###### Proof.

###### Quantitative CLTs on the Poisson space
via Skorohod estimates and $p$-Poincaré inequalities

###### Abstract

We establish new explicit bounds on the Gaussian approximation of Poisson
functionals based on novel estimates of moments of Skorohod integrals.
Combining these with the Malliavin-Stein method, we derive bounds in the
Wasserstein and Kolmogorov distances whose application requires minimal moment
assumptions on add-one cost operators – thereby extending the results from
(Last, Peccati and Schulte, 2016). Our applications include a CLT for the
Online Nearest Neighbour graph, whose validity was conjectured in (Wade, 2009;
Penrose and Wade, 2009). We also apply our techniques to derive quantitative
CLTs for edge functionals of the Gilbert graph, of the $k$-Nearest Neighbour
graph and of the Radial Spanning Tree, both in cases where qualitative CLTs
are known and unknown.

Keywords: CENTRAL LIMIT THEOREM; GILBERT GRAPH; KOLMOGOROV DISTANCE; MALLIAVIN
CALCULUS; NEAREST NEIGHBOUR GRAPH; ONLINE NEAREST NEIGHBOUR GRAPH; POINCARÉ
INEQUALITY; POISSON PROCESS; SKOROHOD INTEGRAL; STEIN’S METHOD; STOCHASTIC
GEOMETRY; RADIAL SPANNING TREE; WASSERSTEIN DISTANCE.

Mathematics Subject Classification (2020): 60F05, 60H07, 60G55, 60D05, 60G57

###### Contents

  1. 1 Introduction
    1. 1.1 Overview
    2. 1.2 Main contributions
  2. 2 Framework and notations
    1. 2.1 Notation
  3. 3 Second order $p$-Poincaré inequalities in Wasserstein and Kolmogorov distances
  4. 4 Ancillary results: new estimates for Skorohod integrals
    1. 4.1 A version of Itô formula
    2. 4.2 Moment Inequalities
  5. 5 Applications
    1. 5.1 Online Nearest Neighbour Graph
    2. 5.2 Gilbert Graph
    3. 5.3 $k$-Nearest Neighbour graphs
    4. 5.4 Radial Spanning Tree
  6. A Background on Malliavin Calculus
  7. B Proof of Theorem 4.1
  8. C Proofs of Theorem 4.2 and Corollaries 4.3 and 4.7
  9. D Proofs of Theorems 3.2, 3.3 and 3.4
  10. E Online Nearest Neighbour Graph
    1. E.1 Moment estimates of conditional expectations of add-one costs
    2. E.2 Moment estimates of add-one costs
    3. E.3 The orders of the variances
    4. E.4 Proof of Theorem 5.3
      1. E.4.1 Wasserstein distance when $\alpha=\frac{d}{2}$
      2. E.4.2 Wasserstein and Kolmogorov distance when $\alpha<\frac{d}{2}$
  11. F Gilbert Graph
  12. G $k$-Nearest Neighbour Graphs
  13. H Radial Spanning Tree

##  1 Introduction

###  1.1 Overview

The aim of this paper is to establish a new collection of probabilistic
inequalities, yielding quantitative CLTs for sequences of Poisson functionals
under minimal moment assumptions. We will see below that our findings
substantially extend and refine the second order Poincaré inequalities proved
in [LPS16] (see also [LRSY19, SY19, SY21]), and heavily rely on new moment
inequalities for Skorohod integrals (see Theorem 4.2) that we believe to be of
independent interest. As demonstrated in Section 5, our findings are
specifically tailored to deriving quantitative CLTs for functionals of spatial
random graphs based on Poisson inputs, in critical or near-critical regimes.

One prominent example dealt with in the present work is the Online Nearest
Neighbour Graph (ONNG), devised by Berger et al. in [BBB+07] as a simplified
version of the FKP model of the internet graph (see [FKP02]). The set-up is
the following: Take a Poisson measure on $\mathbb{R}^{d}\times[0,1]$, in such
a way that each point of the measure has a spatial coordinate in
$\mathbb{R}^{d}$ and an arrival time in $[0,1]$. Within a bounded observation
window, connect each point to its nearest neighbour in space which has smaller
arrival time. The resulting ONNG is a tree growing in time – a simple model
for an expanding network. Other graphs of interest include the Gilbert graph,
where two points are connected if they are close enough to one another, or
nearest-neighbour type graphs like the $k$-Nearest Neighbour graph and the
Radial Spanning Tree (see Section 5).

For graphs like these, one quantity of interest is the total edge-length, or
more generally, the $\alpha$-power weighted total edge-length:

$F=\sum_{e\text{ edge}}|e|^{\alpha}.$ (1.1)

The main question is to understand how such a sum fluctuates as the graph
expands. For the ONNG, convergence to the normal law was shown by Penrose
[Pen05] in the exponent range $\alpha\in\left(0,\frac{d}{4}\right)$ and
conjectured in [Wad09, PW09] for the range
$\alpha\in\left[\frac{d}{4},\frac{d}{2}\right]$. This conjecture has remained
open until now: as an application of our abstract bounds, we settle it in this
article by providing a quantitative central limit theorem for the centred and
rescaled sum of power-weighted edge-lengths with powers
$\alpha\in\left.\left(0,\frac{d}{2}\right.\right]$ (see Theorem 5.3).

###  1.2 Main contributions

As announced in Section 1.1, our theoretical findings refine the main bounds
in [LPS16], where the authors proved second order Poincaré inequalities on
configuration spaces – thus extending the Gaussian second order Poincaré
estimates established in [Cha09, NPR09, Vid20, ERTZ21] to the Poisson case.
The results of [LPS16] are based on the combination of Stein’s method [CGS11,
NP12, PR16] and Malliavin Calculus on configuration spaces [Las16, LP18].

