the expression
“small jumps” of the Lévy process X at time t when referring to one of the following objects:
if ν is of finite variation, “small jumps” stand for the sum of ∆Xs , the jumps of X at times
s ≤ t, that are smaller than ε in absolute value. If ν is of infinite variation, they correspond to
the centered martingale at time t that is associated with the sum of the jumps the magnitude
of which is less than ε in absolute value.
The observations larger than ε in absolute value are in some sense close to the increments
of the compound Poisson process Z(ε) associated with the Lévy density I|x|>ε ν(dx)
dx =: λε hε (x),
where λε := ν(R \ (−ε, ε)) and hε (x) :=

1 ν(dx)
λε dx I|x|>ε .

f (x) = lim λε hε (x)I|x|>ε
ε→0

It immediately follows that
∀x 6= 0.

Therefore, one can construct an estimator of f on A(ε) by constructing estimators for λε
and hε separately. However, estimating hε and λε from observed increments larger than ε is
not straightforward due to the presence of the small jumps. In particular, if i0 is such that
|Xi0 ∆ − X(i0 −1)∆ | > ε, it is not automatically true that there exists s ∈ ((i0 − 1)∆, i0 ∆] such
3

that |∆Xs | > ε (or any other fixed positive number). Ignoring for a moment this difficulty
and reasoning as if the increments of X larger than ε are increments of a compound Poisson
process with intensity λε and jumps density hε , the following estimators are constructed.
First, a natural choice when estimating λε is
bn,ε = n(ε) .
λ
(2)
n∆
In the special case where the Lévy measure ν is finite, we are allowed to take ε = 0 and the
estimator (2), as ∆ → 0, gets close to the maximum likelihood estimator. The study of the
risk of this estimator with respect to an Lp norm is the subject of Theorem 1. Estimators of
the cumulative distribution function of f have also been investigated in [30, 31], which is an
estimation problem that is closely related to the estimation of λε . However, as we detail in
Section 3.2.2 below, the methodology used there does not apply in our setting: it cannot be
adapted to Lp -risks and the results of [30, 31] are established for non-vanishing ε whereas we
are also interested in the case ε → 0.
Second, for the estimation of the jump density hε , we fully exploit the fact that the
observations are in high frequency. Under some additional constraints on the asymptotic of ε
and ∆, we make the approximation hε ≈ L(X∆ |X∆ | > ε) and we apply a wavelet estimator
to the increments larger than ε, in absolute value. The resulting estimator b
hn,ε is close to
hε in Lp loss on A(ε) (see Proposition 2 and Corollary 2). As mentioned above, the main
difficulty in studying such an estimator is due to the presence of small jumps that are difficult
to handle and limit the accuracy of the latter approximation. Also, we need to take into
account that b
hn,ε is constructed from a random number of observations n(ε).
bn,ε and b
Thirdly, making use of the estimators λ
hn,ε we derive an estimator of f under very
bn,εb
hn,ε and we study its risk in Lp norm. Our
weak regularity conditions: we define fbn,ε := λ
main result is then a consequence of Theorem 1 and Corollary 2 and it is stated in Theorem
2. Under very mild conditions on the Lévy density, it is easy to show that the upper bound
we provide tends to 0, regardless of the rate at which ∆ tends to 0 (see Theorem 2). However,
in this upper bound, terms depending on the behavior of the small jumps appear, making
it difficult to derive an explicit rate of convergence. Under additional assumptions on the
Lévy density, we provide an explicit rate of convergence (see Theorem 3 and Corollary 3).
In particular, Theorem 3 ensures that our estimator is minimax-optimal for smooth Lévy
densities when ∆ goes to 0 rapidly enough, for instance n∆2 ≤ 1, at least when ε is fixed and
p = 2 in which cases we can compare to the existing literature (see e.g. [9, 11]).
The fact that known optimal results can be recovered as special cases of Theorem 3 justify
the correctness of our approach. At the same time, however, it has the advantage of providing
upper bounds for any Lp -risk, p ≥ 1; furthermore, it allows the estimation set to get close to
the origin. To our knowledge, these are both novel results.
The rate presented in Theorem 3 depends explicitly on the behavior of the small jumps.
The rate is optimal and independent of the activity of the small jumps if the sampling rate
∆ is small enough (e.g. n∆2 < 1) and the Lévy density is sufficiently regular; otherwise, the
more important is the activity of the small jumps, the slower is the rate of convergence (see
the term v2 (β) in Theorem 3). However, the convergence rates are explicit regardless of how
slowly ∆ goes to 0, provided that we remain under a high frequency regime. The case of finite
4

variation Lévy processes, and more particularly subordinators, is considered separately (see
Corollary 3).
The paper is organized as follows. Preliminary Section 2 provides the statistical context
as well as the necessary definitions and notations. An estimator of the intensity λε is studied
in Section 3.2 and a wavelet density estimator of the density hε in Section 3.3. Our main
Theorems 2 and 3 are given in Section 4. In the present article, for the sake of clarity,
we consider the case where the Lévy process does not have a Gaussian component, but our
procedure generalizes quite easily to the case where the Gaussian part is nonzero. The details
are given in Section 5. Finally, Section 6 contains the proofs of the main results while Appendix
7 collects the proofs of the auxiliary results.

2

Notations and preliminaries

We consider the class of pure jump Lévy processes with Lévy triplet (γν , 0, ν) where
(R
R
if
|x|≤1 |x|ν(dx) < ∞,
|x|≤1 xν(dx)
γν :=
0
otherwise.
We recall that almost allR paths of a pure jump Lévy process with Lévy measure ν have finite
variation if and only if |x|≤1 |x|ν(dx) < ∞. Thanks to the Lévy-Itô decomposition one can
write a Lévy process X of Lévy triplet (γν , 0, ν) as the sum of two independent Lévy processes:
for all ε ∈ (0, 1]
Xt = tbν (ε) + lim

η→0

X
s≤t

∆Xs 1(η,ε] (|∆Xs |) − t

Z

η<|x|≤ε

=: tbν (ε) + Mt (ε) + Zt (ε)

 NX
t (ε)
xν(dx) +
Yi (ε)
i=1

(3)

where
• The drift bν (ε) is defined as
(R
bν (ε) :=

|x|≤ε xν(dx)
R
− ε≤|x|<1 xν(dx)

if

R

|x|≤1 |x|ν(dx)

otherwise;

< ∞,

(4)

• ∆Xr denotes the jump at time r of the càdlàg process X: ∆Xr = Xr − lims↑r Xs ;
• M (ε) = (Mt (ε))t≥0 and Z(ε)
R = (Zt (ε))t≥0 are two independent Lévy processes of Lévy
triplets (0, 0, I|x|≤ε ν) and ( ε≤|x|<1 xν(dx), 0, I|x|>ε ν), respectively;

• M (ε) is a centered martingale consisting of the sum of the jumps of magnitude smaller
than ε in absolute value;

5

• Z(ε) is a compound Poisson process
defined as follows: N (ε) = (Nt (ε))t≥0 is a PoisR
son process of intensity λε := |x|>ε ν(dx) and (Yi (ε))i≥1 are i.i.d. random variables
independent of N (ε) such that P(Y1 (ε) ∈ A) =

ν(A)
λε ,

for all A ∈ B(R \ (−ε, ε)).

The advantage of defining the drift bν (ε) as in (4) lies in the fact that this definition allows
us to consider both the class of processes
X
∆Xs ,
Xt =
s≤t

when ν is of finite variation, and the class of Lévy processes with Lévy triplet (0, 0, ν), when
ν is of infinite variation. Furthermore, when ν(R) < ∞, we can also take ε = 0 and then
Equation (3) reduces to a compound Poisson process
Xt =

Nt
X

Yi

i=1

where the intensity of the Poisson process is λ = λ0 = ν(R \ {0}) = ν(R) and the density of
the i.i.d. random variables (Yi )i≥0 is f (x)/λ.
We also recall that the characteristic function of any Lévy process X as in (3) can be
expressed using the Lévy-Khintchine formula. For all u in R, we have

Z

 iuXt 
iuy
E e
= exp ituγν + t
(e − 1 − iuyI|y|≤1 )ν(dy) ,
(5)
R

where ν is a measure on R satisfying
ν({0}) = 0

and

Z

R

(|y|2 ∧ 1)ν(dy) < ∞.

(6)

In the sequel we shall refer to (γ ν , 0, ν) as the Lévy triplet of the process X and to ν as the
Lévy measure. This triplet characterizes the law of the process X uniquely.
Let us assume that the Lévy measure ν is absolutely continuous with respect to the
Lebesgue measure and denote by f (resp. fε ) the Lévy density of X (resp. Z(ε)), i.e.
I
ν(dx)
(fε (x) = |x|>εdx
). Let hε be the density, with respect to the Lebesgue
f (x) = ν(dx)
dx
measure, of the random variables (Yi (ε))i≥0 , i.e.
fε (x) = λε hε (x)1(ε,∞) (|x|).
We are interested in estimating f in any set of the form A(ε) := (A, −a(ε)] ∪ [a(ε), A) where,
for all 0 ≤ ε ≤ 1, a(ε) is a non-negative real number satisfying a(ε) ≥ ε and A ∈ [1, ∞] (the
case a(ε) = ε = 0 is not excluded). The latter condition is technical, if X is a compound
Poisson process we may choose A := +∞, otherwise we work under the simplifying assumption
that A(ε) is a bounded set. Observe that, for all A ⊂ A(ε) we have
f (x)1A (|x|) = λε hε (x)1A (|x|).
6

(7)

In general, the Lévy density f goes to infinity as x ↓ 0. It follows that if a(ε) ↓ 0, for instance
a(ε) = ε, we estimate a quantity that gets larger and larger. In theR decomposition (7) of
the Lévy density, the quantity that increases as ε goes to 0 is λε = |x|>ε f (x)dx, whereas

the density hε := λfεε may remain bounded in a neighborhood of the origin. The intensity λε
carries the information on the behavior of the Lévy measure f around 0.

Suppose we observe X on [0, T ] at the sampling rate ∆ > 0, without loss of generality, we
set T := n∆ with n ∈ N. Define
Xn,∆ := (X∆ , X2∆ − X∆ , . . . , Xn∆ − X(n−1)∆ ).

(8)

We consider the high frequency setting where ∆ → 0 and T = n∆ → ∞ as n → ∞. The
assumption n∆ → ∞ is necessary to construct a consistent estimator of f . To build an
estimator of f on the set A(ε), we do not consider all the increments (8), but only those
larger than ε in absolute value. Define the dataset Dn,ε := Xi∆ − X(i−1)∆ , i ∈ Iε , where
Iε is the subset of indices such that

Iε := i = 1, . . . , n : |X(i−1)∆ − Xi∆ | > ε .
Furthermore, denote by n(ε) the cardinality of Iε , i.e.:
n(ε) :=

n
X
i=1

1R\[−ε,ε](|Xi∆ − X(i−1)∆ |),

(9)

which is random.
We examine the properties of our estimation procedure in terms of Lp loss functions,
restricted to the estimation set A(ε), for all 0 ≤ ε ≤ 1. Let p ≥ 1,
1
n
Z
o
p
|g(x)|p dx < ∞ .
Lp,ε = g : kgkLp ,ε :=
A(ε)

Define the loss function


ℓp,ε fb, f := E fb − f

1/p
p
Lp,ε

=

Z

A(ε)


 1/p
E |fb(x) − f (x)|p dx
,

where fb is an estimator of f built from the observations Dn,ε .
Finally, denote by P∆ the distribution of the random variable X∆ and by Pn the law of
the random vector Xn,∆ as defined in (8). Since X is a Lévy process, its increments are i.i.d.,
hence
n
O
⊗n
, where Pi,∆ = L (Xi∆ − X(i−1)∆ ).
Pi,∆ = P∆
Pn =
i=1

We consider also the following family of product measures:
O
Pn,ε =
Pi,∆ .
i∈Iε

7

In the following, whenever confusion may arise, the reference probability in expectations is
explicitly stated, for example, writing EPn . The indicator function will be denoted equivalently
as 1A (x) or Ix∈A .

