Concentration inequalities for Poisson point processes with
application to adaptive intensity estimation
Martin Kroll∗

arXiv:1612.07901v1 [math.PR] 23 Dec 2016

April 2, 2018

Synopsis
We derive concentration inequalities for maxima of empirical processes associated with independent but not necessarily identically distributed Poisson point processes. The proofs are
based on a careful application of Ledoux’s entropy method. We demonstrate the utility of
the obtained concentration inequalities by application to adaptive intensity estimation.

Key words and phrases: Poisson point process, concentration inequality, adaptive estimation
AMS 2010 Subject Classification: 60E15, 62G05

1. Introduction
Poisson point processes (PPPs) are of fundamental importance in probability theory and statistics, both from a theoretical and an applied point of view. For instance, they serve as the
elementary building blocks for complex point process models which are used in stochastic geometry [Sto+13] and a wide range of applications including, amongst others, extreme value
theory [Res87], ﬁnance [BH09], forestry [PS00], and queueing theory [Bré81].
In this paper, we consider a sample N1 , . . . , Nn of independent but not necessarily identically
distributed PPPs. In the ﬁrst part, inspired by results in [KR05], we derive concentration
inequalities for maxima of empirical processes associated with this sample. In the second part,
further assuming that the PPPs are identically distributed, we consider the nonparametric
estimation of the common intensity. We derive optimal rates of convergence in terms of the
sample size n and propose a fully-data driven estimator of the intensity. The theoretical study
of the estimator is essentially based on the concentration inequalities derived in the ﬁrst part.
Concentration inequalities provide upper bounds for the probability that a random variable
deviates from its mean or median by a certain amount: the well-known inequalities by Markov,
Hoeﬀding and Bernstein provide classical examples of such inequalities. Modern research on
concentration inequalities goes back at least to the 1970s when concentration inequalities for
deviations from the mean and median for Lipschitz continuous functions of multivariate Gaussian
random variables were derived: the papers [Bor75] and [ST78] consider deviations from the
median, whereas [CIS76] deals with deviations from the mean. The main advantage of these
results is that they do not depend on the dimension of the underlying Euclidean space and allow
to control suprema of Gaussian processes.
∗

Universität Mannheim. Institut für Mathematik. A5, 6. D-68131 Mannheim, Germany.
B: martin.kroll@math.uni-mannheim.de. Financial support by the Deutsche Forschungsgemeinschaft (DFG)
through the Research Training Group RTG 1953 is gratefully acknowledged. I am indebted to Jan Johannes
and Martin Schlather for helpful comments on the paper.

1

In a series of papers, Talagrand [Tal95; Tal96] developed concentration inequalities for suprema
of empirical processes associated with random variables on general product spaces. His original
proof of the Talagrand inequality (see Theorem 1 in [Mas00], for instance) is rather technical
and essentially based on geometric arguments. Ledoux [Led96] proposed the entropy method as
an essentially diﬀerent and more accessible approach to prove Talagrand’s results. However, the
paper [Led96] regained Talagrand’s result with diﬀerent numerical constants and variance factor
only. Massart’s [Mas00] approach is also based on Ledoux’s method and a careful adaption
of Gross’s logarithmic Sobolev inequality to a non-Gaussian framework and led to reasonable
numerical constants in Talagrand type inequalities when the underlying random variables are
independent but eventually non-identically distributed. Finally, in the same setup, Klein and
Rio [KR05] obtained concentration results by an elaborate application of the entropy method.
In addition, they provided a brief comparison of the numerical constants occurring in diﬀerent
papers on the subject showing that their results are optimal in some sense.
There is already some amount of research papers dealing with concentration inequalities for
point processes, see for instance [BL98; Hou02; HP02] and [Wu00]. As pointed out in [RB03]
(cf. p. 109 therein), the main drawback of all these results is that they provide a variance term
that is diﬃcult to deal with in statistical applications. The scope of the paper [RB03] is similar
to our one: it is inspired by Talagrand type results from [Mas00] and derives analogous results
in a model with PPPs. Similarly, one main intention of the present paper is to transfer the
concentration inequalities from [KR05] to a PPP framework. Let us emphasize that it is not
possible to apply the results from [KR05] immediately, that is, interpreting point processes as
random variables in the space of locally ﬁnite measures, in order to obtain our results in this
paper. The stochastic integral of a constant function with respect to a PPP provides an example
that can be dealt with by our result but does not ﬁt into the framework of [KR05] (cf. Remark 2.2
below). To obtain our results, one key argument of the proof is borrowed from [RB03]: the infinite
divisibility of PPPs is exploited and the underlying probability space split into two parts: on
the ﬁrst one, the proofs from [KR05] can be mimicked to a great extent, whereas the probability
of the second one can be neglected asymptotically. Beyond that a careful adaptation of the
many auxiliary results in [KR05] has to be conducted. The formulation and proofs of most of
these results turn out to be more complicated as in [KR05] due to measurability issues caused
by indicator functions on certain events. It is worth noting that exactly the same numerical
constants from [KR05] reoccur in our theorems which is in complete correspondence with the
phenomenon remarked in [RB03] where the constants are the same as in [Mas00]. Based on our
main results, we derive Proposition C.1 in the appendix as an auxiliary concentration result
which represents a key tool for our statistical application. This auxiliary statement is motivated
by a similar result from the research literature (cf., for instance, Lemma 1 in [CRT06]).
Intensity estimation in parametric and nonparametric models has been dealt with in a wide
range of monographs and research papers. For a general treatment of the subject, we refer
to [Kar91] as an introduction to the statistics of point processes, [Kut98] for examples of intensity estimation in diﬀerent parametric and nonparametric models, and [MW04] for estimation
in general spatial models. Early approaches to nonparametric intensity estimation include kernel [Rud82; Kut98] and histogram estimators [Rud82]. In addition, the paper [Rud82] already
discusses adaptive estimation of the intensity. Baraud and Birgé [BB09] consider a Hellinger
type loss function and propose a histogram estimator for intensity estimation. Other contributions focus on non-linear wavelet thresholding techniques, cf., for instance, the articles [Kol99;
WN07; RBR10; San14] and [Big+13]. The paper [Big+13] proposes a non-linear hard thresholding estimator for intensity estimation from indirect observations. Moreover, there exist other
approaches to nonparametric intensity estimation in more speciﬁc models. Let us mention the
paper [GN00] that proposes a minimum complexity estimator in the Aalen model and [PW04]
that uses a wavelet approach to estimation in a multiplicative intensity model, without making

2

a claim to be exhaustive.
In [RB03], the focus is on a model slightly diﬀerent from the one we will consider: estimation
of the intensity on the interval [0, T ] from only one observation and asymptotics for T → ∞ are
considered whereas we assume the availability of an independent sample of realizations of the
point process on the ﬁxed interval [0, 1]. Common ground of our approach and the one taken
in [RB03] is the use of projection estimators. Besides, the statistical methodology of the present
paper is rather motivated by the procedure developed in [JS13] for circular deconvolution. We
derive a minimax lower bound under abstract smoothness conditions and propose a projection
estimator that can attain this lower bound. We note that the proof of the minimax lower bound
given in [RB03] does not hold for ellipsoids deﬁned in terms of the trigonometric basis (cf. the
remark below Deﬁnition 4 and Proposition 3 in [RB03]), whereas we derive such a minimax lower
bound. In return, the analysis in [RB03] is rather general with a slight emphasis on wavelet
methods and not tailored to the trigonometric basis which we exclusively consider in this paper.
As usual in nonparametric statistics, the performance of our proposed projection estimator
crucially depends on the appropriate selection of a dimension parameter. Since the optimal
choice of this parameter depends on the unknown intensity and is thus unavailable in practise, we
propose a fully-data driven selection of the dimension parameter leading to an adaptive estimator
of the intensity. Our approach is based on model selection by minimization of a penalized
contrast function. The combination of model selection techniques and concentration inequalities
has attracted great attention in nonparametric statistics [BBM99], [Com15]. In particular, a
considerable amount of research has been devoted to Gaussian regression and density estimation
frameworks. We refer to the monograph [Mas07] for results and further references concerning
these two frameworks. Concentration inequalities have already been exploited in the context of
nonparametric intensity estimation in the papers [RB03] and [BB09] mentioned above. Taking a
diﬀerent point of view, the paper [Bir07] introduces an approach to model selection via hypothesis
testing in the framework of intensity estimation. Following the model selection approach as
in [BBM99], we obtain an adaptive estimator that attains the optimal rate of convergence
for intensities belonging either to some Sobolev space or some space of (generalized) analytic
functions.
The paper is organized as follows. In Section 2, we derive concentration inequalities for
deviations from the mean for empirical processes associated with PPPs. Section 3 deals with
nonparametric intensity estimation. Subsection 3.1 introduces the model, in Subsection 3.2 we
derive the minimax theory, and in Subsection 3.3 we study the adaptive estimator. Most of the
proofs deferred to the appendix.

2. Concentration inequalities for Poisson point processes
Let N1 , . . . , Nn be independent PPPs with ﬁnite intensity measures Λ1 , . . . , Λn on some Polish
space X. We denote the common underlying probability space of the PPPs with (Ω, A , P). Let
S be a countable class of measurable functions from X to [−1, 1]n . For s = (s1 , . . . , sn ) ∈ S and
k ∈ {1, . . . , n}, let us deﬁne
k

I (s) :=

Z

X

sk (x)(dNk (x) − dΛk (x))

and

Sn (s) := I 1 (s) + . . . + I n (s).

(2.1)

The ﬁrst aim of this paper is to establish concentration inequalities for Z := sups∈S Sn (s). In
the spirit of [KR05], concentration inequalities for right-hand side and left-hand side deviations
are established separately. As already sketched in the introduction, one key idea for the proofs
of both theorems is borrowed from the article [RB03], namely to exploit the inﬁnite divisibility
of PPPs. This property is used to break the proof for the point process case into handy pieces
such that the proof from [KR05] can be mimicked. However, adapting the variety of auxiliary

3

results used in the proofs of [KR05] to our setup is non-trivial. The analogues of these auxiliary
results are collected in subsections A.1 and B.1 in the appendix.
Since the inﬁnite divisibility is the essential ingredient for the proofs, it is not clear whether
and if yes, how concentration results as given in Theorems 2.1 and 2.3 could be transferred to
point processes which are not inﬁnitely divisible. For Cox processes (which represent a natural
generalization of PPPs), the proof presented here fails and the development of tools to deal
with this case might be worth further investigation. On the contrary, it should be possible to
adapt the proof to frameworks with other inﬁnitely divisible processes such as Poisson cluster
processes (see [Sto+13], p. 151) under additional assumptions (for instance, that the number of
daughter points is uniformly bounded in n).
The following theorem (the proof of which is given in Appendix A) provides concentration
inequalities for right-hand side deviations of Z from its mean.
Theorem 2.1 Let N1 , . . . , Nn be independent PPPs on a Polish space X with ﬁnite intensity
measures Λ1 , . . . , Λn , and S be a countable class of measurable functions from X to [−1, 1]n . For
s ∈ S, define Sn (s) as in (2.1) and consider Z := sups∈S Sn (s). Let LZ (t) := log E[exp(tZ)]
denote the logarithm of the moment-generating function of Z and Vn := sups∈S Var (Sn (s)).
Then, for any non-negative t,
a)

LZ (t) ≤ tEZ +

t
(2EZ + Vn ) (exp((e2t − 1)/2) − 1).
2

Setting υ := 2EZ + Vn , we obtain that, for any non-negative x,
b)





x
P (Z ≥ EZ + x) ≤ exp − log(1 + 2 log(1 + x/υ)) ,
4

and
c)

x2
√
P (Z ≥ EZ + x) ≤ exp −
υ + υ 2 + 3υx + (3x/2)

!

x2
≤ exp −
2υ + 3x

!

.