The starting point of Stein’s method is the fact that a real-valued random
variable $N$ is standard Gaussian if and only if

$\mathbb{E}f(N)N=\mathbb{E}f^{\prime}(N)$ (1.2)

for a suitable collection of functions $f:\mathbb{R}\rightarrow\mathbb{R}$.
This allows one to represent the probability distance between a random
variable $F$ and a standard normal $N$ as

$d(F,N)=\sup_{h\in\mathcal{H}}|\mathbb{E}f_{h}(F)F-\mathbb{E}f_{h}^{\prime}(F)|,$
(1.3)

where $\mathcal{H}$ is a suitable collection of functions depending on the
choice of distance and the function $f_{h}$ is the canonical solution to the
differential equation

$f^{\prime}_{h}(x)=xf_{h}(x)+h(x)-\mathbb{E}h(N).$ (1.4)

The crucial idea behind the results of [LPS16] is that, for random variables
$F=F(\eta)$ depending on a Poisson measure $\eta$ on a space $\mathbb{X}$, one
can control quantities such as (1.3) by using integration by parts formulae
involving the add-one cost operator

$\operatorname{D}_{x}F(\eta):=F(\eta+\delta_{x})-F(\eta),$ (1.5)

where $x\in\mathbb{X}$ and $\delta_{x}$ is the Dirac measure in $x$, as well
as its iteration
$\operatorname{D}^{(2)}_{x,y}=\operatorname{D}_{x}\operatorname{D}_{y}F$. As
demonstrated in [LPS16, LRSY19, SY19, LRPY20, SY21, SBP22]) such an approach
leads to flexible bounds in the Kolmogorov and Wasserstein distances, bounds
that are particularly adapted for dealing with functionals displaying a form
of geometric stabilisation – see e.g. [PY01, Pen05, PY05, LRSY19, LRPY20] for
a discussion of this concept, as well as [KL96] for the first seminal
contribution on the topic.

One of the shortcomings of the bounds established in [LPS16, LRSY19, SY19,
LRPY20, SBP22]) is that their use in concrete applications typically requires
one to uniformly bound over $\mathbb{X}$ the moments of order $(4+\epsilon)$
(with $\epsilon>0$) of $\operatorname{D}F$ and $\operatorname{D}^{(2)}F$. Such
a uniform bound is not achievable in many relevant applications, e.g. for edge
functionals of the ONNG in the exponent range
$\alpha\in\left[\frac{d}{4},\frac{d}{2}\right]$, the range where the central
limit theorem was conjectured to hold. We substantially extend the main
results from [LPS16] in two ways:

  1. 1.

In Theorem 3.4 we establish explicit bounds in the Kolmogorov and Wasserstein
distances, whose use only requires one to uniformly bound the moments of order
$2+\epsilon$ of add-one cost operators. (See also [Tri19] for qualitative CLTs
requiring both bounds on the moments of order $2+\epsilon$ and weak
stabilisation). To motivate the reader, we will now give a simple example of a
consequence of Theorem 3.4.

Let $\nu$ be a centred probability measure on $\mathbb{R}$ such that

$c:=\int_{\mathbb{R}}|u|^{2+\epsilon}\nu(du)<\infty$ (1.6)

for some $\epsilon>0$. Define $\sigma^{2}:=\int_{\mathbb{R}}|u|^{2}\nu(du)$
and let $\eta$ be a Poisson measure on $\mathbb{R}\times[0,\infty)$ with
intensity $\nu(du)\otimes ds$. Let $T>0$ and define

$F_{T}:=\int_{0}^{T}\int_{\mathbb{R}}u\ \eta(du,ds).$ (1.7)

Then $F_{T}$ is equal in law to $\sum_{i=1}^{N(T)}X_{i}$, where
$X_{1},X_{2},...$ are i.i.d random variables distributed according to $\nu$
and independent of $N(T)$, a Poisson distributed random variable with
parameter $T$.

One can compare this to the random variable

$G_{n}:=\sum_{i=1}^{n}X_{i},$ (1.8)

where the number of points is deterministically given by $n$. By an extension
of the classical Berry-Esseen theorem given by Petrov in [Pet75, Theorem 6,
p.115], one has the following bound on the Kolmogorov distance between the
laws of $(n\sigma^{2})^{-1/2}G_{n}$ and a standard Gaussian $N$:

$d_{K}\left(\frac{G_{n}}{\sqrt{n}\sigma},N\right)\leqslant\frac{c}{\sigma^{2+\epsilon}}n^{-\epsilon/2}.$
(1.9)

For the functional $F_{T}$, our Theorem 3.4 implies

$\displaystyle
d_{W}\left(\frac{F_{T}}{\sqrt{T}\sigma},N\right)\leqslant\frac{2c}{\sigma^{2+\epsilon}}T^{-\epsilon/2}$
(1.10) and $\displaystyle
d_{K}\left(\frac{F_{T}}{\sqrt{T}\sigma},N\right)\leqslant\frac{(4c)^{\frac{1}{1+\epsilon/2}}}{\sigma^{2}}T^{-\frac{\epsilon/2}{1+\epsilon/2}}$
(1.11)

for the Wasserstein and Kolmogorov distances respectively. The speed of
convergence in Wasserstein distance corresponds exactly to the one given by
Petrov. For the Kolmogorov distance we find a slightly slower speed, which is
however still converging much faster than the square root of the Wasserstein
distance, which is implied by the classic estimate $d_{K}(.,N)\leqslant
2\sqrt{d_{W}(.,N)}$ (see e.g. [NP12, Remark C.2.2]).

In Section 5, we apply Theorem 3.4 to edge-statistics of the form (1.1) of the
Online Nearest Neighbour Graph in the exponent range
$\alpha\in\left(0,\frac{d}{2}\right)$ and of the Gilbert graph for exponents
$\alpha\in\left(-\frac{d}{2},\infty\right)$ (extending existing results from
[RST17]); we also deal with the $k$-Nearest Neighbour Graph and the Radial
Spanning Tree for a general class of decreasing functions
$\phi:(0,\infty)\rightarrow(0,\infty)$ applied to the edge-lengths. In all our
applications, the speeds we find are the same in the Wasserstein and
Kolmogorov distances. Roughly speaking, a $2p$-moment bound leads to a speed
of convergence of $t^{d(1/p-1)}$, where $t^{d}$ is the order of the variance
and $p\in(1,2]$. If $p=2$, we recover the speed of order ‘square root of the
variance’, which is often presumed to be optimal and has in some contexts been
shown to be optimal. If however $p<2$, the resulting speed is slower.
Comparing with the above example and setting $2p=2+\epsilon$, it corresponds
to $t^{-d\frac{\epsilon/2}{1+\epsilon/2}}$ in both Wasserstein and Kolmogorov
distances. Whether or not this speed is optimal remains an open question.

  2. 2.