3

Preliminary estimators

3.1

Estimation strategy

Contrary to existing results mentioned in Section 1, we adopt an estimation strategy that does
not rely on the Lévy-Khintchine formula (5). A direct strategy based on projection estimators
and their limiting distribution has been investigated in [17, 27]. Here, we consider a sequence
of compound Poisson processes, indexed by 0 < ε ≤ 1, that gets close to X as ε ↓ 0 and we
approximate f using that
f (x) = lim λε hε (x),
εց0

∀x ∈ A(ε).

For each compound Poisson process Z(ε), we build separately an estimator of its intensity,
λε , and a wavelet estimator for its jump density, hε . This leads to an estimator of fε = λε hε .
Therefore, we deal with two types of error; a deterministic approximation error arising when
replacing f by fε and a stochastic error occurring when replacing fε by an estimator fbn,ε .
The main advantage of considering a sequence of compound Poisson processes is that, in
the asymptotic ∆ → 0, we can relate the density of the observed increments to the density
of the jumps without going through the Lévy-Khintchine formula (see [14]). This approach
enables the study of Lp loss functions, 1 ≤ p < ∞. Our estimation strategy is the following.
1. We build an estimator of λε using the following result that is a minor modification of
Lemma 6 in Rüschendorf and Woerner [33]. For sake of completeness we reproduce their
argument in the Appendix.
Lemma
1. Let X be a Lévy process with Lévy measure ν. If g is a function such that
R
g(x)
= 0 and (|x|
g(x)ν(dx)
< ∞, limx→0 g(x)
2 ∧1) is bounded for all x in R, then
|x|≥1
x2
Z
1
g(x)ν(dx).
lim E[g(Xt )] =
t→0 t
R
In particular, Lemma 1 applied to g = 1R\[−ε,ε] implies that
1
P(|X∆ | > ε),
∆→0 ∆

λε = lim

Using this equation, we approximate λε by
terpart of P(|X∆ | > ε).

∀ 0 ≤ ε ≤ 1.

1
∆ P(|X∆ |

> ε) and take the empirical coun-

hn,ε
2. From the observations Dn,ε = (Xi∆ − X(i−1)∆ )i∈Iε we build a wavelet estimator b
of hε relying on the approximation that for ∆ small, the random variables (Xi∆ −
X(i−1)∆ )i∈Iε are i.i.d. with a density close to hε (see Lemma 4).
3. Finally, we estimate f on A(ε) following (7) by
bn,εb
fbn,ε (x) := λ
hn,ε (x)1A(ε) (x),
8

∀x ∈ A(ε).

(10)

3.2
3.2.1

bn,ε
Statistical properties of λ

bn,ε
Asymptotic and non-asymptotic results for λ

First, we define the following estimator of the intensity of the Poisson process Z(ε) in terms
of n(ε), the number of jumps that exceed ε.
bn,ε be the estimator of λε defined by
Definition 1. Let λ
bn,ε := n(ε) ,
λ
n∆

where n(ε) is defined as in (9).

(11)

1
Controlling first the accuracy of the deterministic approximation of λε by ∆
P(|X∆ | > ε)
and second the statistical properties of the empirical estimator of P(|X∆ | > ε), we establish
bn,ε .
the following bounds for λ

bn,ε be the estimator of λε
Theorem 1. For all n ≥ 1, ∆ = ∆n > 0 and ε = εn ∈ [0, 1], let λ
defined in (11). Then, for all p ∈ [1, 2), we have
p
n
po


bn,ε − λε |p ≤ C P(|X∆ | > ε) 2 + λε − P(|X∆ | > ε)
EPn |λ
n∆2
∆
and, for all p ∈ [2, ∞)


p 
n
po


bn,ε − λε |p ≤ C nP(|X∆ | > ε) ∨ P(|X∆ | > ε) 2 + λε − P(|X∆ | > ε)
,
EPn |λ
(n∆)p
n∆2
∆

where C is a constant depending only on p.
Moreover, if nP(|X∆ | > εn ) → ∞ as n → ∞, we have for large enough n that

p 
p


P(|X
|
>
ε)
P(|X
|
>
ε)
2
∆
∆
p
bn,ε − λε | ≤ λε −
, ∀p ∈ [1, ∞).
+O
EPn |λ
∆
n∆2
In general, the quantity

P(|X∆ |>ε)
∆

is not easy to handle. Thanks to Lemma 1, for all ε > 0

lim λε −

∆→0

P(|X∆ | > ε)
=0
∆

but the rate of convergence is not known in general (see Section
R 3.2.2 below). Nevertheless,
one can notice that in many cases of interest, in terms of λε = |x|>ε ν(dx),
λε −

P(|X∆ | > ε)
= O(∆λ2ε ),
∆

as long as

∆λε → 0.

This motivates Corollary 1 below; it is an immediate consequence of Theorem 1 under the
following hypothesis.
Assumption (H1 ): X is a Lévy process with a Lévy measure ν such that

P(|Xt | > ε) = tλε + O t2 λ2ε ,
as tλε → 0.
9

bn,ε be the estimator of λε
Corollary 1. Let n ≥ 1, ∆ = ∆n > 0 and ε = εn ∈ (0, 1], let λ
introduced in Definition 1 and assume that (H1 ) is satisfied. Then, for all p ∈ [1, ∞), we have

3.2.2



2 p


bn,ε − λε |p = O λε + λε ∆ 2 + λ2p ∆p .
EPn |λ
ε
n∆

Some remarks on Theorem 1

On Assumption (H1 ). In the literature, expansions of the cumulative distribution functions of Lévy processes have already been established. This includes a result resembling Assumption (H1 ), namely Theorem 3.2 in [18], where the authors consider a compound Poisson
approximation (slightly different from ours, see their Equation (3.2)) to control the probability
P(Xt > y) of a Lévy process X. More precisely, a Lévy process X with Lévy triplet (b, σ 2 , ν)
e ε with intensity
is
processes: a compound Poisson process X
R decomposed as the sum of two
∞
(1 − cε (x))ν(dx) (here cε ∈ C and it satisfies 1[− 2ε , 2ε ] ≤ cε ≤ 1[−ε,ε]) plus the reminder
e ε . Under certain hypotheses on the Lévy measure ν, [18] establishes that given
Xε = X − X

ȳ > 0 such that 0 < ε < min ȳ3 , 1 , there exists t0 > 0 such that, for any y ≥ ȳ and 0 < t < t0
Z ∞
R
−t (1−cε (x))ν(dx)
P(Xt > y) = te
(1 − cε (x))ν(dx) + Oε,ȳ (t2 ).
(12)
y

However, (12) cannot be applied in our framework: we cannot have y = ε, and even if we
did y cannot tend to 0 in their result. Furthermore, we need an explicit dependence in ε of
Oε,ȳ (t2 ). To our knowledge the proof of (12) cannot be easily adapted to obtain an expansion
for P(Xt > ε). It is still an open question to understand under which hypotheses on ν one
has
P(|Xt | > ε) = λε t + O(t2 λ2ε ), for tλε small enough.
Relation to other works. In [30] and [31], the authors propose estimators of the cumulative distribution function of the Lévy measure, which is closely related to λε . Indeed, following
their notation, the authors estimate the quantity
(R
t
ν(dx), if t < 0
N (t) = R−∞
∞
if t > 0.
t ν(dx),

Then, for all ε ∈ (0, 1], we have λε = N (−ε) + N (ε). The low frequency case is investigated
in [30] (∆ > 0) whereas [31] considers the high frequency setting (∆ → 0) and includes the
possibility that the Brownian part is nonzero. In both cases, an estimator of N based on a
spectral approach, relying on the Lévy-Khintchine formula (5), is studied. In [31] a direct
approach equivalent to our estimator is also proposed and studied.
For each of these estimators the performances are investigated in L∞ (V ), where the domain
V is bounded away from 0: V = (−∞, −ζ] ∪ [ζ, ∞), ζ > 0 fixed, and functional central
limit theorems are derived. In the high frequency setting, the expansion of the cumulative
distribution of a Lévy process presented in [18] is used to derive such results. However, as
explained earlier, the result of Figueroa-López and Houdré [18] cannot be applied in our
10

context. In conclusion, up to the knowledge of the authors, the study of the Lp -risk of λε
with ε possibly vanishing, has never been developed and therefore Theorem 1 is new. It is
worth mentioning that the ideas contained in the proof of Proposition 17 in [31] can be used
to derive an upper bound for |λε − ∆−1 P(|X∆ | > ε)| in terms of ∆ and ν. However, the upper
bound obtained in this way is not optimal in general and it is therefore not exploited here.
bn,ε . A
A corrected estimator. In some cases, it is possible to improve the estimator λ
trivial example is the case where X is a compound Poisson process. Then, one should set
ε = 0, as we have exactly
F∆ (0) = P(|X∆ | > 0) = P(N∆ (0) 6= 0) = 1 − e−λ0 ∆ .
Replacing F∆ (0) with its empirical counterpart
Fbn,∆ (0) and inverting the equation, one ob√
tains an estimator of λ0 converging at rate n∆ (see e.g. [14]). A more interesting example is
the case of subordinators, i.e. pure jump Lévy processes of finite variation and Lévy measure
concentrated on (0, ∞). If X is a subordinator of Lévy measure ν = 1(0,∞) ν, using the fact
that P(Z∆ (ε) > ε|N∆ (ε) 6= 0) = 1, we get
P(X∆ > ε) = P(M∆ (ε) + ∆bν (ε) > ε)e−λε ∆ + 1 − e−λε ∆ ,

ε > 0.

(13)

Suppose we know additionally that
v∆ (ε) := P(M∆ (ε) + ∆bν (ε) > ε) = o F∆ (ε)K
for some integer K. Note that, by Lemma 1, we have
∀ε ∈ (0, 1],



v∆ (ε)
=0
∆→0 ∆
lim

(14)

(15)

since the process (Mt (ε) + tbν (ε))t≥0 is a Lévy process with Lévy measure 1[−ε,ε](x)ν(dx).
Equation (15) as well as F∆ (ε) = O(λε ∆) ensures that K ≥ 1 (neglecting the influence of
λε with respect to ∆). Using the same notations as above, define the corrected estimator at
order K
k
K
1 X Fbn,∆ (ε)
K
e
, K ≥ 1.
λn,ε :=
∆
k
k=1

bn,ε . For 1 ≤ p < ∞, straightforward computations lead to
e1 = λ
If K = 1 we have λ
n,ε
E

eK
(λ
n,ε

p

− λε )



k
k
k
K
K
n 1  X
po
Fbn,∆ (ε)
F∆ (ε) p 
1 X F∆ (ε)
≤ Cp
−
− λε
E
+
∆p
k
k
∆
k
k=1
k=1

k
K
n
 1 − v (ε)  p o
E |Fbn,∆ (ε) − F∆ (ε)|p
1 X F∆ (ε)
∆
≤ Cp CK,p
+
−
log
∆p
∆p
k
1 − F∆ (ε)
k=1

11

where we used (13). Finally, using the proof of Theorem 1, expansion at order K of log(1 − x)
in 0 and assumption (14) we easily derive

p 
(K+1)p 
 K

e − λε )p ≤ C F∆ (ε) 2 ∨ F∆ (ε)
E (λ
.
n,ε
n∆2
∆p

In the case of a subordinator,
taking advantage of (13) and λε ∆

−1
O ∆ F∆ (ε) − λε , when λε 6= 0. In many examples ∆−1 F∆ (ε) −
fore v∆ (ε) = O(λ2ε ∆2 ). In these cases one should prefer the estimator
3.2.3

Examples

→ 0 we get v∆ (ε) =
λε = O(λ2ε ∆2 ), theree2 .
λ
n,ε

Compound Poisson process. Let X be a compound Poisson process with Lévy measure
ν and intensity λ (i.e. 0 < λ = ν(R) < ∞). As ν is a finite Lévy measure, we take ε = 0 in
(11) that is,
Pn
1R\{0} (|Xi∆ − X(i−1)∆ |)
b
λn,0 = i=1
.
n∆
In this case, it is straightforward that (H1 ) is satisfied and therefore the rate of convergence
is given by Corollary 1, with λ0 = λ being constant.
This rate depends on the rate at which ∆ goes to 0 and the bound of ∆p might, in some
cases, be slower than the parametric rate in (n∆)−p/2 = T −p/2 . In the compound Poisson case
another estimator of λ converging at parametric rate can be constructed using the Poisson
structure of the problem (see e.g. [14], one may also use the corrected estimator discussed
above).
Gamma process. Let X be a Gamma process of parameter (1, 1), that is a finite variation
R ∞ −x
−x
Lévy process with Lévy density f (x) = e x 1(0,∞) (x), λε = ε e x dx and
P(|Xt | > ε) = P(Xt > ε) =

Z

ε

∞

xt−1 −x
e dx,
Γ(t)

∀ε > 0,

R∞
where Γ(t) denotes the Γ function, i.e. Γ(t) = 0 xt−1 e−x dx. The following lemma, proven
in the Appendix, ensures that Assumption (H1 ) is satisfied and Corollary 1 provides the rate
of convergence with λε = − log(ε).
Lemma 2. Let X be a Gamma process of parameter (1,1), ε > 0 and ∆ > 0. It holds that
λε −
Cauchy process.