Remark 2.2 We emphasize that Theorem 2.1 cannot be immediately
deduced from Theorem 1.1
R
in [KR05]. For instance, for sk ≡ 1 the stochastic integral X sk (x)dNk (x) is an unbounded
function of Nk but obviously sk ≡ 1 ﬁts into the framework of Theorem 2.1.
The quantity Vn is usually referred to as the wimpy variance (cf. [BLM16], Chapter 11). In the
proof of Proposition C.1 in Appendix C, a suitable bound for the quantity υ will be determined.
The following theorem (the proof of which is given in Appendix B) provides concentration
inequalities for left-hand side deviations of Z from its mean.
Theorem 2.3 Under the assumptions of Theorem 2.1, for any non-negative t, it holds
a)

LZ (−t) ≤ −tEZ +

υ 3t
(e − 3t − 1).
9

Consequently, for any non-negative x,
b)





υ
3x
P (Z ≤ EZ − x) ≤ exp − h
9
υ



,

where h(x) = (1 + x) log(1 + x) − x, and
c)

x2
√
P (Z ≤ EZ − x) ≤ exp −
υ + υ 2 + 2υx + x

4

!

x2
≤ exp −
2υ + 2x

!

.

Remark 2.4 The concentration inequalities in Theorems 2.1 and 2.3 translate literally (that is,
with exact coincidence of the numerical constants) the ones obtained in [KR05] to our framework
with PPPs. This observation is in line with the remark made in [RB03] where the derived
concentration inequalities translate literally previous results due to [Mas00].
Remark 2.5 In many situations of interest, it is possible to apply the concentration inequalities
proved in this section (and the one proved in Appendix C) to non-countable classes of measurable functions. A rigorous foundation of this practice can be based on density arguments (see
Remarque 2.1 in [Cha13]).

3. Nonparametric intensity estimation
3.1. Model assumptions
Let N1 , . . . , Nn be i.i.d. realizations of a PPP on [0, 1] with square-integrable intensity function
λ ∈ L2 := L2 ([0, 1], dx). The Ni can be interpreted as N0 -valued random measures which
P
motivates the notation Ni = j δxij (here, δx denotes the Dirac measure with mass concentrated
at x). Our aim is to estimate the intensity function λ nonparametrically from the sample
N1 , . . . , Nn . Let us consider the orthonormal basis {ϕj }j∈Z of L2 which is given by
√
√
ϕ0 = 1, ϕj (t) = 2 cos(2πjt) for j > 0, ϕ−j (t) = 2 sin(2πjt),
j = 1, 2, . . . ,
and deﬁne the sequence ([λ]j )j∈Z of Fourier coeﬃcients via [λ]j :=
the L2 -convergent representation
X
[λ]j ϕj .
λ=

R1
0

λ(t)ϕj (t)dt which yields
(3.1)

j∈Z

e of λ, let us consider the
In order to evaluate the performance of an arbitrary estimator λ
2
e
mean integrated squared error E[kλ − λk ] (where, as usual, the expectation is taken under the
P
true intensity function λ and k · k denotes the L2 -norm). Note that we have k · k2 = j∈Z |[ · ]j |2
thanks to Parseval’s identity. We hold the minimax point of view and consider the maximum
e − λk2 ] for some smoothness class Λ of potential intensity functions.
risk deﬁned by supλ∈Λ E[kλ
e − λk2 ] where the inﬁmum is
The corresponding minimax risk is deﬁned by inf eλ supλ∈Λ E[kλ
e of λ based on the sample N1 , . . . , Nn . An estimator λ∗ is
taken over all potential estimators λ
∗
e − λk2 ], where the notation an . bn
called rate optimal if supλ∈Λ E[kλ − λk2 ] . inf eλ supλ∈Λ E[kλ
means that an ≤ Cbn for some numerical constant that does not depend on n. The speciﬁc form
of the class Λ will be introduced in the following subsection.
For the moment, let us introduce the
general type of projection estimator we will consider
R1
throughout this work: Since we have E[ 0 ϕj (t)dNi (t)] = [λ]j for all j ∈ Z by Campbell’s theorem
c := 1 Pn R 1 ϕ (t)dN (t) is an unbiased estimator of
(cf., for instance, [Str10], Chapter 2), [λ]
i
i=1 0 j
j
n
[λ]j , Equation (3.1) strongly suggests to consider orthogonal series estimators of the form
b :=
λ
k

X

0≤|j|≤k

c ϕ ,
[λ]
j j

(3.2)

where the dimension parameter k ∈ N0 has to be chosen appropriately.
b is by deﬁnition not guaranteed to attain non-negative values
Remark 3.1 The estimator λ
k
only (which, of course, holds for the true intensity λ). In practise, this undesirable feature can
b
b
b
avoided by considering the estimator λ
k+ deﬁned via λk+ (t) := λk (t) ∨ 0, the risk of which is
b .
evidently bounded from above by the one of λ
k

5

3.2. Minimax theory
In order to deﬁne the class of admissible intensity functions in the deﬁnition of the minimax
risk, let γ = (γj )j∈Z be a strictly positive symmetric sequence of weights and r > 0. Set
Λrγ := {λ ∈ L2 : λ ≥ 0 and

X

j∈Z

γj |[λ]j |2 =: kλk2γ ≤ r}.

In the following, our aim is to study the minimax risk with respect to the function class Λrγ .
Our results will be obtained under the following mild regularity assumptions on the sequence γ.
Assumption A γ = (γj )j∈Z is a strictly positive symmetric sequence such that γ0 = 1 and
(γn )n∈N0 is non-decreasing.
b deﬁned in (3.2)
The following proposition provides an upper risk bound for the estimator λ
k
under an appropriate choice of the dimension parameter k.
b ∗ with dimension parameter
Proposition 3.2 Let Assumption A hold. Consider the estimator λ
kn
}.
Then,
for
all
n
∈
N,
kn∗ := argmink∈N0 max{γk−1 , 2k+1
n


∗
2
−1 2kn + 1
b
∗
:=
sup E[kλkn − λk ] . Ψn
max γk∗ ,
.
n

λ∈Λrγ

Proof. Introduce the function λkn∗ :=

P

∗ [λ]j ϕj
0≤|j|≤kn

n

which suggests the decomposition

b ∗ − λk2 ] = kλ − λ ∗ k2 + E[kλ
b ∗ − λ ∗ k2 ]
E[kλ
kn
kn
kn
kn

of the considered risk into squared bias and variance. Using the smoothness assumption λ ∈ Λrγ ,
b ∗ − λ ∗ k]2 ≤ √r · 2kn∗ +1 and the statement of
it is easy to see that kλ − λkn∗ k2 ≤ rγk−1
∗ and E[kλkn
kn
n
n
the theorem follows.


b ∗ considered in Proposition 3.2 is demonstrated by
The rate-optimality of the estimator λ
kn
means of the following theorem which is valid under mild additional assumptions. The proof
makes use of an adaption of standard techniques in nonparametric statistics for the derivation
of minimax lower bounds to our point process framework and is deferred to the appendix.

Theorem 3.3 Let Assumption A hold and further assume that
(C1) Γ :=

P

−1
j∈Z γj

< ∞, and
∗

2kn +1
(C2) 0 < η −1 := inf n∈N (Ψn )−1 min{γk−1
∗ ,
n } for some η ≥ 1,
n

where the quantities kn∗ and Ψn are defined in Proposition 3.2. Then, for all n ∈ N,
e − λk2 ] & Ψn
inf sup E[kλ
e
λ

λ∈Λrγ

e of λ based on the sample N1 , . . . , Nn .
where the infimum is taken over all estimators of λ
P

Remark 3.4 The mild assumption (C1) on the convergence of the series j∈Z γj−1 is needed
only in order to guarantee the non-negativity of the candidate intensities considered in the proof.
On the whole, the proof is very much in line with the proof of Theorem 2.1 in [JS13] expanded
with the essential ingredient that the Hellinger distance between two PPPs is bounded by the
Hellinger distance of the corresponding intensity measures (see Theorem 3.2.1 in [Rei93]).

6

Remark 3.5 Note that the lower bound proof given in [RB03] is based on a speciﬁc property
called localization and is not valid for ellipsoids expressed in terms of the trigonometric basis.
Example 3.6 (Sobolev ellipsoids) Let γ0 = 1, γj = |j|2p for j 6= 0. This setting corresponds to λ
belonging to a Sobolev ellipsoid. Then, Assumption A is satisﬁed and elementary computations
show that kn∗ ≍ n1/(2p+1) as well as Ψn ≍ n−2p/(2p+1) . Furthermore, the additional conditions of
Theorem 3.3 are satisﬁed if p > 1/2 holds.
Example 3.7 (Analytic functions) Let γj = exp(2p|j|) for j ∈ Z for some p > 0. This belongs
to λ belonging to a class of analytic functions. Assumption A is also fulﬁlled in this case and
we obtain kn∗ ≍ log(n) and Ψn ≍ log n/n. The additional assumption of Theorem 3.3 does not
impose any additional restriction on p.
Example 3.8 (Generalized analytic functions) Let γj = exp(2β|j|p ) for β, p > 0. Note that
in this case the Fourier coeﬃcients of λ obey a power exponential decay and λ belongs to a
class of generalized analytic functions. Assumption A is satisﬁed in this case and there are no
additional restrictions on p due to Theorem 3.3. We have kn∗ ≍ (log n)1/p resulting in the rate
Ψn ≍ (log n)1/p /n.

3.3. Adaptive estimation
The optimal choice of the dimension parameter k stated in Proposition 3.2 depends on the
smoothness characteristics of the intensity via the sequence γ. However, such an a priori knowledge is a strong assumption and usually not available in practise. Thus, there is a demand for a
data-driven choice of the dimension parameter which hopefully does not deteriorate the quality
of the upper risk bound or at least leads to worse numerical constants, merely.
This data-driven choice of the dimension parameter and the resulting upper risk bound are
investigated now. For this purpose, we follow a model selection approach which has been successfully applied to a wide range of estimation problems in nonparametric statistics (cf., for
instance, [BBM99; Com15] for general accountsR to this model selection paradigm).
For s, t ∈ L2 , introduce the notation hs, ti := 01 s(x)t(x)dx and consider the contrast function
b n , ti,
Υn (t) := ktk2 − 2hλ

t ∈ L2 .

Deﬁne the random sequence of penalties (penk )k∈N via
c ∨ 1) ·
penk := 12η −1 · ([λ]
0

2k + 1
n

for some η ∈ (0, 1) (the dependence of the estimator on the parameter η will be suppressed for
the sake of convenience from now on).Building on the deﬁnitions made until now, we deﬁne the
data-driven selection kbn of the dimension parameter as the minimizer of the penalized contrast
b ) + pen }.
kbn := argmin{Υn (λ
k
k
0≤k≤n

b . Its proof is
The following theorem provides an upper bound for the risk of the estimator λ
b
kn
given in Appendix E.

Theorem 3.9 Let Assumption A hold. Then, for any n ∈ N, we have



√
1
−1 2k + 1
2
b
+ + exp(−κ n)
sup E[kλbk − λk ] . min max γk ,
n
r
0≤k≤n
n
n

λ∈Λγ

7

where κ > 0 is a numerical constant.
Remark 3.10 The penalty term used in the deﬁnition of kbn is non-deterministic which is in
contrast to penalty terms occuring, for instance, in density estimation or deconvolution problems.
The need for randomization is due to the quantity [λ]0 in the deﬁnition of H in Lemma E.1.
If r (but not γ) was known, one could proceed without randomization by choosing the penalty
√
√
proportional to r(2k + 1)/n. However, the factor r in this deﬁnition cannot be replaced by
√
√
an estimate of r because a reasonable estimator of r is not reachable from the data. Note
that the penalty terms considered in [RB03] in a point process framework similar to ours are
also non-deterministic.
−1 2k+1
b attains the rate Ψ if and only if min
The adaptive estimator λ
n
0≤k≤n max{γk , n } has
b
kn
the same order as Ψn . Since under Assumption A it holds that kn∗ . n, we immediately obtain
the following result.

b is rate optimal over the class Λr .
Corollary 3.11 Under Assumption A, the estimator λ
γ
b
kn

b is rate optimal in the framework of Examples 3.6, 3.7 and 3.8
In particular, the estimator λ
b
k
n

where kn∗ ≍ n1/(2p+1) , kn∗ ≍ log(n), and kn∗ ≍ log(n)1/p , respectively.