The case of an edge functional of the type (1.1) with $\alpha=\frac{d}{2}$ for
the ONNG is of a different nature. The variance contains an additional
logarithmic factor (as conjectured and partially shown in [Wad09], and fully
established in Theorem 5.3) and a $2+\epsilon$ moment bound proves too strong
a condition. To deal with this particular case, we develop in Theorem 3.3 an
estimate of the Wasserstein distance that depends on a time parameter. Taking
$\eta$ to be a Poisson measure on a space $\mathbb{X}\times[0,1]$ with
intensity $\lambda\otimes ds$, the estimate contains moments of
$\mathbb{E}\big{[}\operatorname{D}_{(x,s)}F\big{|}\eta_{|\mathbb{X}\times[0,s)}\big{]}$
and
$\mathbb{E}\big{[}\operatorname{D}^{(2)}_{(x,s),(y,u)}F\big{|}\eta_{|\mathbb{X}\times[0,s\vee
u)}\big{]}$ instead of $\operatorname{D}_{(x,s)}F$ and
$\operatorname{D}^{(2)}_{(x,s),(y,u)}F$. This distinction is crucial and leads
to a quantitative central limit theorem in the critical case
$\alpha=\frac{d}{2}$, stated in Theorem 5.3.

The underlying result making these improved estimates possible is a new
inequality, stated in Theorem 4.2, providing a bound on the $p$th moment of a
Skorohod integral $\delta(h)$ (see Section 2 for a definition), with
$p\in[1,2]$. Even more generally, a bound is provided for quantities of the
type $|\mathbb{E}\phi(\delta(h))|$, where
$\phi:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable with $(p-1)$-Hölder
continuous derivative for some $p\in(1,2]$. To the best of our knowledge, no
comparable inequality exists to date and this result is of independent
interest. In particular, Theorem 4.2 implies generalisations of the classical
Poincaré inequality according to which for a Poisson functional $F\in
L^{2}(\mathbb{P}_{\eta})$, it holds that

$\mathbb{E}\left[F^{2}\right]-\mathbb{E}[F]^{2}\leqslant\mathbb{E}\int\left(\operatorname{D}_{x}F\right)^{2}\lambda(dx).$
(1.12)

As shown in Corollary 4.3 and Remark 4.4, the Poincaré inequality remains true
(up to a multiplying constant) if the exponent $2$ is replaced by $p$, thus
for $p\in[1,2]$:

$\mathbb{E}|F|^{p}-|\mathbb{E}F|^{p}\leqslant
2^{2-p}\mathbb{E}\int_{\mathbb{X}}\left|\operatorname{D}_{x}F\right|^{p}\lambda(dx).$
(1.13)

If the functional $F$ is non-negative or centred, this inequality follows from
modified log-Sobolev type inequalities shown in [Cha04] and also [APS22]. We
stress, however, that the bound in Theorem 4.2 is much more general and not
directly deducible from [Cha04, APS22].

If we work over the time-augmented space $\mathbb{X}\times[0,1]$, it has been
shown by Last and Penrose in [LP11b, Thm. 1.5] that

$\operatorname{Var}(F)=\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}\mathbb{E}[\operatorname{D}_{(x,t)}F|\eta_{|\mathbb{X}\times[0,t)}]^{2}\lambda(dx)dt.$
(1.14)

Correspondingly, the inequality in Corollary 4.3 can be refined even further
by conditioning on the information up to time $t$:

$\mathbb{E}|F|^{p}-|\mathbb{E}F|^{p}\leqslant
2^{2-p}\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}\left|\mathbb{E}[\operatorname{D}_{(x,t)}F|\eta_{|\mathbb{X}\times[0,t)}]\right|^{p}\lambda(dx)dt.$
(1.15)

This inequality is of great importance for improving estimates of both
Wasserstein and Kolmogorov distances. Another consequence of Theorem 4.2 is a
more technical inequality given in Corollary 4.7, this one being crucial when
refining the bound on the Kolmogorov distance.

The proof of Theorem 4.2 relies on a new version of Itô formula, shown in
Theorem 4.1. In contrast to the classical Itô formula for Poisson point
processes as given in [IW81, Theorem II.5.1], our version does not assume the
process to be a semi-martingale or the integrand to be predictable. In turn,
we only use the term corresponding to the integral with respect to a
compensated Poisson measure. A detailed discussion of differences and
similarities with the classical Itô formula and comparable results in the
literature is provided in Section 4.1. We believe this result also to be of
independent interest, as to the best of our knowledge no such formula for
anticipative integrands and general Poisson processes exists in the
literature.

######  Remark 1.1.

As demonstrated in the forthcoming Section 5, the principal achievement of
this paper is the derivation of probabilistic bounds requiring minimal moment
assumptions, that one can directly apply to a variety of models without
implementing truncation or smoothing procedures. That said, it is plausible to
expect that alternate bounds to some of those derived in Section 5 could be
derived by combining the results of [LPS16] with a truncation procedure
similar to the ones implemented e.g. in [Wu00, proof of Theorem 1.1] or
[NPY20, proof of Corollary 3.2]. In order to keep the length of this paper
within reasonable limits, the comparison between the two approaches (in
situations where both apply) will be discussed elsewhere.

Plan of the paper: In Section 2 we provide a short background on Poisson
processes and Malliavin Calculus, with a more detailed account to be found in
Appendix A. In Section 3, we discuss second order $p$-Poincaré inequalities,
while Section 4 contains our version of Itô formula and the new estimates for
Skorohod integrals. Applications are discussed in Section 5, in particular the
ONNG is dealt with in Section 5.1. All proofs can be found in the appendices:
in Appendix B we present the proof of Itô formula (Theorem 4.1), Appendix C
contains the proofs for Theorem 4.2 and its corollaries, and the second order
Poincaré inequalities (Theorems 3.2, 3.3 and 3.4) are shown in Appendix D. The
ONNG is discussed in Section E and the proofs for the Gilbert graph, the
$k$-Nearest Neighbour graphs and the Radial Spanning Tree can be found in
Appendices F, G and H respectively.

Acknowledgment. I would like to thank my supervisor Giovanni Peccati for
extensive discussions and his invaluable help on this project. I would also
like to thank Pierre Perruchaud for his contribution to the computation of the
constant discussed in Lemmas E.24 and E.25. I am grateful to Günter Last,
Matthias Schulte and Mark Podolskij for useful discussions and comments.

##  2 Framework and notations

We provide here an overview of the most relevant (in the context of this
article) properties of Poisson point processes and elements of Poisson
Malliavin calculus. Further definitions and properties that will be necessary
for the proofs can be found in Appendix A. We refer the reader to [Las16,
LP18] for an exhaustive discussion of the material presented below.