P(X∆ > ε)
= O log(ε)2 ∆ .
∆

Let X be a 1-stable Lévy process with

1
1
f (x) =
πx2 R\{0}

and

P(|X∆ | > ε) = 2

12

Z

∞
ε
∆

dx
.
π(x2 + 1)

Then, under the asymptotic ∆/ε → 0, we have
 ∆2 
P(|X∆ | > ε)
− λε = O 3 .
∆
ε

(16)


 ε 
2 ε3 π
ε2
< ∞.
−
arctan
−
3 2
∆
∆
∆2
→0 π ∆
ε

(17)

2
and, furthermore,
Indeed, observe that with such a
choice of the Lévy density we have λε = πε
ε
2 π
P(|X∆ | > ε) = π 2 − arctan ∆ . Hence, in order to prove (16), it is enough to show that

lim

To that purpose, we set y =
rule:
 1 



2
1 π
lim
− arctan
π y→0 y 3 2
y

∆
ε

and we compute the limit in (17) by means of de l’Hôpital

2
1
− 2 = lim
y
π y→0

π
2

− arctan
y3

 
1
y

−y

y2
< ∞.
y→0 (1 + y 2 )3πy 2

= lim

In the case where X is a Cauchy process of parameters (1, 1), Assumption (H1 ) is suboptimal,
therefore an application of Theorem 1 permits to derive a faster rate than the one implied by
Corollary 1.
n
Proposition 1 (Cauchy Process). Let 0 < ε = εn ≤ 1 and ∆ = ∆n be such that limn→∞ ∆
εn =
0. Then, for all p ≥ 1, there exist constants C1 , C2 and n0 , depending only on p, such that
∀n ≥ n0


2p
2


bn,ε − λ|p ≤ C1 ∆ + (n∆)− p2 1 + C2 ∆ .
EPn |λ
ε3p
ε
ε3

Let X be an inverse Gaussian process of parameter (1, 1), i.e.

Inverse Gaussian process.
f (x) =

e−x
3

x2

1(0,∞) (x)

√
2∆ π

and

P(X∆ > ε) = ∆e

Z

∞

e−x−

π∆2
x

3

x2

ε

dx.

Then,
√
P(X∆ > ε)
− λε ≤ e2∆ π
∆

Z

∞

√

π∆2
x

x

ε

After writing the exponential e−
0. Expanding e2∆

e−x e−

π∆2
x

3
2


−1

dx + e2∆

√

π


−1

Z

∞

3

x2

ε

as an infinite sum, we get I = O

π

e−x

∆2
3
ε2



dx =: I + II.
∆
√
→
ε

O √∆ε .

if ∆λε ∝

one finds that, under the same hypothesis, II = O(∆λε ) =
bn,ε is given by Corollary 1 with
Therefore, Assumption (H1 ) is satisfied and the rate of λ
1
λε = √ε .

13

3.3
3.3.1

Statistical properties of b
hn,ε
Construction of b
hn,ε

We estimate the density hε using a linear wavelet density estimator and study its performances
uniformly over Besov balls (see Kerkyacharian and Picard [26] or Härdle et al. [22]). We state
the result and assumptions in terms of the Lévy density f as it is the quantity of interest.
Preliminary on Besov spaces. Let (Φ, Ψ) be a pair of scaling function and mother wavelet
which are compactly supported, of class C r and generate a regular wavelet basis adapted to
the estimation set A(ε) (e.g. Daubechie’s wavelet). Moreover suppose that {Φ(x − k), k ∈ Z}
is an orthonormal family of L2 (R). For all f ∈ Lp,ε we write for j0 ∈ N
X
X X
βjk (f )Ψjk (x), ∀x ∈ A(ε)
f (x) =
αj0 k (f )Φj0 k (x) +
j≥j0 k∈Λj

k∈Λj0

j0

j

where Φj0k (x) = 2 2 Φ(2j0 x − k), Ψjk (x) = 2 2 Ψ(2j x − k) and the coefficients are
Z
Z
Ψjk (x)f (x)dx.
Φj0 k (x)f (x)dx and βjk (f ) =
αj0 k (f ) =
A(ε)

A(ε)

As we consider compactly supported wavelets, for every j ≥ j0 , the set Λj incorporates boundary terms that we choose not to distinguish in notation for simplicity. In the sequel we
apply this decomposition to hε . This is justified because fε ∈ Lp,ε implies hε ∈ Lp,ε and the
coefficients of its decomposition are αj0 k (hε ) = αj0 k (f )/λε and βj0 k (hε ) = βj0 k (f )/λε . The
latter can be interpreted as the expectations of Φj0 k (U ) and Ψjk (U ) where U is a random
variable with density hε with respect to the Lebesgue measure.
We define Besov spaces in terms of wavelet coefficients as follows. For r > s > 0, p ∈ [1, ∞)
s (A(ε)) if the norm
and 1 ≤ q ≤ ∞ a function f belongs to the Besov space Bp,q
s (A(ε)) :=
kf kBp,q

 X

k∈Λj0

p

|αj0 k (f )|

1

p

+

X
j≥j0

j(s+1/2−1/p)

2

X

k∈Λj

p

|βjk (f )|

 1 q  q1
p

(18)

is finite, with the usual modification if q = ∞. We consider Lévy densities f with respect to
the Lebesgue measure, whose restriction to the set A(ε) lies into a Besov ball:

(19)
F (s, p, q, Mε , A(ε)) = f ∈ Lp,ε : kf kBp,q
s (A(ε)) ≤ Mε ,

where Mε := Mλε , for a fixed constant M. Note that the regularity assumption is imposed
on f|A(ε) viewed as a Lp,ε function. Therefore the dependency in A(ε) (hence a(ε)) lies in Mε .
Also, the parameter p measuring the loss of our estimator is the same as the one measuring
the Besov regularity of the function, this is discussed in Section 3.3.2. Lemma 3 below follows
immediately from the definitions of hε and the Besov norm (18).
Lemma 3. For all 0 < ε ≤ 1, let fε be in F (s, p, q, Mε , A(ε)). Then, hε =

class F s, p, q, M, A(ε) .
14

fε
λε

belongs to the

Construction of b
hn,ε . Consider Dn,ε , the increments larger than ε. We need to estimate the
jump density hε but we only have access to the indirect observations {Xi∆ −X(i−1)∆ , i ∈ Iε },
where for each i ∈ Iε , we have
Xi∆ − X(i−1)∆ = Mi∆ (ε) − M(i−1)∆ (ε) + ∆bν (ε) + Zi∆ (ε) − Z(i−1)∆ (ε).
The problem is twofold. First, there is a deconvolution problem as the information on hε is
contained in the observations {Zi∆ (ε) − Z(i−1)∆ (ε), i ∈ Iε }. The distribution of the noise
M∆ (ε) + ∆bν (ε) is unknown, but since this quantity is small we neglect this noise and make
the approximation:
Xi∆ − X(i−1)∆ ≈ Zi∆ (ε) − Z(i−1)∆ (ε),

∀i ∈ Iε .

(20)

Second, even overlooking that it is possible that for some i0 ∈ Iε , |Xi0 ∆ − X(i0 −1)∆ | > ε and
Zi0 ∆ − Z(i0 −1)∆ = 0, the common density of Zi∆ − Z(i−1)∆ |Zi∆ − Z(i−1)∆ 6= 0 is not hε but it
is given by
p∆,ε (x) =

∞
X
k=1

P(N∆ (ε) = k|N∆ (ε) 6= 0)h⋆k
ε (x) =

∞
X
k=1

(λε ∆)k
h⋆k (x),
k!(eλε ∆ − 1) ε

∀x ∈ R,

(21)

where ⋆ denotes the convolution product. However, in the asymptotic ∆ → 0, we can neglect
the possibility that more than one jump of N (ε) occurred in an interval of length ∆. Indeed,
we have the following lemma.
Lemma 4. If λε ∆ → 0, then for all p ≥ 1 there exists some constant C > 2 such that:
p∆,ε − hε

Lp,ε

≤ C∆kf kLp,ε .

Finally, our estimator is based on the chain of approximations
hε ≈ p∆,ε ≈ L(X∆ ||X∆ | > ε).
Therefore, we consider the following estimator
X
b
hn,ε (x) =
α
bJ,k ΦJk (x),
k∈ΛJ

x ∈ A(ε),

(22)

where J is an integer to be chosen and
α
bJ,k :=

1 X
ΦJk (Xi∆ − X(i−1)∆ ).
n(ε)
i∈Iε

We work with a linear estimator, despite the fact that they are not always minimax for
s , 1 ≤ π, q ≤ ∞ (π 6= p). Our choice is motivated by the fact
general Besov spaces Bπ,q
that, contrary to adaptive optimal wavelet threshold estimators, linear estimators permit to
estimate densities on non-compact sets. But, most importantly, to evaluate the loss due to
the fact that we neglect the small jumps M∆ (ε) (see (20)), we make an approximation at
order 1 of our estimator b
hn,ε . We need our estimator to depend smoothly on the observations,
which is not the case if we consider usual thresholding methods. Finally, we recall that on the
class F (s, p, q, Mε , A(ε)) this estimator is optimal in the context of density estimation from
direct i.i.d. observations (see Kerkyacharian and Picard [26], Theorem 3).
15

3.3.2

Upper bound results

Adapting the results of [26], we derive the following conditional upper bound for the estimation
of hε when the Lévy measure is infinite. The case where X is a compound Poisson process is
illustrated in Proposition 3. Recall that A(ε) = (−A, −a(ε)] ∪ [a(ε), A) with A ∈ [1, ∞].
Proposition 2. For all ε ∈ (0, 1] assume fε belongs to the functional class F (s, p, q, Mε , A(ε))
defined in (19), for some 1 ≤ q ≤ ∞, 1 ≤ p < ∞, ε ∈ (0, 1] and A < ∞. Let r > s > p1 and let
b
hn,ε be the wavelet estimator of hε on A(ε), Rdefined in (22). Let v∆ (ε) := P(|M∆ (ε)+∆bν (ε)| >
ε), F∆ (ε) := P(|X∆ | > ε) and σ 2 (ε) := |x|≤ε x2 ν(dx). Suppose that there exist sequences
ε = εn (possibly converging to zero) and ∆ = ∆n → 0 as n → ∞, such that λε ∆ → 0 and
v∆ (ε)
1
F∆ (ε) ≤ 3 as n → ∞. Then, the following inequality holds. For all J ∈ N and for all finite
p ≥ 2, there exists a positive constant C > 0 such that:

h v (ε) p/2  v (ε) p i

∆
∆
p
b
+
E khn,ε ({Xi∆ − X(i−1)∆ }i∈Iε ) − hε kLp,ε |Iε ≤ C 22Jp
n(ε)F∆ (ε)
F∆ (ε)
+ 2−Jsp + 2Jp/2 n(ε)−p/2 + (∆kf kLp,ε )p
i
h
p/2
1−p
−p/2
2
p
J(5p/2−1)
n(ε) ∆ + n(ε)
σ (ε)∆
+ (bν (ε)∆)
+2
,

where n(ε) denotes the cardinality of Iε and C only depends on s, p, khε kLp,ε , khε kLp/2,ε ,
kΦk∞ , kΦ′ k∞ , kΦkp and M. For 1 ≤ p < 2 this bound still holds if one requires in addition
that hε (x) ≤ w(x), ∀x ∈ R for some symmetric function w ∈ Lp/2 .
1
The assumption Fv∆∆(ε)
(ε) ≤ 3 is not restrictive: this term is required to tend to 0 to get
a consistent procedure. Also, the constraint on hε ≤ w for 1 ≤ p < 2, is classical and it
is imposed in [26]. For instance, it is satisfied if hε is compactly supported. An immediate
consequence of the proof of Proposition 2 is the following.