8

Appendix A. Proof of Theorem 2.1
A.1. Notation and preparatory results
In this subsection, we introduce some notation and state preliminary results. The proof of
Theorem 2.1, based on these results, is given in Subsection A.2. The key property used to prove
Theorem 2.1 is the infinite divisibility of the PPPs N1 , . . . , Nn : for every k ∈ {1, . . . , n} and
ℓ ≥ 1, there exist i.i.d. PPPs Nkj such that
d

Nk =

ℓ
X

Nkj .

(A.1)

j=1

The common intensity measure of the Nkj in this representation is Λk /ℓ. Throughout this work,
the dependence of Nkj , Λkj , and derived quantities on ℓ is often suppressed for the sake of
convenience.
R
Deﬁne Λ := supk=1,...,n Λk (X) and ∆ = ∆(ℓ) := Λ/ℓ. For s ∈ S, let I kj (s) := X sk (x)(dNkj (x)−
dΛkj (x)). We deﬁne the random variable Xkj := Nkj (X) (that is, Xkj is the total number of
points of the point process Nkj ) and the event Ωkj via Ωkj := {Xkj ≤ 1}.
Remark A.1 A natural interpretation of the proof of Theorem 2.1 given below is to consider
the result being obtained in a setup with a triangular array of point processes
N1 , . . . , Nn
N11 , N12 , N21 , N22 , . . . , Nn1 , Nn2
N11 , N12 , N13 , N21 , N22 , N23 , . . . , Nn1 , Nn2 , Nn3
..
.
where the point processes in each row are independent and the intensity measures of the single
point processes in a row tend to zero when the row index tends to inﬁnity. All asymptotic
considerations will be obtained under the equivalent regimes ℓ → ∞ and ∆ → 0, respectively.
Lemma A.2 P(Ωckj ) ≤ ∆2 /2.
Proof. The function h : N0 → R, n 7→ n2 − n is non-negative and non-decreasing. Since
Ωckj = {Xkj ≥ 2} the claim estimate follows from Markov’s inequality.

Let us deﬁne the σ-ﬁelds
Fn := σ({N11 , . . . , Nnℓ })

Fnkj := σ ({N11 , . . . , Nnℓ }\{Nkj }) .

and

kj
kj
kj
Further, let Ekj
n [ · ] := E[ · |Fn ] and Pn (A) := En [1A ]. In addition, for the rest of Appendix A
we denote f = f (t) := exp(tZ) and fkj = fkj (t) := Ekj
n [f ].
It will turn out to be suﬃcient to prove the results of this subsection under the following
ﬁniteness assumption.

Assumption F Let S = {s1 , . . . , sm } be a ﬁnite set of measurable functions from the Polish
space X to [−1, 1]n , and let τ denote the minimal value of i such that Z = Sn (si ).
Lemma A.3 Let Assumption F hold. Then, for any non-negative t,
a) f /fkj ≤ exp(tI kj (sτ )), and in addition

√
b) exp(−2(1 + ∆)t)(1 − e(2+3∆)t exp(∆(e2t − 1)/2) · ∆/ 2) ≤ f /fkj on Ωkj .

9

Proof of Lemma A.3. In order to prove statement a), set Snkj (s) := Sn (s)−I kj (s) and Zkj :=
sups∈S Snkj (s). Moreover, deﬁne τkj to be the minimal i such that Snkj (si ) = Zkj . Then, Zkj is
Fnkj -measurable, and we have
exp(t(Zkj + Xkj + ∆)) ≥ f ≥ exp(tZkj ) · exp(tI kj (sτkj )).

(A.2)

kj
The random variable τkj is Fnkj -measurable which implies Ekj
n [I (sτkj )] = 0. Thus, by Jensen’s
inequality, we obtain from the second estimate in (A.2) that
kj
kj
fkj ≥ exp(tZkj ) · Ekj
n [exp(tI (sτkj ))] ≥ exp(tZkj ) ≥ exp(tSn (sτ )),

and consequently fkj ≥ f · exp(−tI kj (sτ )) which implies statement a).
For the proof of b), we retain the notation introduced in the proof of statement a). From the
left-hand side inequality in (A.2), we obtain
fkj ≤ et(Zkj +∆) · E[etXkj 1Ωkj ] + et(Zkj +∆) · E[etXkj 1Ωckj ]
≤ et(Zkj +1+∆) + et(Zkj +∆) · E[e2tXkj ]1/2 P(Ωckj )1/2 .

√
Multiplication with 1Ωkj on both sides, using the estimate P(Ωckj )1/2 ≤ ∆/ 2 from Lemma A.2,
and recalling the formula for the moment-generating function of a Poisson distributed random
variable, yields
√
fkj 1Ωkj ≤ et(Zkj +1+∆) 1Ωkj + et(Zkj +∆) · exp(∆(e2t − 1)/2) · ∆/ 2 · 1Ωkj ,
from which we conclude by exploiting the right-hand side inequality of (A.2) and the deﬁnition
of Ωkj that
√
fkj 1Ωkj ≤ f e2(1+∆)t 1Ωkj + f e(1+2∆)t · exp(∆(e2t − 1)/2) · ∆/ 2 · 1Ωkj ,
and hence by elementary transformations
√
(1 − f /fkj · e(1+2∆)t exp(∆(e2t − 1)/2) · ∆/ 2) · 1Ωkj ≤ f /fkj · e2(1+∆)t 1Ωkj .
Now, by the statement of assertion a) and the deﬁnition of Ωkj
√
(1 − e(2+3∆)t exp(∆(e2t − 1)/2) · ∆/ 2) · 1Ωkj ≤ f /fkj · e2(1+∆)t · 1Ωkj ,
which yields the claim assertion after division by e2(1+∆)t .


√
In the sequel, we use the abbreviation c(t, ℓ) := 1−e(2+3∆)t exp(∆(e2t −1)/2)·∆/ 2. Note that
c(t, ℓ) ≤ 1 and, for any ﬁxed non-negative t, c(t, ℓ) → 1 as ℓ → ∞. In particular, c(t, ℓ) ∈ [1/2, 1],
for suﬃciently large ℓ, say ℓ ≥ ℓ0 = ℓ0 (t). Under the validity of Assumption F, we consider for
k ∈ {1, . . . , n} and j ∈ {1, . . . , ℓ} the positive and Fnkj -measurable random variables hkj deﬁned
by
hkj :=

m
X

kj
kj
kj
Pkj
n (τ = i) exp(tSn (si )) = En [exp(tSn (sτ ))].

(A.3)

i=1

From now on, we denote by C a numerical constant independent of ℓ (but certainly depending
on the ﬁxed value of t considered) whose value may change depending on the context. The
following lemma summarises estimates which are used for the rest of this section.
Lemma A.4 Let Assumption F hold and let η(x) = 1 − exp(−x) − e2(1+∆)t−log c(t,ℓ) x for ℓ ≥ ℓ0 .
Then, the estimate E[X] ≤ C holds true, where X can be replaced by any of the following random

10

variables:
a) h4kj ,
b) (fkj − f )4 ,

c) (f log(f /fkj ))4 ,
d) (f η(tI kj (sτ )))4 ,
e) (I kj (s))4 ,
f) (I kj (s))2 ,
g) exp(tSnkj (sτ )), and
h) exp(4tSnkj (sτ )).
Here, Snkj is defined as in the proof of Lemma A.3. The constant C can be chosen independent
of k and j, and in statements d)–h), it can, in addition, be chosen independent of s and sτ ,
respectively.
Proof. Let us only mention that for the proof of statement c), it is useful to apply statement a)
from Lemma A.3. Then, all the estimates are easy to derive and we thus omit the proof.

Lemma A.5 Let Assumption F hold, and let hkj be defined as in (A.3). Then, for all ℓ ≥ ℓ0 ,
we have
ℓ
n X
X

k=1 j=1

E[(f − hkj )1Ωkj ] ≤ e2(1+∆)t−log c(t,ℓ) E [f ] log E[f ] + C · ℓ−1/2 .

Proof of Lemma A.5. We begin the proof with the observation that
E[(f − hkj )1Ωkj ] = E[f − hkj ] + E[(hkj − f )1Ωckj ] ≤ E[f − hkj ] + E[hkj 1Ωckj ]

(A.4)

where the last estimate is due to the fact that f is non-negative. Thanks to (A.3), we obtain
the decomposition
E[f − hkj ] = E[f (1 − exp(−tI kj (sτ )) − te2(1+∆)t−log c(t,ℓ) I kj (sτ )] + te2(1+∆)t−log c(t,ℓ) E[f I kj (sτ )]
= E[f η(tI kj (sτ ))1Ωkj ] + E[f η(tI kj (sτ ))1Ωckj ] + te2(1+Λ/ℓ)t−log c(t,ℓ) E[f I kj (sτ )],

where the function η is deﬁned via η(x) = 1 − exp(−x) − e2(1+∆)t−log c(t,ℓ) x. Note that η is nonincreasing on the interval [−2(1+∆)t+log c(t, ℓ), ∞). This fact in combination with Lemma A.3
implies that
E[f η(tI kj (sτ ))1Ωkj ] ≤ E[(f − fkj − e2(1+∆)t−log c(t,ℓ) f log(f /fkj ))1Ωkj ].
By the identities 1Ωkj = 1 − 1Ωckj and E[f − fkj ] = 0, we thus obtain
E[f η(tI kj (sτ ))1Ωkj ] ≤ E[(fkj − f )1Ωckj ] + e2(1+∆)t−log c(t,ℓ) E[f log(f /fkj )1Ωckj ]
− e2(1+∆)t−log c(t,ℓ) E[f log(f /fkj )].

Using Hölder’s inequality and Lemma A.4, we obtain the estimate
E[(fkj − f )1Ωckj ] ≤ E[(fkj − f )4 ]1/4 · P(Ωckj )3/4 ≤ C · ℓ−3/2 ,
and by the same argument E[f log(f /fkj )1Ωckj ] ≤ C · ℓ−3/2 , E[f η(tI kj (sτ ))1Ωckj ] ≤ C · ℓ−3/2 , and

11

E[hkj 1Ωckj ] ≤ C · ℓ−3/2 . Putting the obtained estimates into (A.4), we obtain
E[(f − hkj )1Ωkj ] ≤ e2(1+∆)t−log c(t,ℓ) (tE[f I kj (sτ )] − E[f log(f /fkj )]) + C · ℓ−3/2 .
Summation over k and j yields
n X
ℓ
X

k=1 j=1



E[(f − hkj )1Ωkj ] ≤ e2(1+∆)t−log c(t,ℓ) tE[f Z] −

n X
ℓ
X

k=1 j=1

By application of Proposition 4.1 from [Led96], we have
−
and thus

n X
ℓ
X

k=1 j=1

n X
ℓ
X

k=1 j=1



E[f log(f /fkj )] + C · ℓ−1/2 .

E[f log(f /fkj )] ≤ −E[f log f ] + E [f ] · log E[f ],

E[(f − hkj )1Ωkj ] ≤ e2(1+∆)t−log c(t,ℓ) E [f ] · log E[f ] + C · ℓ−1/2 .


Lemma A.6 Consider the function r defined through r(t, x) := x log x + (1 + t)(1 − x). For any
s ∈ S and t ≥ 0, it holds
E[r((1 + ∆)t, exp(tI kj (s))) 1Ωkj ] ≤ Ct2 · ℓ−3/2 +

t2
E[(I kj (s))2 ].
2

Proof of Lemma A.6. For ﬁxed non-negative t consider the functions η, δ deﬁned through
2
η(x) := r((1 + ∆)t, etx ) = etx tx + (1 + (1 + ∆)t)(1 − etx ) and δ(x) = η(x) − xη ′ (0) − (tx)
2 ,
respectively. We have δ(0) = 0 and δ′ (x) = t2 (x − (1 + ∆))(etx − 1). Thus, the sign of δ′ (x)
coincides with the one of x(x − (1 + ∆)). This implies that δ(x) ≤ δ(0) = 0 for all x ≤ 1 + ∆,
and hence η(x) ≤ xη ′ (0) + (tx)2 /2. Since the estimate I kj (s) ≤ 1 + ∆ holds on Ωkj , by the
preceeding arguments we obtain
r((1 + ∆)t, etI

kj (s)

)1Ωkj ≤ (−(1 + ∆)t2 I kj (s) + (tI kj (s))2 /2)1Ωkj .