Poisson random measure. Let $(\mathbb{W},\mathcal{W},\nu)$ be a
$\sigma$-finite measure space and let $\mathbf{N}_{\mathbb{W}}$ be the set of
$\mathbb{N}_{0}\cup\\{\infty\\}$-valued measures on
$(\mathbb{W},\mathcal{W})$. Define the $\sigma$-algebra
$\mathcal{N}_{\mathbb{W}}$ on $\mathbf{N}_{\mathbb{W}}$ as the smallest
$\sigma$-algebra such that $\forall\,W\in\mathcal{W}$, the map
$\mathbf{N}_{\mathbb{W}}\ni\xi\mapsto\xi(W)\in\mathbb{N}\cup\\{\infty\\}$ is
measurable. If it is clear from context which space we refer to, we will write
$\mathbf{N}$ and $\mathcal{N}$ instead of $\mathbf{N}_{\mathbb{W}}$ and
$\mathcal{N}_{\mathbb{W}}$.

A Poisson random measure with intensity $\nu$ is a
$(\mathbf{N},\mathcal{N})$-valued random element $\chi$ defined on some
probability space $(\Omega,\mathcal{F},\mathbb{P})$ such that

  * •

for all $W\in\mathcal{W}$ and all $k\in\mathbb{N}_{0}$, we have
$\mathbb{P}(\chi(W)=k)=\exp(-\nu(W))\frac{\nu(W)^{k}}{k!}$ (with the
convention that $\chi(W)=\infty\ \mathbb{P}$-a.s. if $\nu(W)=\infty$);

  * •

for $W_{1},...,W_{n}\in\mathcal{W}$ disjoint, the random variables
$\chi(W_{1}),...,\chi(W_{n})$ are mutually independent.

Existence and uniqueness of such a measure is shown in [LP18, Chapter 3]. We
denote by $\mathbb{P}_{\chi}$ the law of $\chi$ in $(\mathbf{N},\mathcal{N})$
and we say that $\chi$ is a $(\mathbb{W},\nu)$-Poisson measure.

In view of the $\sigma$-finiteness of $(\mathbb{W},\nu)$ and using [LP18,
Corollary 6.5] we can and will assume throughout the paper that the Poisson
measure $\chi$ is proper, i.e. that there exist independent random elements
$W_{1},W_{2},...\in\mathbb{W}$ and an independent
$\mathbb{N}_{0}\cup\\{\infty\\}$-valued random variable $\kappa$ such that
$\mathbb{P}$-a.s.

$\chi=\sum_{n=1}^{\kappa}\delta_{W_{n}},$ (2.1)

where $\delta_{w}$ is the Dirac mass at the point $w\in\mathbb{W}$. All our
results only depend on the law of $\chi$, hence this assumption has no impact
on them. In this context, we will often identify $\chi$ with its support, i.e.
with the random collection of points $\\{W_{1},W_{2},...\\}$.

Poisson functionals. For $p\geqslant 0$, denote by $L^{p}(\mathbb{P}_{\chi})$
the set of random variables $F$ such that there is a measurable function
$f:\mathbf{N}\rightarrow\mathbb{R}$ such that $F=f(\chi)$ $\mathbb{P}$-a.s.
and, if $p>0$, such that $\mathbb{E}|F|^{p}<\infty$. We call $F$ a Poisson
functional and $f$ a representative of $F$. All results that follow do not
depend on the choice of the representative $f$ and hence, throughout the
article, we will use the symbol $F$ indiscriminately to represent both $f$ and
$F$.

Add-one cost and Malliavin derivative. For a Poisson functional $F\in
L^{0}(\mathbb{P}_{\chi})$ and $w\in\mathbb{W}$, define the add-one cost
operator of $F$ as

$\operatorname{D}_{w}F:=F(\chi+\delta_{w})-F(\chi),$ (2.2)

and inductively set
$\operatorname{D}^{(n)}_{w_{1},...,w_{n}}F:=\operatorname{D}_{w_{n}}\operatorname{D}^{(n-1)}_{w_{1},...,w_{n-1}}F$
for $n\geqslant 1$ and $w_{1},...,w_{n}\in\mathbb{W}$, where
$\operatorname{D}^{(0)}F=F$ and $\operatorname{D}^{(1)}F=\operatorname{D}F$.
It can be shown that $\operatorname{D}^{(n)}F$ is jointly measurable in all
variables and symmetric in $w_{1},...,w_{n}$ (cf. [Las16, p. 5]). We denote by
$\operatorname{dom}\operatorname{D}$ the set of all $F\in
L^{2}(\mathbb{P}_{\chi})$ such that

$\mathbb{E}\int_{\mathbb{W}}(\operatorname{D}_{w}F)^{2}\nu(dx)<\infty.$ (2.3)

The restriction of the operator $\operatorname{D}$ to
$\operatorname{dom}\operatorname{D}$ is called the Malliavin derivative of $F$
(see [Las16, Theorem 3]). Note that for $F\in L^{1}(\mathbb{P}_{\eta})$, the
LHS of (2.3) is well-defined and (2.3) is sufficient for $F$ to be in
$\operatorname{dom}\operatorname{D}$ (as follows from the
$L^{1}(\mathbb{P}_{\eta})$-Poincaré inequality as stated in [Las16, Cor. 1]).

For $F,G\in L^{0}(p_{\chi})$, we have the following formula for the add-one
cost of a product:

$\operatorname{D}(FG)=(\operatorname{D}F)G+F(\operatorname{D}G)+(\operatorname{D}F)(\operatorname{D}G).$
(2.4)

Chaotic decomposition. For a function $g\in L^{2}(\mathbb{W}^{n},\nu^{(n)})$,
denote by $\operatorname{I}_{n}(g)$ the $n$th Wiener-Itô integral of $g$. Then
for $F\in L^{2}(\mathbb{P}_{\chi})$, we have the Wiener-Itô chaos expansion

$F=\sum_{n=0}^{\infty}\operatorname{I}(f_{n}),$ (2.5)

where
$f_{n}(w_{1},...,w_{n})=\frac{1}{n!}\mathbb{E}\operatorname{D}^{(n)}_{w_{1},...,w_{n}}F$
and the series converges in $L^{2}(\mathbb{P}_{\chi})$ (cf. [Las16, Theorem
2]).

Mecke formula. Denote by $L^{p}(\mathbf{N}\times\mathbb{W})$ the quotient set
of all measurable functions
$h:\mathbf{N}\times\mathbb{W}\rightarrow\mathbb{R}$ such that, if $p>0$, one
has $\mathbb{E}\int_{W}|h(\chi,w)|^{p}\nu(dw)<\infty$.