Proposition 3. Assume that f is the Lévy density of a compound Poisson process and that
it belongs to the functional class F (s, p, q, M0 , R \ {0}) defined in (19), for some 1 ≤ q ≤ ∞,
hn,0 be the wavelet estimator of h0 on R \ {0},
1 ≤ p < ∞. Take A = ∞ and let r > s > p1 , b
defined in (22). Then, for all J ∈ N and p ∈ [2, ∞), there exists C > 0 such that:
i
h

E kb
hn,0 ({Xi∆ −X(i−1)∆ }i∈I0 ) − h0 kpLp,0 |I0 ≤ C 2−Jsp + 2Jp/2 n(0)−p/2 + (∆kf kLp,0 )p ,
where n(0) is the cardinality of I0 and C depends on s, p, kh0 kLp,0 , kh0 kLp/2,0 , λ0 , M,
kΦk∞ , kΦ′ k∞ and kΦkp . For 1 ≤ p < 2 this bound still holds if one requires in addition that
h0 (x) ≤ w(x), ∀x ∈ R for some symmetric function w ∈ Lp/2 .
1

s

Taking J such that 2J = n(0) 2s+1 leads to an upper bound in n(0)− 2s+1 ∨ ∆, where
s
n(0)− 2s+1 is the optimal rate of convergence for the density estimation problem from n(0)
i.i.d. direct observations. The error rate ∆ is due to the omission of the event that more than
one jump may occur in an interval of length ∆.
To get an unconditional bound we introduce the following result.
16

Lemma 5. Let F∆ (ε) := P(|X∆ | > ε). For all r ≥ 0 we have
 3nF (ε) −r

∆
≤ E n(ε)−r ≤ 2 exp
2

−3nF∆ (ε) 
32

+

 nF (ε) −r
∆
.
2

Using Lemma 5, together with (15), we can remove the conditioning on Iε and we get an
unconditional upper bound for b
hn,ε .

Corollary 2. For all ε ∈ (0, 1] assume fε belongs to the functional class F (s, p, q, Mε , A(ε))
hn,ε
defined in (19), for some 1 ≤ q ≤ ∞, 1 ≤ p < ∞ and A < ∞. Let r > s > 1p and let b
be the wavelet estimator of hε on A(ε), defined in (22). Suppose that there exist sequences
ε = εn (possibly converging to zero) and ∆ = ∆n → 0 as n → ∞, such that
• λε ∆ → 0 as n → ∞;
•

v∆ (ε)
F∆ (ε)

≤

1
3

as n → ∞;

• nF∆ (ε) → ∞ as n → ∞.
Then, for all J ∈ N and p ∈ [2, ∞) the following inequality holds:

h v (ε) p/2  v (ε) p i

∆
∆
p
b
+
E khn,ε ({Xi∆ −X(i−1)∆ }i∈Iε ) − hε kLp,ε ≤ C 22Jp
nF∆ (ε)2
F∆ (ε)
−p/2
+ 2−Jsp + 2Jp/2 nF∆ (ε)
+ (∆kf kLp,ε )p
i
h
p/2
1−p
−p/2
p
2
J(5p/2−1)
(nF∆ (ε)) ∆ + (nF∆ (ε))
+ (bν (ε)∆)
σ (ε)∆
+2
,

for some C > 0 depending only on s, p, khε kLp,ε , khε kLp/2,ε , kΦk∞ , kΦ′ k∞ , kΦkp and M.
For 1 ≤ p < 2 this bound still holds if one requires in addition that hε (x) ≤ w(x), ∀x ∈ R for
some symmetric function w ∈ Lp/2 .
The various terms appearing in this upper bound are discussed in Section 4.1.1.

4
4.1

Statistical properties of fbn,ε
Main Theorem: general result

Combining the results in Theorem 1 and Corollary 2 we derive the following upper bound for
the estimator fbn,ε of the Lévy density f , when ν(R) = ∞. The case where X is a compound
Poisson process is illustrated in Proposition 4 below.
Theorem 2. For all ε ∈ (0, 1], let fε belong to the functional class F (s, p, q, Mε , A(ε)) defined
in (19), for some 1 ≤ q ≤ ∞, 1 ≤ p < ∞ and A < ∞. Let r > s > p1 and let fbn,ε be the
estimator of f on A(ε), defined in (10). Suppose that there exist sequences ε = εn (possibly
converging to zero) and ∆ = ∆n → 0 as n → ∞, such that
• λε ∆ → 0 as n → ∞;
17

•

v∆ (ε)
F∆ (ε)

≤

1
3

as n → ∞;

• nF∆ (ε) → ∞ as n → ∞.
Then, for all J ∈ N and p ∈ [2, ∞), there exists C > 0 such that the following inequality holds:

p
h v (ε) p/2  v (ε) p i
n
p

F∆ (ε)  2
F∆ (ε) p
∆
∆
2Jp
p
≤C
ℓp,ε fbn,ε , f
2
+
λ
−
+
λ
+
ε
ε
n∆2
∆
nF∆ (ε)2
F∆ (ε)
−p/2
+ 2−Jsp + 2Jp/2 nF∆ (ε)
+ (∆kf kLp,ε )p
io
h
p/2
1−p
−p/2
2
p
J(5p/2−1)
(nF∆ (ε)) ∆ + (nF∆ (ε))
σ (ε)∆
+ (bν (ε)∆)
+2

R
where v∆ (ε) := P(|M∆ (ε) + ∆bν (ε)| > ε), F∆ (ε) := P(|X∆ | > ε), σ 2 (ε) := |x|≤ε x2 ν(dx) and
C depends on s, p, khε kLp,ε , khε kLp/2,ε , kΦk∞ , kΦ′ k∞ , kΦkp and M. For 1 ≤ p < 2 this
bound still holds if one requires in addition that fε (x) ≤ w(x), ∀x ∈ R for some symmetric
function w ∈ Lp/2 .
4.1.1

Discussion

Theorem 2 gives an explicit upper bound for the Lp -risk restricted to the estimation set A(ε)
for the estimation of the Lévy density f = ν(dx)
dx of a pure jump Lévy process with Lévy triplet
(γν , 0, ν). To ensure that the rate of Theorem 2 tends to 0, it is sufficient to notice that the
quantity v∆ (ε)/F∆ (ε) tends to 0. This is indeed the case for every Lévy processes due to
Lemma 1 (see also Equation (15)). If we decompose F∆ (ε) on the number N∆ (ε) of jumps of
the compound Poisson process Z(ε) we get, for all 0 < ε ≤ 1,
F∆ (ε) ≤ P(|M∆ (ε) + ∆bν (ε)| > ε)e−λε ∆ + e−λε ∆ λε ∆ + P(N∆ (ε) ≥ 2)
≤ v∆ (ε) + λε ∆ +

(23)

λ2ε ∆2
,
2

where v∆ (ε) := P(|M∆ (ε) + ∆bν (ε)| > ε). Therefore, together with (15), it is straighforward
to get that
v∆ (ε)
v∆ (ε)
=
(1 + o(1)) → 0 as ∆ → 0.
F∆ (ε)
∆
To understand the rate given by Theorem 2, we give a rough intuition of the different
terms. Thinking back on our estimation strategy, we made different approximations that
entail four different sources of errors (points 2-3-4 hereafter are related to the estimation of
hε whereas point 1 to the estimation of λε ).
1. Estimation of λε : The approximation strategy for the intensity λε leads to the error
(see Section 3.2.2)
 F (ε)  p
F∆ (ε) p
2
∆
:= E1 .
+ λε −
2
n∆
∆
18

2. Neglecting the event {|M∆ (ε) + ∆bν (ε)| > ε}: We consider that each time an increment X∆ exceeds the threshold ε the associated Poisson process N∆ (ε) is nonzero. This
leads to the error
s

v∆ (ε)
v∆ (ε)
v∆ (ε)
2J
2
:= E2 .
+
≍ 22J
2
nF∆ (ε)
F∆ (ε)
F∆ (ε)
This error is unavoidable as we do not observe M (ε) and Z(ε) separately.
3. Neglecting the presence of M∆ (ε) + ∆bν (ε): In (20) we ignore the convolution
structure of the observations. This produces the error in
o
n
1/2
+ (bν (ε)∆)p := E3 .
2J(5/2−1/p) (nF∆ (ε))−1+1/p ∆ + (nF∆ (ε))−1/2 σ 2 (ε)∆

It would have been difficult to have a better strategy than neglecting M∆ (ε)+bν (ε)∆: the
distribution of M∆ (ε) is unknown, therefore we cannot take into account the convolution
structure of the observations. Moreover, even if we did know it (or could estimate it),
deconvolution methods are essentially adapted to L2 losses.