Taking expectations on both sides yields
E[r((1 + ∆)t, exp(tI kj (s)))1Ωkj ] ≤ E[(−(1 + ∆)t2 I kj (s) + (tI kj (s))2 /2)1Ωkj ].
Therefrom, by means of the relation 1Ωkj ≤ 1, we obtain
E[r((1 + ∆)t, exp(tI kj (s)))1Ωkj ] ≤ −(1 + ∆)t2 E[I kj (s)1Ωkj ] +

t2
E[(I kj (s))2 ].
2

The decomposition 1 = 1Ωkj + 1Ωckj , Hölder’s inequality and Lemma A.4 imply that
E[r((1 + ∆)t, exp(tI kj (s)))1Ωkj ] ≤ Ct2 ℓ−3/2 +

t2
E[(I kj (s))2 ],
2

(recall that E[I kj (s)] = 0 for all s ∈ S) which ﬁnishes the proof of the lemma.



Remark A.7 There is a correspondence between some of the auxiliary results proved above
and results appearing in [KR05]. Lemmata 3.1, 3.2, and 3.3 therein correspond to our Lem-

12

mata A.3, A.5, and A.6, respectively. Both, results and proofs turn out to be more intricate in
the considered PPP setup.

A.2. Proof of Theorem 2.1
First note that it is suﬃcient to prove statements a)–c) of Theorem 2.1 for the case of ﬁnite S.
Based on this, the case of countable S follows using the monotone convergence theorem. Thus, we
assume from now on without loss of generality that S = {s1 , . . . , sm }, and the preceeding results
from Section A.1 (which were mostly obtained under the validity of Assumption F) are available.
For ﬁxed t and ℓ ≥ ℓ0 = ℓ0 (t) (here, ℓ0 (t) is deﬁned as in the preceding subsection), let us
represent the PPP Nk as in (A.1) as the superposition of ℓ i.i.d. PPPs Nkj with intensity measures
Λk /ℓ, respectively. Then, application of Proposition 4.1 from [Led96] and the decomposition
1Ω = 1Ωkj + 1Ωckj yield
E [f log f ] − E[f ] log E[f ] ≤
=

n X
ℓ
X

n X
ℓ
X

E[f log(f /fkj )]

k=1 j=1

E[f log(f /fkj ) 1Ωkj ] +

k=1 j=1

{z

|

}

=: 

n X
ℓ
X

E[f log(f /fkj ) 1Ωckj ],

k=1 j=1

|

{z

=: 

(A.5)

}

and we investigate the two terms separately.
Examination of : For k ∈ {1, . . . , n} and j ∈ {1, . . . , ℓ}, consider the strictly positive random
variables gkj deﬁned through
gkj :=

m
X

Pkj
n (τ = i) exp (tSn (si )) .

i=1

We have the elementary decomposition
E[f log(f /fkj )1Ωkj ] = E[gkj log(f /fkj )1Ωkj ] + E[(f − gkj ) log(f /fkj )1Ωkj ].

(A.6)

Note that Ekj
n [f /fkj ] = 1, and thus
h
E[gkj log(f /fkj )1Ωkj ] ≤ sup{E[gkj h1Ωkj ] : h is Fn -measurable with Ekj
n [e ] ≤ 1}.

Thus, due to the duality formula for the relative entropy (cf., for instance, p. 83 in [Led96] or
Proposition 2.12 in [Mas07]), we obtain
E[gkj log(f /fkj )1Ωkj ] ≤ E[gkj 1Ωkj log(gkj 1Ωkj )] − E[gkj 1Ωkj log Ekj
n [gkj 1Ωkj ]].
Putting this estimate into equation (A.6) yields
E[f log (f /fkj ) 1Ωkj ] ≤ E[gkj 1Ωkj log(gkj 1Ωkj )] − E[gkj 1Ωkj log Ekj
n [gkj 1Ωkj ]]
+ E[(f − gkj ) log(f /fkj )1Ωkj ],

and by summation over k and j we obtain
≤

n X
ℓ
X

k=1 j=1

E[gkj 1Ωkj log(gkj 1Ωkj )] −

13

n X
ℓ
X

k=1 j=1

E[gkj 1Ωkj log Ekj
n [gkj 1Ωkj ]]

+

n X
ℓ
X

k=1 j=1

E[(f − gkj ) log(f /fkj ) 1Ωkj ].

(A.7)

Lemma A.3, combined with the facts that f − gkj ≥ 0 and tI kj (sτ )1Ωkj ≤ (1 + ∆)t1Ωkj , implies
E[(f − gkj ) log(f /fkj )1Ωkj ] ≤ (1 + ∆)tE[(f − gkj )1Ωkj ].

(A.8)

For k ∈ {1, . . . , n} and j ∈ {1, . . . , ℓ}, consider the positive and Fnkj -measurable random variables hkj as deﬁned in (A.3). By the variational deﬁnition of relative entropy (see Equation (1.5)
in [Led96] or Proposition 2.12 in [Mas07]), we obtain
kj
kj
Ekj
n [gkj 1Ωkj log(gkj 1Ωkj )] − En [gkj 1Ωkj log En [gkj 1Ωkj ]]

≤ Ekj
n [(gkj log(gkj /hkj ) − gkj + hkj )1Ωkj ].

By taking expectations on both sides of the last estimate, and combining the result with (A.8)
we obtain from (A.7) that
≤

n X
ℓ
X

k=1 j=1

E[(gkj log(gkj /hkj ) + (1 + (1 + ∆)t)(hkj − gkj ))1Ωkj ]
+ (1 + ∆)t

n X
ℓ
X

k=1 j=1

E[(f − hkj ) 1Ωkj ] =: 1 + 2 .

In order to bound 1 from above, introduce the function r deﬁned via
r(t, x) = x log x + (1 + t)(1 − x).
By the deﬁnition of gkj and hkj we have
gkj log(gkj /hkj ) + (1 + (1 + ∆)t)(hkj − gkj ) = hkj r((1 + ∆)t, gkj /hkj ),
and the convexity of r with respect to x yields
hkj r((1 + ∆)t, gkj /hkj ) ≤

m
X

kj
kj
Pkj
n (τ = i) exp(tSn (si ))r((1 + ∆)t, exp(tI (si ))).

i=1

Hence, multiplication with 1Ωkj and application of the Ekj
n operator yield
Ekj
n [hkj r((1 + ∆)t, gkj /hkj )1Ωkj ]
≤

m
X

kj
kj
Pkj
n (τ = i) exp(tSn (si ))E[r((1 + ∆)t, exp(tI (si )))1Ωkj ].

i=1

The expectation on the right-hand side can be bounded by means of Lemma A.6, and we obtain
Ekj
n [hkj r((1 + ∆)t, gkj /hkj )1Ωk j ]

"

#

m
t2 kj X
2 −3/2 kj
kj
1{τ =i} exp(tSnkj (si ))E[(I kj (si ))2 ] .
≤ Ct ℓ
En [exp(tSn (sτ ))] + En
2
i=1

(A.9)

In order to further bound the second term on the right-hand side of the last estimate, we consider

14

the decomposition
Ekj
n

"

m
X
i=1

#

1{τ =i} exp(tSnkj (si ))E[(I kj (si ))2 ]
+

Ekj
n

=

Ekj
n

"m
X
i=1

"m
X

#

1{τ =i} exp(tSnkj (si ))1Ωckj E[(I kj (si ))2 ]

i=1

#

1{τ =i} exp(tSnkj (si ))1Ωkj E[(I kj (si ))2 ]

, (A.10)

and we bound the two terms on the right-hand side of (A.10) separately. In order to treat the
ﬁrst one, note that on Ωkj we have exp(tSnkj (si )) ≤ exp (2t(1 + ∆) + tSn (si )), from which we
conclude that
Ekj
n

"

m
X
i=1

#

1{τ =i} exp(tSnkj (si ))1Ωkj E[(I kj (si ))2 ]
2(1+∆)t

≤e

Ekj
n

"m
X
i=1

kj

2

#

1{τ =i} exp(tSn (si ))E[(I (si )) ] .

(A.11)

For the second term on the right-hand side of (A.10), we have by Lemma A.4 that
Ekj
n

"m
X
i=1

#

1{τ =i} exp(tSnkj (si ))1Ωckj E[(I kj (si ))2 ]

kj
≤ C · Ekj
n [exp(tSn (sτ ))1Ωckj ],

and thus by putting this last estimate and (A.11) into (A.10) we obtain
Ekj
n

"m
X
i=1

#

1{τ =i} exp(tSnkj (si ))E[(I kj (si ))2 ]

2(1+∆)t

≤e

Ekj
n

"

m
X
i=1

kj

2

#

kj
1{τ =i} exp(tSn (si ))E[(I (si )) ] + C · Ekj
n [exp(tSn (sτ ))1Ωckj ].

By taking expectations on both sides of (A.9) and summation over k and j, we obtain by means
of the derived estimates in combination with Lemma A.4 that




m
n X
ℓ
X
X
t2
E[(I kj (si ))2 ]
1 ≤ Ct2 ℓ−1/2 + e2(1+∆)t E  1{τ =i} exp(tSn (si ))
2
i=1
k=1 j=1

+
P

n X
ℓ
Ct2 X
E[exp(tSnkj (sτ ))1Ωckj ].
2 k=1 j=1

P

Since nk=1 ℓj=1 E[(I kj (si ))2 ] ≤ Vn and E[exp(tSnkj (sτ ))1Ωckj ] ≤ C · ℓ−3/2 (the last estimate
follows from HÃűlder’s inequality and Lemma A.4), we obtain
1 ≤ Ct2 ℓ−1/2 +

t2 2(1+∆)t
e
Vn E[f ].
2

A suitable bound for 2 follows directly from Lemma A.5. By combining the derived estimates
for 1 and 2 , we obtain
 ≤ Ct2 ℓ−1/2 +

t2 2(1+∆)t
e
Vn E[f ] + (1 + ∆)te2(1+∆)t−log c(t,l) E [f ] log E[f ] + C · ℓ−1/2 . (A.12)
2

15

Examination of : By Hölder’s inequality, Lemmata A.2, A.3, and A.4, we have
≤

n X
ℓ
X

k=1 j=1

E[(tf I kj (sτ ))4 ]1/4 P(Ωckj )3/4 ≤ C · ℓ−1/2 .

(A.13)

We now merge the examinations of the terms  and . More precisely, by combining (A.5)
with (A.12) and (A.13) and letting ℓ tend towards inﬁnity we obtain that
tL′ (t) − (te2t + 1)L(t) ≤ t2 e2t (Vn /2).
Now, the rest of the proof follows in complete analogy to the one of Theorem 2.1 in [KR05]. 

Appendix B. Proof of Theorem 2.3
B.1. Notation and preparatory results
We maintain a large part of the notation introduced in Section A.1 for the proof of Theod P
rem 2.1. In particular, we use again the representation Nk = ℓj=1 Nkj of the PPPs Nk as the
superposition of independent PPPs Nkj with intensity Λk /ℓ and use the shorthand notations
Λ := supk=1,...,n Λk (X) and ∆ = ∆(ℓ) := Λ/ℓ. Besides, we retain the deﬁnition Ωkj := {Xkj ≤ 1}
where Xkj := Nkj (X). Let us further assume that Assumption F holds, that is, S = {s1 , . . . , sm }
is ﬁnite. Deﬁne now
Li (t) :=

n X
ℓ
X

log E[exp(−tI kj (si ))],

k=1 j=1

i ∈ {1, . . . , m}.