Next, we introduce the so-called Mecke formula (cf. [Las16, (7)]), which holds
for $h\in L^{1}(\mathbf{N}\times\mathbb{W})$ and for
$h:\mathbf{N}\times\mathbb{W}\rightarrow[0,\infty)$ measurable:

$\mathbb{E}\int_{\mathbb{W}}h(\chi,w)\chi(dw)=\mathbb{E}\int_{\mathbb{W}}h(\chi+\delta_{w},w)\nu(dw).$
(2.6)

In particular, combined with the fact that $\chi$ is assumed to be proper,
this implies that for a function $h\in L^{1}(\mathbf{N}\times\mathbb{W})$, the
integral

$\int_{\mathbb{W}}h(\chi-\delta_{w},w)\chi(dw)$ (2.7)

is well-defined.

Skorohod integrals. If $h\in L^{2}(\mathbf{N}\times\mathbb{W})$, then for
$\nu$-a.e. $w\in\mathbb{W}$, we have $h(.,w)\in L^{2}(\mathbb{P}_{\chi})$ and
thus we can write

$h(\chi,w)=\sum_{n=0}^{\infty}\operatorname{I}_{n}(h_{n}(w,.)),$ (2.8)

with
$h_{n}(w,w_{1},...,w_{n})=\frac{1}{n!}\mathbb{E}\operatorname{D}^{(n)}_{w_{1},...,w_{n}}h(\chi,w)$
(cf. [Las16, (42)]). We say that $h\in\operatorname{dom}\delta$ if

$\sum_{n=0}^{\infty}(n+1)!\int_{\mathbb{W}^{n+1}}\tilde{h}_{n}^{2}d\nu^{n+1}<\infty,$
(2.9)

where $\tilde{h}_{n}$ is the symmetrisation of $h_{n}$ given by

$\tilde{h}_{n}(w_{1},...,w_{n+1})=\frac{1}{n+1}\sum_{k=1}^{n+1}h_{n}(w_{k},w_{1},...,w_{k-1},w_{k+1},...,w_{n+1}).$
(2.10)

For $h\in\operatorname{dom}\delta$ we define the Skorohod integral of $h$ by

$\delta(h):=\sum_{n=0}^{\infty}\operatorname{I}_{n+1}(h_{n}),$ (2.11)

which converges in $L^{2}(\mathbb{P}_{\chi})$. Note that, by [Las16, Theorem
5], the following condition is sufficient for $h\in
L^{2}(\mathbf{N}\times\mathbb{W})$ to be in $\operatorname{dom}\delta$:

$\mathbb{E}\int_{\mathbb{W}}\int_{\mathbb{W}}\left(\operatorname{D}_{x}h(\chi,y)\right)^{2}\nu(dx)\nu(dy)<\infty.$
(2.12)

If $h\in L^{1}(\mathbf{N}\times\mathbb{W})\cap\operatorname{dom}\delta$, then
by [Las16, Theorem 6], we have $\mathbb{P}$-a.s.

$\delta(h)=\int_{\mathbb{W}}h(\chi-\delta_{w},w)\chi(dw)-\int_{\mathbb{W}}h(\chi,w)\nu(dw),$
(2.13)

where the RHS is well-defined for any $h\in L^{1}(\mathbf{N}\times\mathbb{W})$
by (2.7).

Extension to a marked space. It will often be convenient to endow the space
$\mathbb{W}$ with marks representing time. As we are only interested in the
law of the Poisson functionals in question, we can always suppose that the
$(\mathbb{W},\nu)$-Poisson measure $\chi$ is the marginal of a
$(\mathbb{W}\times[0,1],\nu\otimes ds)$-Poisson measure $\eta$. Indeed,
$\eta(.\times[0,1])$ has the same law on $\mathbf{N}_{\mathbb{W}}$ as $\chi$.
For a functional $F\in L^{0}(\mathbb{P}_{\chi})$, define

$G(\eta):=F(\eta(.\times[0,1])).$ (2.14)

Then $G(\eta)$ has the same law under $\mathbb{P}_{\eta}$ as $F(\chi)$ under
$\mathbb{P}_{\chi}$. Moreover, for any $(x,s)\in\mathbb{W}\times[0,1]$,

$\operatorname{D}_{(x,s)}G(\eta)=\operatorname{D}_{x}F(\eta(.\times[0,1])),$
(2.15)

which is equal in law to $\operatorname{D}_{x}F(\chi)$.

Predictability. We call a measurable function
$h:\mathbf{N}_{\mathbb{W}\times[0,1]}\times\mathbb{W}\times[0,1]\rightarrow\mathbb{R}$
predictable if for all $(y,s)\in\mathbb{W}\times[0,1]$ and all
$\nu\in\mathbf{N}_{\mathbb{W}\times[0,1]}$

$h(\nu,y,s)=h(\nu_{|\mathbb{W}\times[0,s)},y,s).$ (2.16)

This definition of predictability appears e.g. in [LP11a, (2.5)], where it is
argued that this version of predictability is comparable to predictability in
the classical sense (as defined e.g. in [IW81, Definition I.5.2]). It is also
shown in [LP11a, Proposition 2.4] that if $h\in
L^{2}(\mathbf{N}\times\mathbb{W}\times[0,1])$ satisfies (2.16), then
$h\in\operatorname{dom}\delta$.

Conditional expectations and Clark-Ocône formula. Let $\eta$ be a
$(\mathbb{W}\times[0,1],\nu\otimes ds)$-Poisson measure. Using that the
measures $\eta_{|\mathbb{W}\times[0,s)}$ and $\eta_{|\mathbb{W}\times[s,0]}$
are independent, one can define a version of conditional expectation for any
non-negative or integrable random variable $G\in L^{0}(\mathbb{P}_{\eta})$ by

$\mathbb{E}[G|\eta_{|\mathbb{W}\times[0,s)}]:=\int
G(\eta_{|\mathbb{W}\times[0,s)}+\xi)\Pi_{s}(d\xi),$ (2.17)

where $\Pi_{s}$ is the law of $\eta_{|\mathbb{W}\times[s,1]}$. If it is
finite, the conditional expectation
$\mathbb{E}[G|\eta_{|\mathbb{W}\times[0,s)}]$ is predictable. In particular,
for $F\in L^{2}(\mathbb{P}_{\eta})$ the quantity
$\mathbb{E}[\operatorname{D}_{(x,s)}F|\eta_{|\mathbb{W}\times[0,s)}]$ is well-
defined, finite and predictable and the following Clark-Ocône type formula is
shown in [LP11a, Theorem 2.1] (see also [Wu00, HP02]):

$F=\mathbb{E}F+\delta(\mathbb{E}[\operatorname{D}_{(x,s)}F|\eta_{|\mathbb{W}\times[0,s)}])\qquad\mathbb{P}_{\chi}-\text{a.s.}$
(2.18)

This formula will be essential in the proof of Corollary 4.3.