4. Estimation of the compound Poisson Z(ε): This estimation problem is solved in
two steps. First, we neglect the event {N∆ (ε) ≥ 2} which generates the error:
∆kf kLp,ε := E4 .
This error could have been improved considering a corrected estimator as in [14], but this
would have added even more heaviness in the final result. Second, we recover an estimation error that is classical for the density estimation problem from i.i.d. observations
in
−1/2
:= E5 .
2−Js + 2J/2 nF∆ (ε)

Theorem 2 ensures that our estimator is consistent under very weak hypothesis on the
Lévy measure ν. Namely, f has a Besov norm restricted on A(ε) that does not grow more
1
than a constant times λε and, for n large enough, Fv∆∆(ε)
(ε) ≤ 3 and nF∆ (ε) → ∞ as n → ∞.
These are very mild assumptions on ν and ∆ (e.g. if Assumption (H1 ) is satisfied and λε 6= 0,
these conditions hold). However, the rate of convergence is not explicit in terms of n and ∆,
as it depends on the quantities F∆ (ε) and v∆ (ε) that depend implicitly on the Lévy measure
ν and are in general difficult to handle. Consequently, we cannot say in general which one of
the above error terms E1 , E2 , E3 , E4 or E5 is predominant and therefore which one is the
leading term. A more readable result is presented
in Theorem 3 below, where an explicit rate

b
of convergence for the Lp -risk ℓp,ε fn,ε , f is given.
4.1.2

Compound Poisson process

In the case where X is a compound Poisson process, Theorem 2 simplifies and it is easy to
derive the rate of convergence of our estimator. More general exemples, such as subordinators,
can be handled with Theorem 3 below.
19

Let X be a compound Poisson process, set ε = 0 as λ := λ0 < ∞ and we have
F∆ (0) = P(|X∆ | > 0) = 1 − e−λ∆ = O(∆).
It is straightforward to see that nF∆ (0) → ∞. Moreover, the choice ε = 0, simplifies the proof
e (0) and
of Proposition 2 significantly. Indeed, we have that v∆ (0) = 0, I0 = K0 , n(0) = n
that X∆ has distribution p∆,0 (see Section 6.2 for notations). Proposition 3 and Lemma 5 lead
then to the following upper bound. For all J ∈ N, ∀h0 ∈ F s, p, q, M, R \ {0}), 1 ≤ q ≤ ∞
and p ≥ 1, there exists a constant C > 0 such that:


E kb
hn,0 − h0 kpLp,0 ≤C 2−Jsp + 2Jp/2 (n∆)−p/2 + ∆p
where C depends on λ, M, s, p, kh0 kLp,0 , kΦk∞ , kΦ′ k∞ and kΦkp . Choosing J such that
1

2J = (n∆) 2s+1 we get



sp
E kb
hn,0 − h0 kpLp,0 ≤ C (n∆)− 2s+1 + ∆p ,

where the first term is the optimal rate of convergence to estimate p∆,0 from the observations
Dn,0 and the second term is the deterministic error of the approximation of h0 by p∆,0 . This
result is consistent with the results in [14] and is more general in the sense that the estimation
set is unbounded.
Concerning the estimation of the Lévy density f = f0 , we apply a slight modification of Theorem 2 (due to the simplifications that occur when taking ε = 0), and we use
Corollary 1 to derive the following result. Let ε = 0, assume that f0 belongs to the class
F (s, p, q, M0 , [−A, A] \ {0}) defined in (19), where 1 ≤ q ≤ ∞, 1 ≤ p < ∞ and A < ∞.
Here we consider a bounded set A(0) for technical reasons (see the proof of Theorem 2), this
assumptions might be removed at the expense of additional technicalities. Let J be such that
1
2J = (n∆) 2s+1 , then for p ≥ 2 we have


sp

sp

− p
p
ℓp,0 fbn,0 , f
= O (n∆)1−p ∨ n∆ 2 + (n∆)− 2s+1 + ∆p = O (n∆)− 2s+1 + ∆p .
The case p ∈ [1, 2) can be treated similarly and leads to the same rate. As earlier, the first
term is the optimal rate of convergence to estimate p∆,0 from the observations Dn,0 and the
second term gathers the deterministic errors of the approximations of h0 by p∆,0 and λ0 by
1
∆ P(|X∆ | > 0). We therefore established the following result.

Proposition 4. Let f ∈ F (s, p, q, M0 , [−A, A] \ {0}), 1 ≤ q ≤ ∞, 1 ≤ p < ∞ and A < ∞, be
the Lévy density of a compound Poisson process. Let r > s > 1p and let fbn,0 be the estimator
of f = f0 on [−A, A] \ {0}, defined in (10). Then, for all p ∈ [1, ∞) and n big enough, there
exists a constant C > 0 such that


p
sp

≤ C (n∆)− 2s+1 + ∆p ,
ℓp,0 fbn,0 , f
where C depends on λ, M, s, p, kh0 kLp,0 , kΦk∞ , kΦ′ k∞ and kΦkp .
20

4.2

An explicit rate under additional assumptions on ν

Without any specific assumption on the Lévy density, the rate of Theorem 2 is not explicit.
However, simplifying the rate is doable if one assumes (H1 ) and that v∆ (ε) can be controlled
(see Assumption (H2 (β)) below). We present the resulting rates in Theorem 3. The proof of
this theorem is omitted, as it is a straightforward computation: it suffices to apply Theorem
2 under Assumptions (H1 ) and (H2 (β)).
Assumption (H2 (β)): X is a Lévy process with a Lévy triplet (γν , 0, ν) such that

P(|Mt (ε) + tbν (ε)| > ε) = O tβ λjε ,
as tλε → 0,

for some β > 1 and j ≥ 0.

Theorem 3. Let X be a Lévy process with Lévy triplet (γν , 0, ν) and let ν be a Lévy measure
admitting a density f with respect to the Lebesgue measure. For all ε ∈ (0, 1], suppose that
fε belongs to the functional class F (s, p, q, Mε , A(ε)) defined in (19), for some 1 ≤ q ≤ ∞,
1 ≤ p < ∞, and A < ∞. Let r > s > p1 and let fbn,ε be the estimator of f on A(ε), defined in
(10). Suppose that there exist sequences ε = εn (possibly converging to zero) and ∆ = ∆n → 0
as n → ∞, such that
• Assumptions (H1 ) and (H2 (β)) hold;

• λkεn ∆n → 0, as n → ∞, for all k ∈ N;
• For 1 ≤ p < 2, fε (x) ≤ w(x), ∀x ∈ R for some symmetric function w ∈ Lp/2 .
p

is bounded, up to a polynomial factor depending
Then, for all p ≥ 1, the Lp -risk ℓp,ε fbn,ε , f
only on log ε, as follows:

sp
3(s+1)
2s+4
(n∆n)− 2s+1
and n∆2n ≤ 1,
if s ≥ 3−2/p


2 , 2s+1 < β < 2s+1


sp


2

max (n∆n )− 2s+1 , v1 (β), v2 (β), v3
if s < 3−2/p

2 , p ≥ 2 and n∆n ≤ 1,




sp
2
max (n∆n )− 2s+1 , v1 (β), v2 (β), v4
if s < 3−2/p
2 , 1 ≤ p < 2 and n∆n ≤ 1,



sp



if s ≥ 3−2/p
max (n∆n )− 2s+1 , v2 (β), v5
and n∆2n > 1,

2





max v (β), v
and n∆2n > 1,
if s < 3−2/p
2
5
2
sp

sp
(β−1) 2+s

where v1 (β) = (n∆n2−β )− 2s+4 , v2 (β) = ∆n
2sp

, v3 = n

sp
− 2s+5−2/p

2s
− 2s+5−2/p

p−1 )
, v4 = (∆−1
n (n∆n )

and v5 = ∆n2s+5−2/p .
In some cases it is easy to see that, under Assumption (H1 ), Assumption (H2 (β)) is
fulfilled. For instance, if X is a subordinator, for all ε = εn > 0 and ∆ = ∆n such that
λεn ∆n → 0 as n → ∞, we have


P(M∆n (εn ) + ∆n bν (εn ) > εn ) = eλεn ∆n P(X∆n > εn ) + e−λεn ∆n − 1 = O(λ2εn ∆2n ).

Therefore, Assumption (H2 (β)) is satisfied for β = 2. The case where X is subordinator is
the subject of Corollary 3 below, which is a straightforward consequence of Theorem 3, taking
β = 2.
21

Corollary 3. Let X be a subordinator with Lévy measure ν and Lévy density (with respect
to the Lebesgue measure) denoted by f . For all ε ∈ (0, 1], suppose that fε belongs to the
functional class F (s, p, q, Mε , A(ε)) defined in (19), for some 1 ≤ q ≤ ∞, 1 ≤ p < ∞, and
A < ∞. Let r > s > 1p and let fbn,ε be the estimator of f on A(ε), defined in (10). Suppose
that there exist sequences ε = εn (possibly converging to zero) and ∆ = ∆n → 0 as n → ∞,
such that
• Assumption (H1 ) holds;
• λkεn ∆n → 0, as n → ∞, for all k ∈ N;
• For 1 ≤ p < 2, fε (x) ≤ w(x), ∀x ∈ R for some symmetric function w ∈ Lp/2 .

The Lp -risk ℓp,ε fbn,ε , f , p ≥ 1, is bounded, up to a polynomial factor depending only on log ε,
as follows:

sp

(n∆n)− 2s+1
∨ 1,
if n∆2n ≤ 1 and s ≥ 3−2/p

2


sp

p
3−2/p
−
2
if n∆n ≤ 1 and s < 2 ∨ 1,
ℓp,ε fbn,ε , f
. max (n∆n ) 2s+1 , v2 (2), v3 , v4



sp

max (n∆ )− 2s+1 , v (2), v
if n∆2 > 1,
2

n

sp

where v2 (2) = ∆n2+s , v3 = n
4.2.1

sp
− 2s+5−2/p

5

n

2s
− 2s+5−2/p

p−1 )
, v4 = (∆−1
n (n∆n )

2sp

and v5 = ∆n2s+5−2/p .

Discussion

On Assumption on (H2 (β)). As pointed out in Section 3.2.2, there are results for the
expansion of the cumulative distribution of a given Lévy process, but they do not apply in
our framework. For instance, in [18] the authors use a compound Poisson approximation to
give an expansion for P(Xt > y) (see Section 3.2.2 for the notation) and the key ingredient
in controlling X ε , the Lévy process taking into account the small jumps, is the following tail
estimate valid for y big enough and bigger than ε (see Remark 3.1 in [18]):
P(Xtε

x
 R 2
e u cε (u)ν(du) ε x
tε,
≥ y) ≤
εx

(24)

whenever x := y−E[Xtε ] > 0 and σ = 0. The quantity Xtε is not exactly equal to Mt (ε)+tbν (ε)
but they are very close in spirit. Again, as it was the case for P(Xt > ε), we need the explicit
rate of convergence of P(|Mt (ε) + tbν (ε)| > ε) in function of ε and t. Thanks to (15) we know
that, as t → 0, there exists a function ρ depending on t and ε such that
P(|Mt (ε) + tbν (ε)| > ε) = tρ(t, ε),

with lim ρ(t, ε) = 0.
t→0

(25)

Unfortunately, the tail estimate (24) is not enough to recover (25) and it is still an open
problem how to write the function ρ(t, ε) in terms of ν, ε and t. That is why we need an
assumption such as (H2 (β)) which is a very mild hypothesis on ν in view of (25).

22

Comments on Theorem 3. Theorem 3 holds true under very weak hypothesis on ν
and it is stated without any assumption on ∆ beyond the fact that we are in a high frequency
regime. It gives an explicit upper bound for the Lp -risk for the estimator fbn,ε of the Lévy
density f restricted to the estimation set A(ε). This latter, contrary to existing results, can
get asymptotically close to the origin. Indeed, we do not require ε to be fixed. We may have
ε → 0 as long as λε goes to infinity slowly with respect to ∆. We also observe that Theorem
3 is minimax-optimal for Lévy densities with Besov regularity s ≥ 3−2/p
under the classical
2
2
assumption n∆ ≤ 1, ε fixed, and if (H2 (β)) is satisfied for β not too small with respect
to the regularity of f . Otherwise, the situation changes and a new term depending on the
behavior of the small jumps appears in the rate via the exponent β in the rates v1 (β) and
v2 (β). Therefore, the rate presented in Theorem 3 explicitly states how the behavior of the
small jumps may modify the rates of convergence. In particular, when β is too small and close
to 1, namely the small jumps have infinite activity, the term v2 (β) tends to 0 very slowly:
the presence of the small
jumps spoils the rates of convergence. In cases where the rate is
s
not given by (n∆n )− 2s+1 , we cannot ensure the minimax-optimality of the procedure as lower
bounds are unknown in these cases. For instance, if X is a subordinator (see Corollary 3),
∨ 1 and n∆2 ≤ 1, which are
then (H2 (2)) holds and our procedure is optimal if s ≥ 3−2/p
2
classical conditions when p = 2 and ε > 0, non vanishing, (see e.g. [9, 11]), but our procedure
has the same properties, up to a factor in log ε, if p ≥ 1 and ε := εn → 0 slowly.