The corresponding exponentially compensated empirical process is Ti (t) := Sn (si ) + t−1 Li (t).
In addition to Z, let us deﬁne Zt := supi∈{1,...,m} Ti (t) (for notational convenience, from now
on we use the shorthand notation supi /inf i when the supremum/inﬁmum over i ∈ {1, . . . , m}
kj
is taken) and redeﬁne f = f (t) := exp(−tZt ) and fkj = fkj (t) := Ekj
n [f ] (the σ-ﬁelds Fn are
deﬁned as in Section A). Finally, we deﬁne F (t) := E[f ] and L(t) := log F (t). The main strategy
of the proof given in the next subsection is to derive a diﬀerential inequality for L. Let τ = τ (t)
denote the minimal value of i ∈ {1, . . . , m} such that Zt = Ti (t). As before, C denotes always
some constant (whose value is independent from ℓ) which might have diﬀerent values in diﬀerent
contexts.
Lemma B.1 Let Assumption F hold. Then, the estimate E[X] ≤ C holds true, where X can be
replaced by any of the following random variables:
a) exp(−4tI kj (sτ )),
b) (fkj − f − ψeℓ (t)f log(fkj /f )),

c) (f (eηkj − 1 − ψeℓ (t)ηkj ))4 ,

d) ((gkj − f ) log(fkj /f ))4 , and
e) (Ikj (si ))4 e−4tIkj (si ) .

Here gkj , ηkj and ψeℓ are defined in Lemma B.3 and its proof, respectively. The constant C can
be chosen independent of k and j, and in statements a) and e), it can in addition be chosen
independently of sτ and si , respectively.
Proof. As for the proof of Lemma A.4, all the estimates are easily derived and we thus omit
the proof.


16

Lemma B.2 Let Assumption F hold and ψℓ (t) = 12 (1+e2(1+∆)t ). Set ℓkji (t) = log E[exp(−tI kj (si ))].
Then, the following estimates hold almost surely.
a) fkj /f ≤ exp(tI kj (sτ ) + ℓkjτ ), and
b) exp(tI kj (sτ ) + ℓkjτ ) ≤ ψℓ (t) · (1 + αℓ ) + β · ℓ−3/2 on Ωkj where αℓ is a monotone sequence
decreasing to 0 as ℓ to ∞ and β > 0.
Proof. For s ∈ S, deﬁne Snkj (s) := Sn (s) − I kj (s) and
Z kj := sup(Snkj (s) + t−1 log E[exp(−tSnkj (s))]).
s∈S

Let τkj be the smallest i ∈ {1, . . . , m} such that
Z kj = Snkj (si ) + t−1 log E[exp(−tSnkj (si ))].
kj
Then, f ≤ exp(−tZ kj ) exp(−tI kj (sτkj ) − ℓkjτkj (t)), which implies Ekj
n [f ] ≤ exp(−tZ ). By
deﬁnition of Z kj , we have exp(−tZ kj ) ≤ f · exp(tI kj (sτ ) + ℓkjτ (t)), and statement a) follows. In
order to proof statement b), ﬁrst note that exp(tI kj (sτ )) ≤ e(1+∆)t on Ωkj , and it remains to
ﬁnd an estimate for exp(ℓkjτ (t)) = E[exp(−tI kj (sτ ))]. Consider the decomposition

E[exp(−tI kj (sτ ))] = E[exp(−tI kj (sτ ))1Ωkj ] + E[exp(−tI kj (sτ ))1Ωckj ].

(B.1)

In order to bound the ﬁrst term on the right-hand side of (B.1), note that E[exp(−tI kj (sτ ))1Ωkj ] ≤
E[etY ] with Y = −I kj (sτ )1Ωkj . By the convexity of the exponential function, we have
E[etY ] ≤

1
1 + ∆ − EY −(1+∆)t EY + 1 + ∆ (1+∆)t
e
+
e
= (e−(1+∆)t + e(1+∆)t )(1 + o(1)). (B.2)
2(1 + ∆)
2(1 + ∆)
2

The second term one the right-hand side of (B.1) is bounded using HÃűlder’s inequality, Lemmata A.2 and B.1 as follows:
E[exp(−tI kj (sτ ))1Ωckj ] ≤ E[exp(−4tI kj (sτ ))]1/4 · P(Ωckj )3/4 ≤ C · ℓ−3/2 ,

(B.3)

and statement b) follows from the combination of (B.2) and (B.3).



Lemma B.3 For k ∈ {1, . . . , n} and j ∈ {1, . . . , ℓ}, define positive random variables gkj via
gkj =

m
X
i=1

Pkj
n (τ = i) exp(−tSn (si ) − Li (t)).

Set ϕℓ := ψeℓ ·log ψeℓ where ψeℓ := ψℓ ·(1+αℓ )+β·ℓ−3/2 with ψℓ , αℓ , and β defined as in Lemma B.2.
For sufficiently large ℓ, let θℓ be the unique positive solution of the equation ϕℓ (t) = 1. Then,
for any t ∈ (0, θℓ ),
n X
ℓ
X

k=1 j=1

E[(gkj − f ) log(fkj /f )]




n X
ℓ
ϕℓ (t)  X
−1/2

E(gkj log(gkj /Ekj
.
≤
n [gkj ])) − E[f log f ] + C · ℓ
1 − ϕℓ (t) k=1 j=1

17

Proof. Since Snkj is Fnkj -measurable, it is easy to verify that
kj
kj
Ekj
n [gkj ] = En [f exp(tI (sτ ) + ℓkjτ (t))],

and hence,
n X
ℓ
X

k=1 j=1

n X
ℓ
X

E[gkj − f ] =

k=1 j=1

E[f (exp(tI kj (sτ ) + ℓkjτ (t)) − 1)].

Set ηkj = tI kj (sτ ) + ℓkjτ (t). Then,
n X
ℓ
X

k=1 j=1

E[gkj − f ] =
=

n X
ℓ
X

k=1 j=1
n X
ℓ
X

k=1 j=1

P

P

E[f (eηkj − 1 − ψeℓ (t)ηkj )] + ψeℓ (t)E[f

n X
ℓ
X

ηkj ]

k=1 j=1

E[f (eηkj − 1 − ψeℓ (t)ηkj )] − ψeℓ (t)E[f log f ],

(B.4)

since nk=1 ℓj=1 ηkj = − log f . Consider the ﬁrst term on the right-hand side of (B.4). First,
by HÃűlder’s inequality and Lemma B.1
n X
ℓ
X

k=1 j=1

P

P

E[f (eηkj − 1 − ψeℓ (t)ηkj )1Ωckj ] ≤ C · ℓ−1/2 .

In order to bound nk=1 ℓj=1 E[f (eηkj − 1 − ψeℓ (t)ηkj )1Ωkj ] from above, note that the function
x 7→ ex − 1 − xψeℓ (t) is non-increasing on the interval (−∞, log ψeℓ (t)]. Hence, we obtain by
Lemma B.1 that
n X
ℓ
X

k=1 j=1

E[f (eηkj − 1 − ψeℓ ηkj )1Ωkj ] ≤ ψeℓ
−

n X
ℓ
X

n X
ℓ
X

k=1 j=1

≤ ψeℓ

E[f log(f /fkj )]

k=1 j=1

E[(fkj − f − ψeℓ f log(fkj /f ))1Ωckj ]

n X
ℓ
X

k=1 j=1

E[f log(f /fkj )] + C · ℓ−1/2 .

Putting the obtained estimates into (B.4) yields
n X
ℓ
X

k=1 j=1



E[gkj − f ] ≤ ψeℓ (t) 

n X
ℓ
X

k=1 j=1



E[f log(f /fkj )] − E[f log f ] + C · ℓ−1/2 .

Using the same argument as in the proof of Theorem 2.1 yields
n X
ℓ
X

k=1 j=1

E[gkj − f ]


≤ ψeℓ (t) 

ℓ
n X
X

k=1 j=1

(B.5)


−1/2

.
E[gkj log(gkj /Ekj
n [gkj ]) + (gkj − f ) log(fkj /f )] − E[f log f ] + C · ℓ

18

Now, in order to prove the claim assertion of the lemma, take note of the decomposition
n X
ℓ
X

k=1 j=1

E[(gkj − f ) log(fkj /f )] =

n X
ℓ
X

k=1 j=1

+

E[(gkj − f ) log(fkj /f )1Ωkj ]

n X
ℓ
X

k=1 j=1

E[(gkj − f ) log(fkj /f )1Ωckj ].

(B.6)

Using statement b) of Lemma B.2, the estimate (B.5) and the deﬁnition of ϕℓ , we can bound
the ﬁrst term as follows (note that gkj − f ≥ 0):
ℓ
n X
X

k=1 j=1

E[(gkj

− f ) log(fkj /f )1Ωkj ] ≤ log ψeℓ (t)



≤ ϕℓ (t) 

ℓ
n X
X

ℓ
n X
X

k=1 j=1

E[gkj − f ]


−1/2

.
[gkj log(gkj /Ekj
n [gkj ]) + (gkj − f ) log(fkj /f )] − E[f log f ] + C · ℓ

k=1 j=1

The second summand on the right-hand side of (B.6) can be bounded using Hölder’s inequality,
Lemma A.2 and Lemma B.1 once more:
n X
ℓ
X

k=1 j=1

E[(gkj − f ) log(fkj /f )1Ωcℓ ] ≤

n X
ℓ
X

k=1 j=1

E[(gkj − f )4 (log(fkj /f ))4 ]1/4 P(Ωcℓ )3/4 ≤ C · ℓ−1/2 .

Combining the bounds obtained for the two terms in (B.6) implies the assertion of the lemma. 
Remark B.4 Both ψeℓ (t) and ϕℓ (t) are non-increasing in ℓ and non-decreasing in t. Hence, the
solution θℓ of the equation ϕℓ = 1 (which exists for suﬃciently large ℓ) is non-decreasing in ℓ and
the limit θ∞ := limℓ→∞ θℓ satisﬁes θ∞ ∈ [0.46, 0.47] (cf. p. 1075 in [KR05]). The approximate
value of θ∞ is of interest for the proof of Theorem 2.3 which is done by considering diﬀerent
cases for the value of t (cf. [KR05] for details).
Lemma B.5 Let Y be a random variable with values in (−∞, 1 + ∆] and E[Y 2 ] < +∞. Then,
for any positive t,
E[tY etY ] − E[etY ] log E[etY ] ≤

E[Y 2 ]
(1 + ((1 + ∆)t − 1)e(1+∆)t ).
(1 + ∆)2

Proof. The proof follows completely along the lines of the one of Lemma 4.4 in [KR05], and
we thus omit it.

Remark B.6 Again, there is a correspondence between some of the auxiliary results here and
the ones used in [KR05]. Lemmata B.2 and B.3 are versions of Lemmata 4.2 and 4.3 in [KR05]
tailored to our framework. As already mentioned above, Lemma B.5 is nearly the same as
Lemma 4.4 in [KR05].

B.2. Proof of Theorem 2.3
The essential arguments of the proof follow along the proof of Theorem 1.2 in [KR05]. Since the
random functions Ti (t) are analytic in t, the random function f (t) is continuous and piecewise
analytic as a function in t. Its derivative with respect to t satisﬁes
f ′ = −(Zt + tZt′ )f

19

where tZt′ = L′τ (t) − t−1 Lτ (t). Thus, by the Fubini’s theorem, we have
F (t) = 1 −

Z

t

0

E[(Zu + uZu′ )f (u)]du.

Hence, F is absolutely continuous with respect to the Lebesgue measure, with a.e. derivative
in the sense of Lebesgue given by F ′ (t) = −E[(Zt + tZt′ )f ]. Moreover, the function Λ = log F
has the a.e. derivative F ′ /F . As in the proof of Theorem 2.1, application of Proposition 4.1
from [Led96] yields
E[f log f ] − E[f ] log E[f ] ≤

n X
ℓ
X

E[gkj log(gkj /Ekj
n [gkj ])]

k=1 j=1

+

n X
ℓ
X

k=1 j=1

(B.7)
E[(gkj − f ) log(f /fkj )]

for any positive integrable random variables gkj such that E[gkj log[gkj ]] < ∞. On the other
hand,
E[f log f ] − E[f ] log E[f ] = E[t2 Zt′ f ] + tF ′ (t) − F (t) log F (t) a.e.
(B.8)
Combining (B.7) and (B.8) yields
tF ′ (t) − F (t) log F (t) ≤ −E[t2 Zt′ f ] +

n X
ℓ
X

E[gkj log(gkj /Ekj
n [gkj ])]

k=1 j=1

+

n X
ℓ
X

k=1 j=1

E[(gkj − f ) log(fkj /f )].