Generic sets. Let $\mu\subset\mathbb{R}^{d}$ be a finite set. We say that
$\mu$ is generic if all pairwise distances between points are distinct. We say
that a set $\mu\subset\mathbb{R}^{d}$ is generic with respect to points
$x,y\in\mathbb{R}^{d}$ if $x,y\notin\mu$ and $\mu\cup\\{x,y\\}$ is generic.
Note that for compact sets $H\subset\mathbb{R}^{d}$, any $(H,dx)$-Poisson
measure $\chi$ can a.s. be identified with its support and this support is
a.s. generic. To simplify the presentation, we will at times adopt the
notation

$F(\mu):=F(\xi_{\mu}),\qquad\text{where }\xi_{\mu}=\sum_{x\in\mu}\delta_{x}$
(2.19)

for a finite set $\mu\in\mathbb{R}^{d}$ and a measurable functional
$F:\mathbf{N_{\mathbb{R}^{d}}}\rightarrow\mathbb{R}$. Similar notation will be
used for $\operatorname{D}F(\mu)$, $\operatorname{D}^{(2)}F(\mu)$ etc.

###  2.1 Notation

For $x\in\mathbb{R}^{d}$ and $r>0$, we write $B^{d}(x,r)$ to indicate the
(open) ball of centre $x$ and radius $r$. For a measurable set
$A\subset\mathbb{R}^{d}$, we denote by $|A|$ the Lebesgue measure of $A$,
unless $A$ is finite, in which case $|A|$ denotes the number of elements in
$A$. We use $\mkern 1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu$ to
denote the closure of $A$. Throughout this paper, $\kappa_{d}=|B^{d}(0,1)|$.
We use the symbols $\wedge$ (resp. $\vee$) to denote a minimum (resp. maximum)
of two elements. We shall use LHS and RHS to denote ‘left hand side’ and
‘right hand side’ and use $|x|$ to denote the Euclidean norm of
$x\in\mathbb{R}^{d}$. The supremum norm of a function
$f:\mathbb{R}\rightarrow\mathbb{R}$ is denoted by $\lVert f\rVert_{\infty}$.
By $\stackrel{{\scriptstyle d}}{{=}}$ and $\stackrel{{\scriptstyle
d}}{{\longrightarrow}}$ we mean equality and convergence in distribution
respectively. We use the symbol $\simeq$ (resp. $\lesssim$) if there is
equality (resp. inequality) up to multiplication by a positive constant.

##  3 Second order $p$-Poincaré inequalities in Wasserstein and Kolmogorov
distances

In this section we state our new bounds on the distance between the
distribution of a Poisson functional and the Normal law. These bounds are
called second order $p$-Poincaré inequalities, following a nomenclature coined
in [Cha09], where bounds of this type were given for the first time in a
Gaussian context. We make use of the well-established Malliavin-Stein method,
which was pioneered in [NP09] in the Wiener case, used for the first time in
the Poisson case in [PSTU10] and subsequently extended and developed in a wide
range of articles – see the references given in [LPS16], the survey [APY18],
the monograph [PR16] and the website [Mal]. Related bounds in the Kolmogorov
distance have been studied in various places [ET14, Sch16, LPS16, LRPY20].

Recall that for an integrable random variable $F$ and a standard Gaussian $N$,
the Wasserstein distance between the distributions of $F$ and $N$ is given by

$d_{W}(F,N)=\sup_{h\in\mathcal{H}}|\mathbb{E}h(F)-\mathbb{E}h(N)|$ (3.1)

where $\mathcal{H}$ is the set of Lipschitz-continuous functions
$h:\mathbb{R}\rightarrow\mathbb{R}$ with Lipschitz constant $\lVert
h\rVert_{L}\leqslant 1$. On the other hand, the Kolmogorov distance between
the distributions of $F$ and $N$ is defined as

$d_{K}(F,N)=\sup_{z\in\mathbb{R}}|\mathbb{P}(F\leqslant
z)-\mathbb{P}(N\leqslant z)|.$ (3.2)

See e.g. [NP12, Appendix C], and the references therein, for a discussion of
the basic properties of $d_{W}$ and $d_{K}$.

######  Remark 3.1.

For the rest of this section, we fix a $\sigma$-finite measure space
$(\mathbb{X},\mathcal{X},\lambda)$. Before we state our main theorems, we
introduce some simplified notation to improve legibility of the following
results. Write $\mathbb{Y}:=\mathbb{X}\times[0,1]$ and
$\bar{\lambda}=\lambda\otimes dt$. We introduce a total order on $\mathbb{Y}$
by saying that $x<y$ if $x=(z,s)$, $y=(w,u)$ and $s<u$. In the following,
$\eta$ will be a $(\mathbb{Y},\bar{\lambda})$-Poisson measure and $\chi$ will
be a $(\mathbb{X},\lambda)$-Poisson measure. We will write $\eta_{x}$ for
$\eta_{|\mathbb{X}\times[0,s)}$. Integrals with respect to $\bar{\lambda}$ are
taken over $\mathbb{Y}$ and integrals with respect to $\lambda$ are taken over
$\mathbb{X}$.

The next statement contains the general abstract bounds on which our analysis
will rely.

######  Theorem 3.2.

Let $F\in L^{2}(\mathbb{P}_{\eta})\cap\operatorname{dom}\operatorname{D}$ such
that $\mathbb{E}F=0$ and $\mathbb{E}F^{2}=1$. Then for any $q\in[1,2]$

$\displaystyle
d_{W}(F,N)\leqslant\sqrt{\frac{2}{\pi}}\,\mathbb{E}\left|1-\int_{\mathbb{Y}}\operatorname{D}_{y}F\,\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\bar{\lambda}(dy)\right|+2\,\mathbb{E}\int_{\mathbb{Y}}\left|\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\right|\cdot|\operatorname{D}_{y}F|^{q}\bar{\lambda}(dy)$
(3.3) and
$\displaystyle\begin{multlined}d_{K}(F,N)\leqslant\mathbb{E}\left|1-\int_{\mathbb{Y}}\operatorname{D}_{y}F\,\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\bar{\lambda}(dy)\right|\\\
+\sup_{z\in\mathbb{R}}\mathbb{E}\int_{\mathbb{Y}}\left|\mathbb{E}\big{[}\operatorname{D}_{y}F\big{|}\eta_{y}\big{]}\right|\operatorname{D}_{y}F\cdot\operatorname{D}_{y}(Ff_{z}(F)+\mathbbm{1}_{\\{F>z\\}})\bar{\lambda}(dy).\end{multlined}d_{K}(F,N)\leqslant\mathbb{E}\left|1-\int_{\mathbb{Y}}\operatorname{D}_{y}F\,\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\bar{\lambda}(dy)\right|\\\
+\sup_{z\in\mathbb{R}}\mathbb{E}\int_{\mathbb{Y}}\left|\mathbb{E}\big{[}\operatorname{D}_{y}F\big{|}\eta_{y}\big{]}\right|\operatorname{D}_{y}F\cdot\operatorname{D}_{y}(Ff_{z}(F)+\mathbbm{1}_{\\{F>z\\}})\bar{\lambda}(dy).$
(3.6)