5

Estimation in presence of a Gaussian component

It is worth mentioning that in case the Lévy process has a Brownian component the estimator
presented here applies and that the results established can be generalized without technical
e be a Lévy process with Lévy triplet (γν , σ, ν), σ ≥ 0. As in (3), X
e can be
difficulties. Let X
rewritten as the sum of three independent Lévy processes: for all ε ∈ (0, 1]
et = tbν (ε) + σWt + Mt (ε) + Zt (ε),
X

where W is a standard Wiener process, independent of M (ε) and Z(ε).


f
ei∆ − X
e(i−1)∆ , i ∈ If
As in Section 3, consider the increments X
ε where Iε := i =
ei∆ − X
e(i−1)∆ | > ε . Constructing estimators (11) and (22) from these increments,
1, . . . , n : |X
d √
one obtains estimators of λε and hε . In the asymptotic ∆ → 0 we have σW∆ = σ ∆N (0, 1)
e Theorem
and the approximation (20) still makes sense when applied to the increments of X.
1 holds true in this setting, as its proof remains unchanged by the additional Brownian part.
1
e∆ | > ε)| needs to be handled differently in examples.
P(|X
However, the quantity |λε − ∆
Moreover, it is easy to check, at the expense of small modifications in its proof, that Corollary
e∆ | > ε) and v∆ (ε) by v∆,σ (ε) :=
2 still holds after replacing in its statement F∆ (ε) by P(|X
P |∆bν (ε) + M∆ (ε) + σW∆ | > ε (both quantities agree with the previous definitions when
σ = 0). Combining those results, we get similarly that Theorem 2 holds.
Finally, we conclude with the following two inequalities, which are useful to apply the
former results in presence of a Gaussian component. First, an analogous decomposition to

23

the one in (23) leads to the inequality
2 2
e∆ | > ε) ≤ v∆,σ (ε)e−λε ∆ + λε ∆ + λε ∆ .
F∆ (ε) = P(|X
2

Moreover, it holds that

v∆,σ (ε) = P(|∆bν (ε) + M∆ (ε) + σW∆ | > ε) ≤ P(|∆bν (ε) + M∆ (ε)| + σ|W∆ | > ε)
r Z

x2
2 ∞
ε
≤ P |∆bν (ε) + M∆ (ε)| >
+
e− 2 dx = VI + VII ,
ε
2
π
√
2σ

∆

where by (15) we get that VI = v∆,0 ( 2ε ) = o(∆) and standard computations lead to VII = o(∆)
if ε−2 ∆ → 0. Therefore, we recover that, without further assumption, F∆ (ε) = O(∆) and
v∆,σ (ε) = o(∆) from which we obtain the consistency of the procedure. Moreover, under
Assumptions (H1 ) and (H2 (β)) the rates obtained in Theorem 3 also apply.

6

Proofs

In the sequel, C denotes a generic constant whose value may vary from line to line. Its
dependencies may be given in indices. The proofs of auxiliary lemmas are postponed to the
Appendix in Section 7.

6.1

Proof of Theorem 1

Let F∆ (ε) := P(|X∆ | > ε) and Fb∆ (ε) :=
E Pn

h

bn,ε
λε − λ

p

i

p

≤2



Pn

1
n

i=1 1(ε,∞) (|Xi∆

F∆ (ε)
λε −
∆

p

− X(i−1)∆ |). The following holds

h
i
1
p
b
.
+ p EPn F∆ (ε) − F∆ (ε)
∆

(26)

To control the second term in (26), we introduce the i.i.d. centered random variables
Ui :=

1(ε,∞) (|Xi∆ − X(i−1)∆ |) − F∆ (ε)
,
n

i = 1, . . . , n.



For p ≥ 2, an application of the Rosenthal inequality together with E |Ui |p = O
ensure the existence of a constant Cp such that
E Pn

 X
n

Ui

i=1

p





≤ Cp n

1−p

 F (ε) p/2 
∆
F∆ (ε) +
.
n

For p ∈ [1, 2), the Jensen inequality and the previous result for p = 2 lead to
E Pn

 X
n
i=1

Ui

p



≤



E Pn

n
h X
i=1

24

Ui

2i

p/2

≤



F∆ (ε)
n

p/2

.

F∆ (ε) 
np

In the asymptotic n → ∞, using (26), we are only left to show that, for p ≥ 2,
1
EP
∆p n

 X
n

Ui

i=1

p

=O




F∆ (ε) p/2
.
n∆2

(27)

An application of the Bernstein inequality (using that |Ui | ≤ n−1 and the fact that the
variance V[Ui ] ≤ F∆n2(ε) ) allows us to deduce that
P

 X
n
i=1

Ui ≥ t



≤ 2 exp



−

t2 n
2F∆ (ε) +

2t
3



.

Therefore,
E Pn

 X
n

Ui

i=1

p

=p

Z

∞

t

p−1

P

0

 X
n
i=1



Ui ≥ t dt ≤ 2p

Z

∞

∞
0

t

p−1

exp



t2 n
−
2F∆ (ε) +

2t
3



dt ≤

Z

3
F (ε)
2 ∆

t

p−1

0

exp

0

Observe that, for t ≤ 32 F∆ (ε), the denominator 2F∆ (ε) +
t ≥ 32 F∆ (ε) we have 2F∆ (ε) + 2t
3 ≤ 2t. It follows that
Z

t

p−1

2t
3



t2 n
−
2F∆ (ε) +

2t
3



dt.

is smaller than 3F∆ (ε) while for

t2 n 
dt +
exp −
3F∆ (ε)


Z

∞
3
F (ε)
2 ∆

tn

tp−1 e− 2 dt.

After a change of variables, the following inequalities hold:
Z

3
F (ε)
2 ∆

0


tp−1 exp −

1  3F∆ (ε) p/2  p 
t2 n 
dt ≤
Γ
3F∆ (ε)
2
n
2

(28)

 2 p  nF (ε) 
∆
.
Γ p,
n
4

(29)

and
Z

∞
3
F (ε)
2 ∆

tn

tp−1 e− 2 dt ≤

R∞
Here, Γ(s, x) = x xs−1 e−x dx denotes the incomplete Gamma function and Γ(s) = Γ(s, 0) is
the usual Gamma function. Equation (28) readily gives the desired asymptotic. To conclude,
we use the classical estimate for the incomplete Gamma function for |x| → ∞:

 1 
s−1
+O 2 .
Γ(s, x) ≈ xs−1 e−x 1 +
x
x

1
−nF∆ (ε) , which goes
In particular, when (29) is divided by ∆p , it is asymptotically O (n∆)
pe
to 0 faster than (27).

25

6.2

Proof of Proposition 2

Preliminary. Since the proof of Proposition 2 is lengthy, to help the reader we enlighten
here the two main difficulties that arise due to the fact that the estimator b
hn,ε uses the
observations Dn,ε , i.e. b
hn,ε = b
hn,ε (Dn,ε ).

1. The cardinality of Dn,ε is n(ε) that is random. That is why in Proposition 2 we study
the risk of this estimator conditionally on Iε . We then get the general result using that
ii
h
h
h

i
hn,ε , hε Iε .
hn,ε , hε = E EPn,ε ℓp,ε b
EPn ℓp,ε b
Once the conditional expectation is bounded, we use Lemma 5 to remove the conditioning and derive Corollary 2.

2. An observation of Dn,ε is not a realization of hε . Indeed, an increment of the process
Z(ε) does not necessarily correspond to one jump, whose density is hε , and, more
demandingly, the presence of the small jumps M (ε) needs to be taken into account. To
do so we split the sample Dn,ε in two according to the presence or absence of jumps
in the Poisson part. On the subsample where the Poisson part is nonzero, we make an
expansion at order 1 and we neglect the presence of the small jumps. This is the subject
of the following paragraph.
Expansion of b
hn,ε . Consider Dn,ε = {Xi∆ − X(i−1)∆ , i ∈ Iε } the increments larger than
ε. Recall that, for each i, we have
Xi∆ − X(i−1)∆ = ∆bν (ε) + Mi∆ (ε) − M(i−1)∆ (ε) + Zi∆ (ε) − Z(i−1)∆ (ε).
We split the sample as follows:
Kε := {i ∈ Iε , Zi∆ (ε) − Z(i−1)∆ (ε) 6= 0}

Kεc := Iε \ Kε .

e (ε) the cardinality of Kε . To avoid cumbersomeness, in the remainder of the proof
Denote by n
J
we write M instead of M (ε) and Z instead of Z(ε). Recall that ΦJk (x) = 2 2 Φ(2J x − k).
Using that Φ is continuously differentiable we can write, ∀k ∈ ΛJ ,
X 
1 X
α
bJ,k =
+
ΦJk (Xi∆ − X(i−1)∆ )
n(ε)
c
i∈Kε

=

1
n(ε)

X

i∈Kε

i∈Kε

ΦJk (Zi∆ − Z(i−1)∆ ) + 23J/2 (Mi∆ − M(i−1)∆ + bν (ε)∆)Φ′ (2J ηi − k)

1 X
ΦJk (Xi∆ − X(i−1)∆ ),
+
n(ε)
c
i∈Kε

26

where ηi ∈ [min{Zi∆ −Z(i−1)∆ , Xi∆ −X(i−1)∆ }, max{Zi∆ −Z(i−1)∆ , Xi∆ −X(i−1)∆ }]. It follows
that
X
b
hn,ε (x, {Xi∆ − X(i−1)∆ }i∈Iε ) =
α
bJ,k ΦJk (x)
k∈ΛJ

e (ε) e
n
hn,ε (x, {Zi∆ − Z(i−1)∆ }i∈Kε )
n(ε)
X
23J/2 X
(Mi∆ − M(i−1)∆ + bν (ε)∆)
Φ′ (2J ηi − k)ΦJk (x)
+
n(ε)
i∈Kε
k∈ΛJ
1 X X
ΦJk (Mi∆ − M(i−1)∆ + bν (ε)∆)ΦJk (x),
+
n(ε)
c

:=

i∈Kε k∈ΛJ

where conditional on Kε , e
hn,ε ({Zi∆ − Z(i−1)∆ }i∈Kε ) is the linear wavelet estimator of p∆,ε
e (ε) direct measurements. Explicitly, it is defined as follows
defined in (21) from n
X
e
hn,ε (x, {Zi∆ − Z(i−1)∆ }i∈Kε ) =
α
eJ,k ΦJk (x),
(30)
k∈ΛJ

P
where α
eJ,k = ne 1(ε) i∈Kε ΦJk (Zi∆ − Z(i−1)∆ ). This is not an estimator as both Kε and
eJ,k approximates the quantity
{Zi∆ − Z(i−1)∆ }i∈Kε are not observed. However, α
Z
ΦJk (x)p∆,ε (x)dx.
(31)
αJ,k :=
A(ε)

Decomposition of the Lp,ε loss.
we get

Taking the Lp,ε norm and applying the triangle inequality

kb
hn,ε ({Xi∆ − X(i−1)∆ }i∈Iε ) − hε kpLp,ε ≤ Cp


e (ε) p
n
n(ε)

ke
hn,ε ({Zi∆ − Z(i−1)∆ }i∈Kε ) − hε kpLp,ε


e (ε) p
n
+ 1−
khε kpLp,ε
n(ε)
Z
3Jp/2
X
X
p
2
′ J
+
(M
−
M
+
b
(ε)∆)
Φ
(2
η
−
k)Φ
(x)
dx
i∆
ν
i
Jk
(i−1)∆
n(ε)p A(ε)
i∈Kε
k∈ΛJ

Z
X X
p
1
dx
Φ
(M
−
M
+
∆b
(ε))Φ
(x)
+
i∆
ν
Jk
Jk
(i−1)∆
n(ε)p A(ε)
i∈Kεc k∈ΛJ

(32)
= Cp T1 + T2 + T3 + T4 .

After taking expectation conditionally on Iε and Kε , we bound each term separately.