P

kj
We now specialize this estimate with the choice gkj = m
i=1 Pn (τ = i) exp(−tSn (si ) − Li (t)),
which coincides with the deﬁnition of gkj in Lemma B.3. Applying Lemma B.3 and algebraic
transformations yields

(1 − ϕℓ (t))(tF ′ (t) − F log F ) ≤ ϕℓ (t) · E[t2 Zt′ f − f log f ]
− E[t2 Zt′ ft ] +

n X
ℓ
X

k=1 j=1

−1/2
E[gkj log(gkj /Ekj
,
n [gkj ])] + C · ℓ

where ϕℓ is deﬁned in Lemma B.3. Using the identity E[t2 Zt′ f − f log f ] = −tF ′ , we obtain
tF ′ − (1 − ϕℓ (t))F log F ≤ −E[t2 Zt′ f ] +

n X
ℓ
X

k=1 j=1

−1/2
E[gkj log(gkj /Ekj
.
n [gkj ])] + C · ℓ

(B.9)

kj
kj
kj
kj
Deﬁne ωkj := gkj /Ekj
n [gkj ]. Then, En [gkj log(gkj /En [gkj ])] = En [gkj ] · En [ωkj log ωkj ]. Using
the convexity of x 7→ x log x, we conclude that

Ekj
n [gkj ]ωkj

log ωkj ≤

m
X
i=1

kj
Pkj
n (τ = i)(−tI (si ) − ℓkji (t)) exp(−tSn (si ) − Li (t)),

and by appling the Ekj
n operator on both sides we obtain
kj
Ekj
n [gkj log(gkj /En [gkj ])]

20

≤

m
X

=

Ekj
n

i=1

kj
′
Pkj
n (τ = i) exp(−tSn (si ) − Li (t) + ℓkji (t))(tℓkji (t) − ℓkji(t))

"m
X

1{τ =i} exp(−tSnkj

i=1

−

Li + ℓkji )(tℓ′kji (t)

#

− ℓkji ) .

Thus, by taking expectations
kj
′
E[gkj log(gkj /Ekj
n [gkj ])] ≤ E[f exp(tI (sτ ) + ℓkjτ )(tℓkjτ − ℓkjτ )].
−3/2 . In
By HÃűlder’s inequality and Lemma B.1, we have E[gkj log(gkj /Ekj
n [gkj ])1Ωckj ] ≤ C · ℓ

order to bound E[gkj log(gkj /Ekj
n [gkj ])1Ωkj ], ﬁrst note that the convexity of the functions together
with the fact that ℓkji(0) = 0 implies tℓ′kjτ − ℓkjτ ≥ 0. Thus, we can use Lemma B.2 in order to
obtain
′
e
E[gkj log(gkj /Ekj
n [gkj ])1Ωkj ] ≤ ψℓ (t) · E[(tℓkjτ (t) − ℓkjτ (t))f ].
By the identity t2 Zt′ = tL′τ − Lτ , we get
−E[t2 Zt′ f ] +

n X
ℓ
X

k=1 j=1

′
e
E[gkj log(gkj /Ekj
n [gkj ])] ≤ (ψℓ (t) − 1)E[(tLτ − Lτ )f ].

(B.10)

In order to bound the expectation on the right-hand side of the last estimate, let us ﬁrst note that
tL′τ − Lτ ≤ supi (tL′i − Li ). In order to bound supi (tL′i − Li ), introduce (for ﬁxed i ∈ {1, . . . , m})
e deﬁned via
the event Ω
kj
e kj := {I kj (si ) ≥ −(1 + ∆)}.
Ω

e , hence Ω
e c ⊆ Ωc . Setting
Thanks to the boundedness of the functions s ∈ S, we have Ωkj ⊆ Ω
kj
kj
kj
Ykj := −I kj (si ), we obtain

tℓ′kji (t) − ℓkji (t) ≤ tE[exp(tYkj )Ykj ] − E[etYkj ] log E[etYkj ]

tYkj 1

tYkj 1
e
Ωkj
e
Ωkj
] log E[e
]. (B.11)

≤ tE[exp(tYkj )Ykj 1Ω
e c ] + tE[exp(tYkj )Ykj 1Ω
e ] − E[e
kj

kj

The ﬁrst term on the right-hand side of (B.11) is bounded using Lemma B.1:

e c 3/4 ≤ C · P(Ωc )3/4 ≤ C · ℓ−3/2 .
tE[exp(tYkj )Ykj 1Ω
kj
e c ] ≤ C · P(Ωkj )
kj

The second and third term on the right-hand side of (B.11) are bounded using Lemma B.5 which
yields
tYkj 1

E[t exp(tYkj )Ykj 1Ω
e ] − E[e
kj

tYkj 1
e
Ωkj
e
Ωkj
] log E[e
]

tYkj 1

= E[t exp(tYkj 1Ω
e )Ykj 1Ω
e ] − E[e
≤

kj

2]
E[Ykj

2 (1

(1 + ∆)

kj

tYkj 1
e
Ωkj
e
Ωkj
] log E[e
]

+ ((1 + ∆)t − 1)e(1+∆)t ).

Hence summing over all k and j in (B.11) yields
tL′i − Li ≤ C · ℓ−1/2 +

Vn
(1 + ((1 + ∆)t − 1)e(1+∆)t ),
(1 + ∆)2

21

and this estimate holds for all i ∈ {1, . . . , m}. Putting the obtained estimates into (B.9)
and (B.10) and letting ℓ tend to ∞, we obtain
tF ′ − (1 − ϕ)F log F ≤ (ψ(t) − 1)F Vn (1 + (t − 1)et ),
where ψ(t) = 12 (1 + e2t ) and ϕ = ψ log ψ. Division by F yields
tL′ − (1 − ϕ)L ≤

Vn 2t
(e − 1)(1 + (t − 1)et ).
2

This diﬀerential inequality for L coincides with equation (4.21) in [KR05] and the rest of the
proof follows along the lines of the one given in that paper (Lemma 4.1 in [KR05] which is
used for the proof translates without changes in the proof to our framework, whereas the purely
analytical Lemmata 4.5 and 4.6 in [KR05] can be borrowed unchanged).


Appendix C. A useful consequence of Theorem 2.1
In this subsection, we state and prove another concentration inequality which is the essential
ingredient to prove the auxiliary Lemma E.1 in Appendix E. As will become clear from the
proof, it can be regarded as an integrated version of statement c) from Theorem 2.1.
Proposition C.1 Let N1 , . . . , Nn be independent PPPs
on some Polish
space X with finite
R
R
1 Pn
intensity measures Λ1 , . . . , Λn . Set νn (r) = n k=1 { X r(x)dNk (x) − X r(x)dΛk (x)} for r contained in a countable class R of real-valued measurable functions.
Then, for any ε > 0, there exist constants c1 , c2 , c3 such that
E

"

2

sup |νn (r)| − c(ε)H

r∈R

≤ c1

2

(

! #
+

υ
nH 2
exp −c2 ε
n
υ

!



√ nH
M2
+ 2 1 2 exp −c3 C(ε) ε
C (ε)n
M1

)

√
where C(ε) = ( 1 + ε − 1) ∧ 1, c(ε) = 2(1 + 2ε) and M1 , H and υ are such that
sup krk∞ ≤ M1 ,

r∈R

"

#

E sup |νn (r)| ≤ H, and sup Var
r∈R

r∈R

Z

X



r(x)dNk (x) ≤ υ ∀k.

Remark C.2 Analogues of Proposition C.1 have been used in the context of adaptive nonparametric estimation at various places, cf., for instance, [CRT06], [Lac08] and [JS13]. The proof
follows along the lines of the proof given in [Cha13] to a great extent and is thus only sketched.
Proof of Proposition C.1. For r ∈ R and k ∈ {1, . . . , n} deﬁne functions skr : X → R via
skr (x) :=

r(x)
M1

Hence, for all r ∈ R and and k ∈ {1, . . . , n}, we have |skr (x)| ≤ 1 and we can apply statement c)
of Theorem 2.1 for S = {(s1r , . . . , snr ) : r ∈ R} (the quantity Z corresponds to Mn1 supr∈R νn (r)).
Application of Theorem 2.1 c) yields for any x > 0 that
P

"

#

n
n
sup νn (r) ≥
E sup νn (r) + x
M1 r∈R
M1
r∈R

22

!

x2
≤ exp −
2υ + 3x

!

with υ = 2EZ + Vn where Vn = supr∈R Var (Sn (sr )) and Sn is deﬁned as in the statement of
Theorem 2.1. Specialising with x = ny/M1 , we have
n2 y 2
P(sup νn (r) ≥ H + y) ≤ P(sup νn (r) ≥ E[sup νn (r)] + y) ≤ exp −
2
2M1 υ + 3M1 ny
r∈R
r∈R
r∈R
Note that on the one hand we have EZ ≤ nH/M1 , and on the other hand Vn ≤

combination imply υ ≤ 2nH/M1 +

nυ/M12 .

We have

ny 2
P(sup νn (r) ≥ H + y) ≤ exp −
2(2M1 H + υ) + 3M1 y
r∈R

nυ
,
M12

!

.

which in

!

which is used to obtain
P(sup |νn (r)| ≥ H + y) ≤ P(sup νn (r) ≥ H + y) + P(sup −νn (r) ≥ H + y)
r∈R

r∈R

r∈R

= P(sup νn (r) ≥ H + y) + P(sup νn (−r) ≥ H + y)
r∈R

r∈R

ny 2

≤ 2 · exp −

2(2M1 H + υ) + 3M1 y

!

.

Below, we will apply this estimate for y = µ + ηH. With this choice of y, following an argument
from [BM98], one can obtain the estimate
#

"

y2
1 µ2 2(η ∧ 1) µ
≥
∧
,
2(2M1 H + υ) + 3M1 y
3 2υ
7
M1
which in turn implies
n
P(sup |νn (r)| ≥ µ + (η + 1)H) ≤ 2 exp −
3
r∈R

(

µ2 2(η ∧ 1) µ
∧
2υ
7
M1

)!

.

The proof of the claim assertion follows now from
E

"

2

sup |νn (r)| − 2(1 + 2ε)H

r∈R

2

! #

=

Z

∞

0

+

2

!

2

P sup |νn (r)| ≥ 2(1 + 2ε)H + t dt
r∈R

and computation of the integral on the right-hand side (see [Cha13] for details).



Appendix D. Proof of Theorem 3.3
1 16δ
Let us deﬁne ζ = min{ Γη
, √r } with δ =
function λθ through

λθ :=

 1/2

r
4

+ θ0



rζ
16n

1/2

+



rζ
16n

1
2

−

1/2

1
√
2 2

∗

and for each θ = (θj )0≤|j|≤kn∗ ∈ {±1}2kn +1 the

X

θj ϕj =

∗
1≤|j|≤kn

 1/2

r
4

Then, the calculation


rζ
16n

1/2

X

∗
0≤|j|≤kn

θj ϕj
∞

≤



rζ
16n

1/2

23

X

∗
0≤|j|≤kn

√

2

+



rζ
16n

1/2

X

∗
0≤|j|≤kn

θj ϕj .