As a next step, we derive the upper bounds we use in applications. Define the
following quantities:

$\displaystyle\beta_{1}:=\frac{2^{2/p}\sqrt{2}}{\sqrt{\pi}}\sigma^{-2}\left(\int_{\mathbb{Y}}\left(\int_{\mathbb{Y}}\mathbb{E}\left[\mathbb{E}\big{[}|\operatorname{D}_{y}F|\big{|}\eta_{y}\big{]}^{2p}\right]^{\frac{1}{2p}}\cdot\mathbb{E}\left[\mathbb{E}\big{[}|\operatorname{D}^{(2)}_{x,y}F|\big{|}\eta_{x\vee
y}\big{]}^{2p}\right]^{\frac{1}{2p}}\bar{\lambda}(dy)\right)^{p}\bar{\lambda}(dx)\right)^{1/p}$
$\displaystyle\beta_{2}:=\frac{2^{2/p}}{\sqrt{2\pi}}\sigma^{-2}\left(\int_{\mathbb{Y}}\left(\int_{\mathbb{Y}}\mathbbm{1}_{\\{x<y\\}}\mathbb{E}\left[\mathbb{E}\big{[}|\operatorname{D}^{(2)}_{x,y}F|\big{|}\eta_{y}\big{]}^{2p}\right]^{1/p}\bar{\lambda}(dy)\right)^{p}\bar{\lambda}(dx)\right)^{1/p}$
$\displaystyle\beta_{3}:=2\sigma^{-(q+1)}\mathbb{E}\int_{\mathbb{Y}}\big{|}\mathbb{E}\left[\left.\operatorname{D}_{y}F\right|\eta_{y}\right]\big{|}^{q+1}\bar{\lambda}(dy),$
$\displaystyle\beta_{4}:=2^{3-q}\sigma^{-(q+1)}\int_{\mathbb{Y}}\int_{\mathbb{Y}}\mathbbm{1}_{\\{y\leqslant
x\\}}\mathbb{E}\left[\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]^{2}\right]^{1/2}\cdot\mathbb{E}\left[\big{|}\mathbb{E}[\operatorname{D}^{(2)}_{x,y}F|\eta_{x}]\big{|}^{2q}\right]^{1/2}\bar{\lambda}(dx)\bar{\lambda}(dy)$

The following statement is our first bound on Wasserstein distances, expressed
in terms of moments of the first and second order add-one costs conditional on
past behaviour.

######  Theorem 3.3.

Let $\eta$ be a $(\mathbb{Y},\bar{\lambda})$-Poisson-measure and let $F\in
L^{2}(\mathbb{P}_{\eta})\cap\operatorname{dom}\operatorname{D}$. Define
$\sigma:=\sqrt{\operatorname{Var}(F)}$ and
$\hat{F}:=(F-\mathbb{E}F)\sigma^{-1}$. Let $p,q\in(1,2]$. Then

$d_{W}(\hat{F},N)\leqslant\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4}.$ (3.7)

The proof can be found in Appendix D.

Now define

$\displaystyle\gamma_{1}:=\frac{2^{2/p}\sqrt{2}}{\sqrt{\pi}}\sigma^{-2}\left(\int_{\mathbb{X}}\left(\int_{\mathbb{X}}\mathbb{E}\left[|\operatorname{D}_{y}F|^{2p}\right]^{\frac{1}{2p}}\cdot\mathbb{E}\left[|\operatorname{D}^{(2)}_{x,y}F|^{2p}\right]^{\frac{1}{2p}}\lambda(dy)\right)^{p}\lambda(dx)\right)^{1/p}$
$\displaystyle\gamma_{2}:=\frac{2^{2/p}}{\sqrt{2\pi}}\sigma^{-2}\left(\int_{\mathbb{X}}\left(\int_{\mathbb{X}}\mathbb{E}\left[|\operatorname{D}^{(2)}_{x,y}F|^{2p}\right]^{1/p}\lambda(dy)\right)^{p}\lambda(dx)\right)^{1/p}$
$\displaystyle\gamma_{3}:=2\sigma^{-(q+1)}\int_{\mathbb{X}}\mathbb{E}\left|\operatorname{D}_{y}F\right|^{q+1}\lambda(dy)$

and

$\displaystyle\gamma_{4}:=\sigma^{-2}\left(4\int_{\mathbb{X}}\mathbb{E}\left[|\operatorname{D}_{y}F|^{2p}\right]\lambda(dy)\right)^{1/p}$
$\displaystyle\gamma_{5}:=\sigma^{-2}\left(4p\int_{\mathbb{X}}\int_{\mathbb{X}}\mathbb{E}\left[|\operatorname{D}^{(2)}_{x,y}F|^{2p}\right]\lambda(dy)\lambda(dx)\right)^{1/p}$
$\displaystyle\gamma_{6}:=\sigma^{-2}\left(2^{2+p}p\int_{\mathbb{X}}\int_{\mathbb{X}}\mathbb{E}\left[|\operatorname{D}^{(2)}_{x,y}F|^{2p}\right]^{1/2}\cdot\mathbb{E}\left[|\operatorname{D}_{x}F|^{2p}\right]^{1/2}\lambda(dy)\lambda(dx)\right)^{1/p}$
$\displaystyle\gamma_{7}:=\sigma^{-2}\left(8p\int_{\mathbb{X}}\int_{\mathbb{X}}\left(\mathbb{E}|\operatorname{D}^{(2)}_{x,y}F|^{2p}\right)^{\frac{1}{2p}}\cdot\left(\mathbb{E}|\operatorname{D}_{x}F|^{2p}\right)^{\frac{1}{2p}}\cdot\left(\mathbb{E}\left|\operatorname{D}_{y}F\right|^{2p}\right)^{1-1/p}\lambda(dy)\lambda(dx)\right)^{1/p}.$

Note that the quantities $\beta_{1},...,\beta_{4}$ and
$\gamma_{1},...,\gamma_{7}$ only contain expressions related to
$\operatorname{D}F$ and $\operatorname{D}^{(2)}F$.