Remark 1. If X is a compound Poisson process and we take ε = 0, then b
hn,ε = e
hn,ε (and
e (0)) and T2 = T3 = T4 = 0.
n(0) = n
27

Control of T1 .

We have


ke
hn,ε ({Zi∆ − Z(i−1)∆ }i∈Kε ) − hε kpLp,ε ≤ Cp ke
hn,ε ({Zi∆ − Z(i−1)∆ }i∈Kε ) − p∆,ε kpLp,ε
+ kp∆,ε − hε kpLp,ε

=: Cp (T5 + T6 ).
The deterministic term T6 is bounded using Lemma 4 by (∆kf kLp,ε )p . Taking expectation
conditionally on Iε and Kε of T5 , we recover the linear wavelet estimator of p∆,ε studied
by Kerkyacharian and Picard [26] (see their Theorem 2). For the sake of completeness we
reproduce the main steps of their proof. First, the control
 of the bias is the same as in [26],
noticing that Lemma 3 implies p∆,ε ∈ F s, p, q, M, A(ε) (see Lemma 5.1 in [14]) we get


X

−Jsp p
J(p/2−1)
p
E T5 |Iε , Kε ≤ Cp 2
M +2
E(|e
αJ,k − αJ,k | |Iε , Kε ) ,
k∈ΛJ

where α
eJ,k and αJ,k are defined in (30) and (31). First consider the case p ≥ 2. We start by
observing that


Z
1 X
E
ΦJk (x)p∆,ε (x) Iε , Kε =
ΦJk (Zi∆ − Z(i−1)∆ ) −
e (ε)
n
A(ε)
i∈Kε
Z
X
1 X
1{I=Kε } E
ΦJk (x)p∆,ε (x) ,
ΦJk (Zi∆ − Z(i−1)∆ ) −
|I|
A(ε)
I⊂{1,...,n}

i∈I

where |I| denotes the cardinality of the set I. To bound the last term we apply the inequality
of Bretagnolle and Huber to the i.i.d. centered random variables ΦJk (Zi∆ − Z(i−1)∆ ) −

E[ΦJk (Zi∆ − Z(i−1)∆ )] i∈I bounded by 2J/2+1 kΦk∞ , conditional to {Kε = I}. We obtain,
1 X
ΦJk (Zi∆ − Z(i−1)∆ ) −
E
|I|
i∈I

Z

A(ε)

ΦJk (x)p∆,ε (x) ≤

ip/2
X  1 h Z
J
J
2
2
Φ(2
x
−
k)
p
(x)dx
+
Cp
∆,ε
p/2
|I|
A(ε)
k∈ΛJ

Z
(p−2)(J/2+1)
2
kΦkp−2
∞
J
J
2
2 Φ(2 x − k) p∆,ε (x)dx .
|I|p−1
A(ε)

Therefore we get
X

k∈ΛJ

p

E(|e
αJ,k − αJ,k | |Iε , Kε ) ≤ Cp

X 

k∈ΛJ

1 h J
2
e (ε)p/2
n

Z

A(ε)

Φ(2J x − k)2 p∆,ε (x)dx

2(p−2)(J/2+1) kΦkp−2
∞
+
e (ε)p−1
n
28

Z

J

A(ε)

J

2

ip/2



2 Φ(2 x − k) p∆,ε (x)dx ,

where, as developed in [26],
X hZ
k∈ΛJ

X Z

and

k∈ΛJ

A(ε)

A(ε)

2J Φ(2J x − k)2 p∆,ε (x)dx

ip/2

p/2

≤ M2J kp∆,ε kLp/2,ε

2J Φ(2J x − k)2 p∆,ε (x)dx ≤ M2J .

We can then conclude that
X

k∈ΛJ

 2J kp kp/2

∆,ε Lp/2,ε
2Jp/2 kΦkp−2
∞
E(|e
αJ,k − αJ,k | |Iε , Kε ) ≤ Cp M
+
.
e (ε)p−1
n
e (ε)p/2
n
p

Plugging this last inequality in T5 we obtain

 2J p/2 
−Jsp p
E(T5 |Iε , Kε ) ≤ C 2
M +M
,
e (ε)
n

where C is a constant depending on p, khε kLp/2,ε , that is an upper bound of kp∆,ε kLp/2,ε , and
kΦk∞ . Gathering all terms we get, for p ≥ 2,


E kb
hn,ε ({Zi∆ − Z(i−1)∆ }i∈Kε ) − hε kpLp,ε |Iε , Kε


p
 2J p/2
p
−Js
+ (∆kf kLp,ε ) ,
M +M
≤C 2
e (ε)
n
where C is a constant depending on p, khε kLp/2,ε and kΦk∞ . For p ∈ [1, 2), together with the
additional assumption of hε , following the lines of the proof of Theorem 2 of Kerkyacharian
and Picard [26] we obtain the same bound as above. Finally, we have established that

n
p
 2J p/2

o
E T1 |Iε , Kε ≤ C 2−Js M + M
+ (∆kf kLp,ε )p ,
e (ε)
n

(33)
1

e (ε) ≤ n(ε). Moreover, taking J such that 2J = n
e (ε) 2s+1 we have,
where we used that n
−s/(2s+1)
e (ε)
uniformly over F (s, p, q, M, A(ε)), an upper bound in n
for the estimation of
e (ε) direct independent
p∆,ε , which is the optimal rate of convergence for a density from n
observations (see [26]).
Note that we did not use that A(ε) is bounded to control this quantity, it was possible to
have A = ∞. This together with Remark 1 lead to Proposition 3.
Using the fact that Φ′ is compactly supported, we get

Control of T3 .
E(T3 |Iε , Kε ) ≤

2

3J p
2

kΦ′ kp∞
n(ε)p

Z

X

R k∈Λ
J

Φ(x − k)

p dx

29

E
2J

 X

i∈Kε

Mi∆ − M(i−1)∆ + bν (ε)∆



p


|Iε , Kε .

Furthermore, we use the following upper bound for the last term in the expression above:
 X

p
E
Mi∆ − M(i−1)∆ + bν (ε)∆ |Iε , Kε
i∈Kε

  X


p
p
e (ε)bν (ε)∆
≤ Cp E
Mi∆ − M(i−1)∆ |Iε , Kε + n
i∈Kε

From Rosenthal’s inequality conditional on Iε and Kε we derive for p ≥ 2
n
 X

p o
p
2
e (ε)E(|M∆ |p ) + n
e (ε)E(M∆
E
) 2
Mi∆ − M(i−1)∆ |Iε , Kε ≤ Cp n
i∈Kε

n
p/2 o
e (ε)∆µp (ε) + n
e (ε)σ 2 (ε)∆
= Cp n

where ∆µp (ε) := E(|M∆ |p ) = O(∆). For p ∈ [1, 2) we obtain the same result using the Jensen
inequality and the latter inequality with p = 2. Next,
Z X
p dx
Φ(x − k) J ≤ 2−J |ΛJ |p kΦkpp .
2
R
k∈ΛJ

As Φ is compactly supported, and since we estimate hε on an set bounded by A, for every
j ≥ 0, the set ΛJ has cardinality bounded by |ΛJ | ≤ C2J , where C depends on the support
of Φ and A. It follows that,

p/2
′ p
p J(5p/2−1)
e (ε)n(ε)−p ∆ + n(ε)−p n
e (ε)σ 2 (ε)∆
n
E(T3 |Iε , Kε ) ≤ CkΦ k∞ kΦkp 2

n
e (ε)bν (ε)∆ p
.
(34)
+
n(ε)
Control of T4 .

Similarly, for the last term we have

e (ε) p
n
.
E(T4 |Iε , Kε ) ≤ 2J(2p−1) kΦkp∞ kΦkpp 1 −
n(ε)

(35)

Deconditioning on Kε . Replacing (33), (34) and (35) into (32), and noticing that T2 is
negligible compared to E(T4 |Iε , Kε ), we obtain



e (ε) p
n
p
b
E khn,ε ({Xi∆ − X(i−1)∆ }i∈Iε ) − hε kLp,ε |Iε , Kε ≤ C 22Jp 1 −
n(ε)
 2J p/2
+ (∆kf kLp,ε )p
+ 2−Jsp +
e (ε)
n

h
p/2  n
e (ε)bν (ε)∆ p i
−p
−p
2
J(3p/2−1)
p
e (ε)n(ε) ∆ + n(ε)
e (ε)σ (ε)∆
n
+
+2
|ΛJ | n
.
n(ε)
where C depends on s, p, khε kLp,ε , khε kLp/2,ε , kΦk∞ , kΦ′ k∞ , kΦkp and M. To remove the
conditional expectation on Kε we apply the following lemma.
30

1
Lemma 6. Let v∆ (ε) = P(|M∆ (ε) + ∆bν (ε)| > ε) and F∆ (ε) = P(|X∆ | > ε). If Fv∆∆(ε)
(ε) ≤ 3 ,
then for all r ≥ 0, there exists a constant C depending on r such that

e (ε)−r Iε ≤ Cn(ε)−r
E n
n

v∆ (ε)e−λε ∆ r o
v∆ (ε)e−λε ∆ r/2 
e (ε))r Iε ≤ C n(ε)
E (n(ε) − n
+ n(ε)
.
F∆ (ε)
F∆ (ε)

e (ε) ≤ n(ε), we conclude:
Finally, using the fact that λε ∆ → 0 and n

h v (ε) p/2  v (ε) p i

∆
∆
p
b
E khn,ε ({Xi∆ − X(i−1)∆ }i∈Iε ) − hε kLp,ε |Iε ≤ C 22Jp
+
n(ε)F∆ (ε)
F∆ (ε)
i
h
+ 2−Jsp + 2Jp/2 n(ε)−p/2 + (∆kf kLp,ε )p
i
h
p/2
+ (bν (ε)∆)p ,
+ 2J(5p/2−1) n(ε)1−p ∆ + n(ε)−p/2 σ 2 (ε)∆

where C depends on s, p, khε kLp,ε , khε kLp/2,ε , kΦk∞ , kΦ′ k∞ , kΦkp and M. The proof is now
complete.
✷

6.3

Proof of Theorem 2

Theorem 2 is a consequence of Theorem 1 and Corollary 2. For all 0 < ε ≤ 1, we decompose
ℓp,ε(fbn,ε , f ) as follows:
Z
h
i
p

p
bn,εb
b
hn,ε (x) − λε hε (x) dx
E Pn λ
=
ℓp,ε fn,ε, f
A(ε)
h
i
h
i
p
bn,ε p b
bn,ε − λε p khε kp + 2p−1 EP λ
h
−
h
≤ 2p−1 EPn λ
n,ε
ε
n
Lp,ε
Lp,ε
=: 2p−1 (I1 + I2 ).

The term I1 is controlled by means of Theorem 1 combined with the fact that if fε ∈
F (s, p, q, Mε , A(ε)) then hε ∈ F (s, p, q, M, A(ε)), which implies khε kLp,ε ≤ M. Concerning the term I2 , the Cauchy-Schwarz inequality gives
Z

p
bn,ε p b
E Pn λ
hn,ε (x) − hε (x) dx
I2 =
A(ε)
Z
q
q
p p



2p 
2p
b
E Pn b
hn,ε (x) − hε (x)
dx = J1 J2 .
≤ EPn |λn,ε |
A(ε)

The term J1 is treated using Theorem 1 and the triangle inequality
bn,ε |2p ] ≤ Cp (λ2p + E[|λ
bn,ε − λε |2p ]).
E[|λ
ε

For the term J2 , notice that as A(ε) is bounded, hence applying the Jensen inequality we get:
Z
q
q



2p 
E Pn b
hn,ε − hk2p
hn,ε (x) − hε (x)
dx ≤ C EPn kb
2p ,
A(ε)

where C depends on A. The rate of the right hand side of the inequality has been studied in
Corollary 2. The proof is now complete.
✷
31

7

Appendix

In this section we collect the proofs of all auxiliary results.