≤



rζ
8



1/2



∗
0≤|j|≤kn

1/2 



X

1/2 

γj−1 

2k∗ + 1
rζΓ
γkn∗ · n
8
n

1/2
 1/2
rζηΓ
r
≤
≤
8
8
≤

X



1/2

∗
0≤|j|≤kn

1/2

γj 
n

√
∗
shows that λθ ≥ r · δ. In particular, λθ is non-negative for all θ ∈ {±1}2kn +1 . Moreover
∗
kλθ k2γ ≤ r holds for each θ ∈ {±1}2kn +1 due to the estimate
kλθ k2γ

=

X

2

∗
0≤|j|≤kn

|[λθ ]j | γj =
≤

" 
1/2

r
4

r
+
2



rζ
8n



+ θ0


+

rζ
16n

1/2 #2

+

rζ
16

X 1
rζ
γkn∗
16
n
1≤|j|≤k ∗

X

∗
1≤|j|≤kn

γj
n

n

r rζ
2k∗ + 1
≤ + γkn∗ n
≤ r.
2
8
n

∗

This estimate and the non-negativity of λθ together imply λθ ∈ Λrγ for all θ ∈ {±1}2kn +1 . Let Pθ
denote the joint distribution of the i.i.d. sample N1 , . . . , Nn when the true parameter is λθ . Let
i
PN
θ denote the corresponding one-dimensional marginal distributions and Eθ the expectation
e be an arbitrary estimator of λ. The key argument of the
with respect to Pθ . From now on, let λ
proof is the reduction scheme
e − λk2 ] ≥
sup E[kλ

λ∈Λrγ

=
=

sup
∗ +1
2kn

θ∈{±1}

e − λ k2 ] ≥
Eθ [kλ
θ

X

1
22kn∗ +1

∗

X

θ∈{±1}2kn +1 0≤|j|≤kn

X

1
22kn∗ +1

∗

X

∗

θ∈{±1}2kn +1 0≤|j|≤kn
∗

X

1

22kn∗ +1

∗

θ∈{±1}2kn +1

e − λ ] |2 ]
Eθ [|[λ
θ j

e − λ k2 ]
Eθ [kλ
θ

1
e − λ (j) ]j |2 ]}, (D.1)
e − λ ]j |2 ] + E (j) [|[λ
{Eθ [|[λ
θ
θ
θ
2

∗

(j)

∗

where for θ ∈ {±1}2kn +1 the element θ (j) ∈ {±1}2kn +1 is deﬁned by θk = θk for k 6= j
Rp
(j)
and θj = −θj . Consider the Hellinger aﬃnity ρ(Pθ , Pθ(j) ) :=
dPθ dPθ(j) . For an arbitrary
e of λ we have
estimator λ
ρ(Pθ , Pθ(j) ) ≤
≤

Z

e − λ ]j | q
|[λ
θ
dPθ dPθ(j) +
|[λθ − λθ(j) ]j |

Z

e − λ ] |2
|[λ
θ j
dPθ
|[λθ − λθ(j) ]j |2

!1/2

+

Z

Z

e − λ (j) ]j | q
|[λ
θ
dPθ dPθ(j)
|[λθ − λθ(j) ]j |
e − λ (j) ] |2
|[λ
j
θ
dP (j)
|[λθ − λθ(j) ]j |2 θ

!1/2

,

from which we conclude by means of the elementary inequality (a + b)2 ≤ 2a2 + 2b2 that
1
e − λ (j) ] |2 ].
e − λ ] |2 ] + E (j) [|[λ
|[λθ − λθ(j) ]j |2 ρ2 (Pθ , Pθ(j) ) ≤ Eθ [|[λ
j
θ j
θ
θ
2

24

Recall that the
Hellinger
√ distance between two probability measures P and Q is deﬁned as
R √
H(P, Q) := ( [ dP − dQ]2 )1/2 . Using Theorem 3.2.1 from [Rei93], we obtain
√
Z
Z p
q
|λθ − λθ(j) |2
1
ζ r
1
2
Ni
2
2 Ni
√
p
H (Pθ , Pθ(j) ) = ( λθ − λθ(j) ) =
≤ √ kλθ − λθ(j) k2 =
≤ .
2
4δ r
16δn
n
( λθ + λθ(j) )
Consequently, with Lemma 3.3.10 (i) from [Rei89] it holds
H 2 (Pθ , Pθ(j) ) ≤

n
X
i=1

Ni
i
H 2 (PN
θ , Pθ (j) ) ≤ 1.

Thus, the relation ρ(Pθ , Pθ(j) ) = 1 − 21 H 2 (Pθ , Pθ(j) ) implies ρ(Pθ , Pθ(j) ) ≥ 21 . Finally, putting the
obtained estimates into the reduction scheme (D.1) leads to
1

e − λk2 ] ≥
sup E[kλ

22kn +1
∗

λ∈Λrγ

X

≥

X

X

∗

θ∈{±1}2kn +1 0≤|j|≤kn

∗
0≤|j|≤kn

∗

o
1n
Eθ [|[λ̃ − λθ ]j |2 ] + Eθ(j) [|[λ̃ − λθ(j) ]j |2 ]
2

ζr X 1
ζr
1
|[λθ − λθ(j) ]j |2 =
≥
· Ψn ,
16
64 0≤|j|≤k∗ n
64η
n

e was arbitrary.
which ﬁnishes the proof of the theorem since λ



Appendix E. Proof of Theorem 3.9

E.1. Proof of Theorem 3.9
c ∨ 1 ≤ η −1 ([λ] ∨ 1)}, the deﬁnition of which
Let us introduce the event Ω := {η([λ]0 ∨ 1) ≤ [λ]
0
0
is used to obtain the decomposition
b − λk2 1Ωc ] .
b − λk2 ] ≤ E[kλ
b − λk2 1Ω ] + E[kλ
E[kλ
b
b
b
k
k
k
n

|

n

{z

}

=:

|

n

{z

}

=:

We establish uniform upper bounds for both terms separately.
b − tk2 − kλ
b k2 holds for all t ∈ L2 ,
Uniform upper bound for : Since the equation Υn (t) = kλ
n
n
b
we obtain that argmint∈Sk Υn (t) = λk for all k ∈ {0, . . . , n} where Sk denotes the linear subspace
of L2 generated by the ϕj with |j| ≤ k. This identity combined with the deﬁnition of kbn yields
for all k ∈ {0, . . . , n} the inequality chain
b ) + pen ≤ Υn (λ
b ) + pen ≤ Υn (λ ) + pen ,
Υn (λ
k
k
k
k
b
b
k
k
n

P

n

where λk := 0≤|j|≤k [λ]j ϕj is the projection of λ on the ﬁnite-dimensional space Sk . Hence,
using the deﬁnition of the contrast, we obtain
b k2 ≤ kλ k2 + 2hλ
bn , λ
b − λ i + pen − pen
kλ
k
k
k
b
b
b
k
k
k
n

n

n

b n − λn that
b n := λ
for all k ∈ {0, . . . , n}, from which we conclude by setting Θ
b − λk2 ≤ kλ − λ k2 + pen − pen + 2hΘ
b −λ i
b n, λ
kλ
k
k
k
b
b
b
k
k
k
n

n

n

(E.1)

for all k ∈ {0, . . . , n}. Consider the set Bk := {λ ∈ Sk : kλk2 ≤ 1}. By means of the inequality

25

2uv ≤ τ u2 + τ −1 v 2 , we obtain for every τ > 0 and t ∈ Sk that
2|hh, ti| ≤ 2 ktk sup |hh, ti| ≤ τ ktk2 + τ −1 sup |hh, ti|2 .
t∈Bk

t∈Bk

b −λ ∈S
)
Combining this estimate with the estimate (E.1), we obtain (note that λ
k
b
k
k∨b
k
n

n

b − λk2 ≤ kλ − λ k2 + pen − pen + τ kλ
b − λ k2 + τ −1 sup |hΘ
b n , ti|2 .
kλ
k
k
k
b
b
b
k
k
k
n

n

n

t∈B

b

k∨kn

b − λ k2 ≤ 2kλ
b − λk2 + 2kλ − λk2 and kλ − λ k2 ≤ rγ −1 for all λ ∈ Λr thanks
We have kλ
k
k
k
γ
k
b
b
kn
kn
to Assumption A. Hence, specializing with τ := 1/4 implies
b − λk2 ≤ 3rγ −1 + 2pen − 2pen + 8 sup |hΘ
b n , ti|2 ,
kλ
k
k
b
b
k
k
n

n

t∈B

b

k∨kn

which is used to obtain





b
b − λk2 ≤ 3rγ −1 + 8  sup |hΘ
b n , ti|2 − 3([λ]0 ∨ 1) · (2(k ∨ kn ) + 1) 
kλ
k
b
kn
n
t∈B
b

k∨kn

+

24([λ]0 ∨ 1) · (2(k ∨ kbn ) + 1)
n

+

+ 2penk − 2penbk .
n

Note that we have 2(k ∨ kbn ) + 1 ≤ 2k + 2kbn + 2. Thus, due to the deﬁnition of both the penalty
and Ω we obtain







b
b − λk2 1Ω ≤ 3rγ −1 + 8  sup |hΘ
b n , ti|2 − 3([λ]0 ∨ 1) · (2(k ∨ kn ) + 1) 
kλ
k
b
kn

n
t∈B
b

k∨kn



+

√
2k + 1
1Ω .
+24(1 + η −2 )( r ∨ 1) ·
n

Since the last estimate holds for all 0 ≤ k ≤ n and λ ∈ Λrγ , we obtain



1 2k + 1
2
−2 √
b
E[kλbk − λk 1Ω ] ≤ (3r + 24(1 + η )( r ∨ 1)) min max
,
n
0≤k≤n
γk
n
"
! #
n
X
3([λ]
∨
1)(2k
+
1)
0
2
b n , ti| −
+8
E
sup |hΘ
.
k=0

t∈Bk

n

(E.2)

+

We now apply Lemma E.1 from Subsection E.2 which using λ ∈ Λrγ yields that
E

"

! #

b n , ti| − 3([λ]0 ∨ 1)(2k + 1)
sup |hΘ
n
t∈Bk
+


√

s
√
√
√ 
2k
+
1
2k
+
1(
r
∨
1)
r
2k
+
1
+
exp −K2
exp −K3 n  ,
≤ K1 
n
r
n2
2

where K1 , K2 and K3 are numerical constants independent of n. The estimate 2k + 1 ≤ 3n for
k ≤ n yields
n
X

k=0

"

b n , ti|2 − 3([λ]0 ∨ 1)(2k + 1)
sup |hΘ

t∈Bk

n

! #
+

26



s
√
√
2k + 1
2k + 1 
.
exp −K2
+ exp(−K3 n).
n
r
k=0
∞
X

Note that we have



P∞ √
k=0

2k + 1 exp −K2

q

2k+1
r



≤ C < ∞ for some numerical constant

C. Since all the computations up to now hold uniformly for all λ ∈ Λrγ , plugging the derived
estimates into (E.2), we obtain


b − λk2 1 ] . min max 1 , 2k + 1
sup E[kλ
Ω
b
kn
0≤k≤n
γk
n
λ∈Λrγ



+

√
1
+ exp(−K3 n).
n

Uniform upper bound for : In order to derive an upper bound for , ﬁrst recall the deﬁnition
P
λk := 0≤|j|≤k [λ]j ϕj from above. We obtain the identity
b − λk2 1 c ] = E[kλ
b − λ k2 1 c ] + E[kλ − λ k2 1 c ].
E[kλ
Ω
Ω
Ω
b
b
b
b
k
k
k
k
n

n

n

n

(E.3)

Since kλ − λbk k2 ≤ kλk2 ≤ r due to Assumption A, the second term on the right-hand side
n
of (E.3) satisﬁes
1
(E.4)
E[kλ − λbk k2 1Ωc ] ≤ rP(Ωc ) . ,
n
n
where the probability estimate for Ωc is proved below. In order to bound the ﬁrst term on the
right-hand side of (E.3), ﬁrst note that
b − λ k2 1Ωc ] ≤
E[kλ
b
b
kn
kn

X

0≤|j|≤n

c − [λ] |2 1 c ] ≤ P(Ωc )1/2
E[|[λ]
j
Ω
j

X

0≤|j|≤n

c − [λ] |4 ]1/2 .
E[|[λ]
j
j

Therefrom, by applying Theorem 2.10 from [Pet95] (with p = 4 in the statement of this theorem),
we can conclude
b − λ k2 1Ωc ] . P (Ωc )1/2 ,
E[kλ
b
b
k
k
n

n

and it remains to ﬁnd a suitable bound for P(Ωc ). We have

c ∨ 1 < η([λ] ∨ 1)) + P([λ]
c ∨ 1 > η −1 ([λ] ∨ 1)),
P(Ωc ) = P([λ]
0
0
0
0

and the probabilities on the right-hand side can be bounded by Chernoﬀ bounds for Poisson
distributed random variables (see [MU05], Theorem 5.4). More precisely, by considering diﬀerent
cases, one can obtain the uniform bounds
c ∨ 1 < η([λ] ∨ 1)) ≤ exp(−ω (η)η −1 n) and
P([λ]
0
1
0

c ∨ 1 > η −1 ([λ] ) ∨ 1) ≤ exp(−ω (η)n)
P([λ]
0
2
0

with ω1 (η) = 1 − η + η log η > 0 and ω2 (η) = 1 − η −1 − η −1 log η > 0 for all η ∈ (0, 1). Hence,
putting together the estimates derived so far, we obtain
b − λ k2 1Ωc ] . 1 .
E[kλ
b
b
kn
kn
n

(E.5)

Putting the estimates (E.4) and (E.5) into (E.3) and noting that all the obtained estimates hold
uniformly for λ ∈ Λrγ , we obtain that
b − λk2 1Ωc ] . 1 .
sup E[kλ
b
kn
n
λ∈Λrγ

27

Combining the derived bounds for  and  implies the statement of the theorem.