7.1

Proof of Lemma 1

Let (tn )n≥0 be any sequence converging to zero and let X n be a sequence of compound Poisson
processes with Lévy measure
νn (A) :=

P(Xtn ∈ A)
,
tn

∀A ∈ B(R \ {0}).

Using the Lévy-Khintchine formula jointly with the fact that any Lévy process has an infinitely
divisible distribution we get:


  iuXtn 
Z
 iuX n 
E e
−1
iux
1
(e − 1)νn (dx) = exp
ϕn (u) := E e
= exp
tn
R\{0}






t
t
n
n
ϕ(u) − 1
E[eiuX1 ] − 1
=: exp
.
(36)
= exp
tn
tn
We then deduce that the characteristic function ϕn of X1n converges to ϕ, the characteristic
function of X1 , as n goes to infinity. Indeed, from (36), we have



exp(tn log(ϕ(u))) − 1
ϕn (u) = exp
= exp log(ϕ(u)) + O(tn ) .
tn

This implies that the law of X1n converges weakly to the law of X1 .
Let us introduce the sequence of measures ρn (dx) = (x2 ∧ 1)νn (dx). RFrom Theorem 8.7 in
[34] it follows that (ρn )n is tight, i.e. supn ρn (R) < ∞ and limℓ→∞ supn |x|>ℓ ρn (dx). So there
exists a subsequence (ρnk )k that converges weakly to a finite measure ρ. Let us introduce the
measure νe(dx) := (x2 ∧ 1)−1 ρ(dx) on R \ {0} and νe({0}) = 0. Then, for any function f such
that f (x)(x2 ∧ 1)−1 is bounded, the following equalities hold:
Z
Z
lim
f (x)νnk (dx) = lim
f (x)(x2 ∧ 1)−1 ρnk (dx)
k→∞
k→∞
Z
Z
= f (x)(x2 ∧ 1)−1 ρ(dx) = f (x)e
ν (dx)
By definition of νn , this implies that

1 
E f (Xtn ) =
lim
n→∞ tn

Z

f (x)e
ν (dx).

The uniqueness of νe (see [34], p. 43) joint with the fact that νn converges weakly to ν (since
the law of X1n converges weakly to the law of X1 ), allow us to conclude that νe ≡ ν.
✷
32

7.2

Proof of Lemma 2

bn,ε − λε |p can be expressed in terms of the
Using Theorem 1, an upper bound for EPn |λ
∆ >ε)
quantities λε − P(X∆
and P(X∆ > ε) that can be made explicit. Let us begin by computing
the first term
Z ∞ −x
P(X∆ > ε)
P(X∆ > ε)
e
λε −
dx −
=
.
(37)
∆
x
∆
ε
R∞
Define Γ(∆, ε) = ε x∆−1 e−x dx, such that Γ(∆, 0) = Γ(∆). Using that Γ(∆, ε) is analytic
we can write the right hand side of (37) as
∞
o
X
P(X∆ > ε)
∆k n ∂ k
1
λε −
Γ(∆,
ε)
=
∆Γ(∆, 0)Γ(0, ε) −
∆=0
∆
∆Γ(∆)
k! ∂∆k
k=0

≤ Γ(0, ε)

X ∆k n ∂ k
1
1 − ∆Γ(∆, 0)
Γ(∆, ε)
+
∆Γ(∆)
∆Γ(∆)
k! ∂∆k
∞

k=1

∆=0

o

.

(38)

As Γ(∆, 0) is a meromorphic function with
pole in 0 and residue 1, there exists a
P a simple
1
k . Therefore,
+ ∞
a
∆
sequence (ak )k≥0 such that Γ(∆) = ∆
k
k=0
1 − ∆Γ(∆, 0) = ∆

and

∞
X

ak ∆k ,

k=0

P
k
∆ ∞
1 − ∆Γ(∆)
k=0 ak ∆
P
=
= O(∆).
∞
∆Γ(∆)
1 + ∆ k=0 ak ∆k

P
∆k ∂ k
Let us now study the term ∞
k=1 k! ∂∆k Γ(∆, ε) ∆=0 . We have:
Z ∞
Z 1
∂k
−1
−1
k
e−x (log(x))k dx
≤ e
x (log(x)) dx +
Γ(∆, ε)
∂∆k
∆=0
1
ε
Z ∞
k+1
|
log(ε)|
= e−1
e−x (log(x))k dx.
+
k+1
1
x0

Let x0 be the largest real number such that e 2 = (log(x0 ))k . This equation has two solutions
if and only if k ≥ 6. If no such point exists, take x0 = 1. Then,
Z ∞
Z x0
Z ∞

x0
x
−x
k
−x
k
e (log(x)) dx +
e− 2 dx ≤ (log(x0 ))k e−1 − e−x0 + 2e− 2
e (log(x)) dx ≤
x0

1

1

≤e

x0
−1
2

−

+e

x0
2

k

≤ k + 1,

where we used the inequality x0 < 2k log k, for each integer k. Summing up, we get
∞
X
∆k n ∂ k
k=1

k!

∂∆

Γ(∆, ε)

∆=0

o

≤ e−1

∞
X
∆k | log(ε)|k+1
k=1

k!

k+1

+

5
X

1

2e− 2

k=1

∞

∆k X ∆k k
+
(k + 1)
k!
k!
k=6

∞
 ∆| log(ε)|
 X
∆ 2  k k
≤ | log(ε)| e
−1 +
+ O(∆)
k! e
k=6

2

≤ (log(ε)) ∆ + O(∆).
33

k

In the last two steps, we have used first that ∆ < e−2 and then the Stirling approximation
1
formula to deduce that the last remaining sum is O(∆3 ). Clearly, the factor ∆Γ(∆)
∼ 1, as
∆ → 0, in (38) does not change the asymptotic. Finally we derive that
λε −


P(X∆ > ε)
= O log(ε)2 ∆ .
∆

Another consequence is that there exists a constant C, independent of ∆ and ε, such that

P(X∆ > ε) ≤ ∆ λε + C log(ε)2 ∆ .

7.3

✷

Proof of Lemma 4

Using the definition (21) we derive that
p∆,ε − hε = hε

∞
 e−λε ∆ λ ∆
 X
e−λε ∆ (λε ∆)k ⋆k
ε
−
1
+
h
1 − e−λε ∆
k!(1 − e−λε ∆ ) ε
k=2

∞ −λε ∆

 e−λε ∆ λ ∆
X
e
(λε ∆)k−2 ⋆k
ε
2
−
1
h
+
(λ
∆)
h .
=
ε
ε
1 − e−λε ∆
k!(1 − e−λε ∆ ) ε
k=2

Taking the Lp norm and using the Young inequality together with the fact that hε is a density
with respect to the Lebesgue measure, i.e. khε kL1,ε ≤ 1, we get
p∆,ε − hε

Lp,ε

≤

 e−λε ∆ λ ∆
(λε ∆)2 
ε
+
−
1
khε kLp,ε .
1 − e−λε ∆
1 − e−λε ∆

Under the regime λε ∆ → 0 we obtain

p∆,ε − hε

Lp,ε

≤ Cλε ∆khε kLp,ε ,

where C > 2 and λε hε = fε , which completes the proof.

7.4

✷

Proof of Lemma 5

We have
n(ε) =

n
X
i=1

bn,ε n∆.
1(ε,∞)(|Xi∆ − X(i−1)∆ |) = λ

We introduce the centered i.i.d. random variables Vi = 1(ε,∞) (|Xi∆ −X(i−1)∆ |)−F∆ (ε), which
are bounded by 2 and such that E[Vi2 ] ≤ F∆ (ε). Applying the Bernstein inequality we have,
P

 n(ε)


nx2
− F∆ (ε) > x ≤ 2 exp −
n
2(F∆ (ε) +

34

2x
3 )


,

x > 0.

(39)

Fix x = F∆ (ε)/2, on the set Ax =
n



n(ε)
n

− F∆ (ε) ≤ x we have

F∆ (ε)
3F∆ (ε)
≤ n(ε) ≤ n
.
2
2

(40)

Moreover it holds that



E n(ε)−r = E n(ε)−r 1Acx + E n(ε)−r 1Ax .

Since r ≥ 0 and n(ε) ≥ 1, using (39) and (40) we get the following upper bound

  nF∆ (ε) −r
3
E n(ε)−r ≤ 2 exp − 32
nF∆ (ε) +
2
and the lower bound


  3nF∆ (ε) −r
E n(ε)−r ≥ E n(ε)−r 1Ax ≥
.
2

This completes the proof.

7.5

✷

Proof of Lemma 6

For the first inequality, the proof is similar to the proof of Lemma 5. Using the definition of
e (ε) we have
n
X
e (ε) =
n
IZi∆ (ε)6=Z(i−1)∆ (ε) .
i∈Iε

For i ∈ Iε , we set Wi := IZi∆ (ε)6=Z(i−1)∆ (ε) . We have


v∆ (ε)e−λε ∆
E(Wi |i ∈ Iε ) = P Zi∆ (ε) 6= Z(i−1)∆ (ε) |Xi∆ − X(i−1)∆ | > ε = 1 −
F∆ (ε)

using the independence of M (ε) and Z(ε). The variables Wi − E(Wi |i ∈ Iε ) are centered
i.i.d., bounded by 2 and such that the following bound on the variance holds: V(Wi |Iε ) ≤
v∆ (ε)e−λε ∆
. Applying the Bernstein inequality we have,
F∆ (ε)
P



e (ε) 
v∆ (ε)e−λε ∆ 
n
> x Iε
− 1−
n(ε)
F∆ (ε)

Fix x = 12 , on the set Ax =

if

v∆ (ε)
F∆ (ε)



e (ε)
n
n(ε)



− (1 −

≤ 2 exp



v∆ (ε)e−λε ∆
)
F∆ (ε)

n(ε)x2

−
≤

2

v∆ (ε)e−λε ∆
F∆ (ε)

1
2

we have

+

n(ε)  1 v∆ (ε)e−λε ∆ 
e (ε),
n(ε) ≤ n
<
−
6
2
F∆ (ε)

e (ε) ≥ 1 that for r ≥ 0
≤ 31 . It follows from (41), (42) and n

e (ε)−r Iε ≤ 2 exp −
E n
35



3
16 n(ε)

+

 n(ε) −r
6



 ,

2x
3 )

x > 0. (41)

(42)

.

Finally, using that for all x > 0 we have xr e−x ≤ Cr := r r e−r we derive
 n(ε) −r

e (ε)−r Iε ≤ Cr n(ε)−r +
E n
,
6

which leads to the first part of the result.
The second part of the result can be obtained by means of Rosenthal’s inequality. For
e (ε),
r ≥ 0, we have, using that n(ε) ≥ n
 o
 
n 
r

−λε ∆ 
v∆ (ε)e−λε ∆ r
e (ε))r Iε ≤ Cr E n(ε) 1 − v∆ (ε)e
e
.
+
n(ε)
E (n(ε) − n
I
−
n
(ε)
ε
F∆ (ε)
F∆ (ε)

The Rosenthal inequality leads to, for r ≥ 2,

E n(ε) 1 −

v∆ (ε)e−λε ∆ 
F∆ (ε)

e (ε)
−n

r




−λε ∆ r/2
.
Iε ≤ Cr n(ε) v∆ (ε)e
F∆ (ε)

Thanks to Jensen’s inequality we can also treat the case 0 < r < 2 recovering the same
inequality. Therefore, it follows that for all r > 0
n

v∆ (ε)e−λε ∆ r/2 
v∆ (ε)e−λε ∆ r o
e (ε))r Iε ≤ C n(ε)
E (n(ε) − n
.
+ n(ε)
F∆ (ε)
F∆ (ε)

This completes the proof.

✷

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