E.2. Auxiliary results
The following lemma is a version of Lemma A4 in [JS13] adapted to our framework. In that
paper, a circular deconvolution model was considered and the same way Lemma A4 in [JS13] is
obtained from a variant of Proposition C.1 (cf. Lemma A3 in [JS13] or Lemma 1 in [CRT06]),
the key ingredient for the proof of Lemma E.1 is Proposition C.1.
Lemma E.1 For all k ∈ {0, . . . , n}, we have
E

"

! #

b n , ti|2 − 3([λ]0 ∨ 1)(2k + 1)
sup |hΘ
n
t∈Bk
+
(√
!
)
√
√ 
2k + 1
2k + 1 · ([λ]0 ∨ 1) kλk
2k + 1
exp −K2 ·
exp −K3 n ,
≤ K1
+
n
kλk
n2

with numerical constants K1 , K2 , and K3 .
P

Proof. For t ∈ Sk , we deﬁne the function rt by rt := kj=−k [t]j ϕj . Then, it is readily veriﬁed
R
R
b n , ti = 1 Pn { 1 rt (x)dNi (x) − 1 rt (x)λ(x)dx}. Hence, building on this deﬁnition of r,
that hΘ
i=1
0
0
n
it remains to ﬁnd constants M1 , H and υ satisfying the preconditions of Proposition C.1.
Condition concerning M1 : We have


sup krt k2∞ = sup sup |rt (y)|2 ≤ sup sup 
t∈Bk y∈[0,1)

t∈Bk y∈[0,1)

t∈Bk

j=−k



≤ sup sup 
t∈Bk y∈[0,1)

≤ 2k + 1 =:

k
X

k
X

j=−k
M12 .

2

|[t]j ||ϕj (y)|


|[t]j |2  

k
X

j=−k



ϕ2j (y)

Condition concerning H: We have


k
X

b n , ti|2 ] ≤ sup 
E[ sup |hΘ
t∈Bk

t∈Bk

≤

k
X

j=−k

1
Var
n j=−k

k
1 X
≤
n j=−k

≤

Z

1

0





|[t]j |2  · E 
Z

0

1

k
X

n Z
1X

n

j=−k

i=1

0

1



ϕj (x)[dNi (x) − dΛi (x)]



2




ϕj (x)dN1 (x)

ϕ2j (x)λ(x)dx

2k + 1
· [λ]0 ,
n

and it follows from Jensen’s inequality that we can choose H := (([λ]0 ∨ 1) · (2k + 1)/n)1/2 .
Condition concerning υ: We have
Var

Z

0

1



rt (x)dN1 (x) =

Deﬁne ej (t) = exp(2πijt) and set hλij =

R1
0

Z

0

1

|rt (x)|2 λ(x)dx.

λ(t)ej (−t)dt using which the identity λ =

28

(E.6)
P

j∈Z hλij ej

holds. We have
*

|rt (x)|2 =

k
X

i=−k

htii ei (x),

k
X

j=−k

+

htij ej (x)

R

=

k
k
X
X

i=−k j=−k



htii htij ei (x)e−j (x),

and thus by means of (E.6) that Var 01 rt (x)dN1 (x) = hAhti, htii, where for t ∈ Bk we denote
by hti the vector (hti−k , . . . , htik ) and by A positive semi-deﬁnite matrix A = ([λ]j−i )j,i=−k,...,k .
Hence,
sup Var
t∈Bk

Z

0

1



rt (x)dN1 (x) ≤ sup hA1/2 hti, A1/2 htii = sup kA1/2 tk2 = kAk.
t∈Bk

t∈Bk

In order to bound kAk, recall for an arbitrary matrix B = (bij ) the deﬁnitions
kBk1 := max
j

X
i

|bij |

and

kBk∞ := max
i

X
j

|bij |.

√
Note
that
by
the
Cauchy-Schwarz
inequality
we
have
both
kAk
≤
2k + 1kλk and kAk∞ ≤
1
√
p
2k + 1kλk and hence
by
the
formula
kAk
≤
kAk
·
kAk
(see
Corollary
2.3.2 in [GVL96])
1
∞
√
√
we obtain kAk ≤ 2k + 1 · kλk. Thus, we can choose υ := 2k + 1 · kλk · ([λ]0 ∨ 1).
The result of the Lemma follows now directly from Proposition C.1 with ε = 14 .


References
[BB09]

Baraud, Y. and Birgé, L. Estimating the intensity of a random measure by histogram type estimators. Probab. Theory Relat. Fields 143 (1) (2009), 239–284.

[BBM99]

Barron, A., Birgé, L., and Massart, P. Risk bounds for model selection via
penalization. Probab. Theory Relat. Fields 113 (3) (1999), 301–413.

[BH09]

Bauwens, L. and Hautsch, N. Modelling Financial High Frequency Data Using
Point Processes. In: Handbook of Financial Time Series. Ed. by Mikosch, T. et al.
Springer, 2009, 953–979.

[Big+13]

Bigot, J. et al. Intensity estimation of non-homogeneous Poisson processes from
shifted trajectories. Electron. J. Stat. 7 (2013), 881–931.

[Bir07]

Birgé, L. Model selection for Poisson processes. In: IMS Lecture Notes - Monograph
Series. Institute of Mathematical Statistics, 2007, 32–64.

[BL98]

Bobkov, S. and Ledoux, M. On Modiﬁed Logarithmic Sobolev Inequalities for
Bernoulli and Poisson Measures. J. Funct. Anal. 156 (2) (1998), 347–365.

[BLM16]

Boucheron, S., Lugosi, G., and Massart, P. Concentration inequalities: a nonasymptotic theory of independence. Oxford University Press, 2016.

[BM98]

Birgé, L. and Massart, P. Minimum contrast estimators on sieves: exponential
bounds and rates of convergence. Bernoulli 4 (3) (1998), 329–375.

[Bor75]

Borell, C. The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30 (2)
(1975), 207–216.

[Bré81]

Brémaud, P. Point processes and queues, martingale dynamics. Springer, 1981.

[Cha13]

Chagny, G. Estimation adaptative avec des données transformées ou incomplètes.
Application à des modèles de survie. PhD thesis. 2013.

29

[CIS76]

Cirel’son, B. S., Ibragimov, I. A., and Sudakov, V. N. Norms of Gaussian
sample function. Proceedings of the Third Japan-U.S.S.R. Symposium on Probability
Theory. Lecture Notes in Math. 550 (1976), 20–41.

[Com15]

Comte, F. Estimation non-paramétrique. Spartacus, 2015.

[CRT06]

Comte, F., Rozenholc, Y., and Taupin, M.-L. Penalized contrast estimator for
adaptive density deconvolution. Can. J. Stat. 34 (3) (2006), 431–452.

[GN00]

Grègoire, G. and Nembè, J. Convergence rates for the minimum complexity estimator of counting process intensities. Nonparametric Statistics 12 (2000), 611–643.

[GVL96]

Golub, G. H. and Van Loan, C. F. Matrix computations. Third edition. John
Hopkins University Press, 1996.

[Hou02]

Houdré, C. Remarks on deviation inequalities for functions of inﬁnitely divisible
random vectors. Ann. Probab. 30 (3) (2002), 1223–1237.

[HP02]

Houdré, C. and Privault, N. Concentration and deviation inequalities in inﬁnite
dimensions via covariance representations. Bernoulli 8 (6) (2002), 697–720.

[JS13]

Johannes, J. and Schwarz, M. Adaptive circular deconvolution by model selection
under unknown error distribution. Bernoulli 19 (5A) (2013), 1576–1611.

[Kar91]

Karr, A. F. Point processes and their statistical inference. Marcel Dekker, 1991.

[Kol99]

Kolaczyk, E. D. Wavelet shrinkage estimation of certain Poisson intensity signals
using corrected thresholds. Stat. Sinica 9 (1) (1999), 119–135.

[KR05]

Klein, T. and Rio, E. Concentration around the mean for maxima of empirical
processes. Ann. Probab. 33 (3) (2005), 1060–1077.

[Kut98]

Kutoyants, Y. A. Statistical Inference for Spatial Poisson Processes. 134. Springer,
1998.

[Lac08]

Lacour, C. Adaptive estimation of the transition density of a particular hidden
Markov chain. J. Multivariate Anal. 99 (5) (2008), 787–814.

[Led96]

Ledoux, M. On Talagrand’s deviation inequalities for product measures. ESAIM:
Probability and Statistics 1 (1996), 63–87.

[Mas00]

Massart, P. About the constants in Talagrand’s concentration inequalities for empirical processes. Ann. Probab. 28 (2) (2000), 863–884.

[Mas07]

Massart, P. Concentration Inequalities and Model Selection. Vol. 1896. Springer,
2007.

[MU05]

Mitzenmacher, M. and Upfal, E. Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, 2005.

[MW04]

Møller, J. and Waagepetersen, R. P. Statistical inference and simulation for
spatial point processes. Chapman & Hall/CRC, 2004.

[Pet95]

Petrov, V. V. Limit theorems of probability theory. Oxford Science Publications,
1995.

[PS00]

Penttinen, A. and Stoyan, D. Recent applications of point process methods in
forestry statistics. Stat. Sci. 15 (1) (2000), 61–78.

[PW04]

Patil, P. N. and Wood, A. T. A. Counting process intensity estimation by orthogonal wavelet methods. Bernoulli 10 (1) (2004), 1–24.

[RB03]

Reynaud-Bouret, P. Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities. Probab. Theory Relat. Fields 126 (1)
(2003), 103–153.

30

[RBR10]

Reynaud-Bouret, P. and Rivoirard, V. Near optimal thresholding estimation
of a Poisson intensity on the real line. Electron. J. Stat. 4 (2010), 172–238.

[Rei89]

Reiss, R.-D. Approximate Distributions of Order Statistics. Springer, 1989.

[Rei93]

Reiss, R.-D. A Course on Point Processes. Springer, 1993.

[Res87]

Resnick, S. I. Extreme Values, Regular Variation and Point Processes. Springer,
1987.

[Rud82]

Rudemo, M. Empirical Choice of Histograms and Kernel Density Estimators. Scand.
J. Stat. 9 (2) (1982), 65–78.

[San14]

Sansonnet, L. Wavelet Thresholding Estimation in a Poissonian Interactions Model
with Application to Genomic Data. Scand. J. Stat. 41 (1) (2014), 200–226.

[ST78]

Sudakov, V. N. and Tsirel’son, B. S. Extremal properties of half-spaces for spherically invariant measures. Journal of Soviet Mathematics 9 (1) (1978), 9–18.

[Sto+13]

Stoyan, D. et al. Stochastic Geometry and its Applications. Third edition. John
Wiley & Sons, 2013.

[Str10]

Streit, R. Poisson Point Processes. Springer, 2010.

[Tal95]

Talagrand, M. Concentration of measure and isoperimetric inequalities in product
spaces. Publications Mathématiques de l’IHÉS 81 (1) (1995), 73–205.

[Tal96]

Talagrand, M. New concentration inequalities in product spaces. Invent. Math.
126 (3) (1996), 505–563.

[WN07]

Willett, R. M. and Nowak, R. D. Multiscale Poisson Intensity and Density Estimation. IEEE T. Inform. Theory 53 (9) (2007), 3171–3187.

[Wu00]

Wu, L. A new modiﬁed logarithmic Sobolev inequality for Poisson point processes
and several applications. Probab. Theory Relat. Fields 118 (3) (2000), 427–438.

31