The next statement contains our main estimates on Wasserstein and Kolmogorov
distances, given in terms of moments of first and second order add-one costs
(without conditioning).

######  Theorem 3.4.

Let $\chi$ be a $(\mathbb{X},\lambda)$-Poisson measure and let $F\in
L^{2}(\mathbb{P}_{\chi})\cap\operatorname{dom}\operatorname{D}$. Define
$\sigma:=\sqrt{\operatorname{Var}F}$ and
$\hat{F}:=(F-\mathbb{E}F)\sigma^{-1}$. Then

$d_{W}(\hat{F},N)\leqslant\gamma_{1}+\gamma_{2}+\gamma_{3}$ (3.8)

and

$d_{K}(\hat{F},N)\leqslant\sqrt{\tfrac{\pi}{2}}\gamma_{1}+\sqrt{\tfrac{\pi}{2}}\gamma_{2}+\gamma_{4}+\gamma_{5}+\gamma_{6}+\gamma_{7}.$
(3.9)

######  Remark 3.5.

Using Hölder’s inequality, one can replace the term $\gamma_{7}$ by the
slightly larger but simpler bound

$\sigma^{-2}\left(\int_{\mathbb{X}}\int_{\mathbb{X}}\left(\mathbb{E}|\operatorname{D}^{(2)}_{x,y}F|^{2p}\right)^{\frac{1}{2p}}\cdot\left(\mathbb{E}|\operatorname{D}_{x}F|^{2p}\right)^{1-\frac{1}{2p}}\lambda(dx)\lambda(dy)\right)^{1/p}.$
(3.10)

We will use this bound in the proof of Theorem 5.8 in the context of the
Radial Spanning Tree.

######  Remark 3.6 (Discussion of literature).

Our results in this section are a substantial extension of [LPS16]. The bounds
given in [LPS16, Theorems 1.1 and 1.2] contain moments of first and second
order add-one costs with exponent $4$ (or even $4+\epsilon$, see [LPS16,
Proposition 1.4]). While this is a very powerful tool for showing
asymptotically Gaussian behaviour, a finite $4$th moment is too strong a
condition for some applications, most notably for the ONNG discussed in
Section 5.1. Our Theorem 3.4 reduces this condition to finite $2p$ moments,
where $p\in(1,2]$, while retaining similar bounds in the case $p=2$. In
particular, [LPS16, Theorem 6.1 and Proposition 1.4] follow from our Theorem
3.4. (See also [Tri19] for qualitative results requiring bounds on moments of
order $2p$ under weak stabilisation assumptions).

The proofs of Theorems 3.2, 3.3 and 3.4 follow in spirit the ideas from
[LPS16] and [LRPY20, Theorem 1.12] (for the Kolmogorov distance). However, we
work on a space $\mathbb{X}\times[0,1]$ extended by a time component and
systematically replace the operator $L^{-1}$ by the conditional expectation
$\mathbb{E}[\operatorname{D}_{(x,t)}.|\eta_{|\mathbb{X}\times[0,t)}]$ (see
[PT13] for a similar approach for Poisson measures on the real line).
Moreover, we apply the inequalities established in Section 4 to achieve the
improvement in the exponent. For the Wasserstein distance, we also use an
improvement due to [BOPT20] to obtain the terms
$\beta_{3},\beta_{4},\gamma_{3}$. For the Kolmogorov distance, our bound in
Theorem 3.4 makes use of an improvement implemented in [LRPY20], but we remove
a strong condition on $F$. The resulting bound is close in spirit to the one
given in [LPS16, Theorem 1.2], but with an improvement from $4$th moments to
$2p$th moments. Moreover, our bound does not need the term corresponding to
[LPS16, term $\gamma_{3}$, p. 670] and replaces the term corresponding to
[LPS16, term $\gamma_{4}$, p. 671] by a term depending only on the add-one
cost operators of $F$ instead of $\mathbb{E}F^{4}$.

In Theorem 3.3, we do not take moments of the first and second order add-one
costs of our functionals, but of their expectation conditional on ‘past
behaviour’. A bound of this type is new and the distinction is crucial to
solve the critical case of the ONNG (see Theorem 5.3). As of now, such a bound
is only available in the Wasserstein distance.

##  4 Ancillary results: new estimates for Skorohod integrals

###  4.1 A version of Itô formula

We start this section by giving a version of Itô formula for Poisson integrals
with anticipative integrands. This is a crucial ingredient for the proof of
the new estimates given in Theorem 4.2. In the following, we will take $\eta$
to be a
$(\mathbb{X}\times[0,1],\mathcal{X}\otimes\mathcal{B}([0,1]),\lambda(dx)\otimes
ds)$-Poisson measure, where $(\mathbb{X},\mathcal{X},\lambda)$ is a
$\sigma$-finite measure space.

######  Theorem 4.1 (Itô formula for non-adapted integrands).

Let $h\in L^{1}(\mathbf{N}\times\mathbb{X}\times[0,1])$ be bounded and let
$X_{0}\in\mathbb{R}$. For $t\in[0,1]$, define

$X_{t}(\eta):=X_{0}+\int_{\mathbb{X}\times[0,t]}h(\eta-\delta_{(y,s)},y,s)\eta(dy,ds)-\int_{\mathbb{X}}\int_{0}^{t}h(\eta,y,s)\lambda(dy)ds.$
(4.1)

Then the process $(X_{t})_{t\in[0,1]}$ is well-defined and $\mathbb{P}$-a.s.
càdlàg. Let $\phi\in\mathcal{C}^{1}(\mathbb{R})$. Then, $\forall\,t\in[0,1]$,

$\phi(X_{t})=\phi(X_{0})+\int_{\mathbb{X}\times[0,t]}\left(\phi\left(X_{s-}+h(\eta-\delta_{(y,s)},y,s)\right)-\phi(X_{s-})\right)\eta(dy,ds)\\\
-\int_{\mathbb{X}}\int_{0}^{t}\phi^{\prime}(X_{s})h(\eta,y,s)\lambda(dy)ds\qquad\mathbb{P}\text{-a.s.}$
(4.2)

and the quantities in (4.2) are well-defined.

In the next three items we compare our version of Itô formula with the
classical one given in [IW81, Theorem II.5.1].

  1. 1.

The main difference between (4.2) and [IW81, Thm. II.5.1] consists in the fact
that we do not assume the integrand $h$ to be predictable. There exist Itô
formulae for anticipative integrands in various settings, e.g. in the Wiener
case in [AN98] and [NP88]) and for pure jump and general Lévy processes in
[DNMBOP05] and [ALV08] respectively. To the best of our knowledge, our setting
of a general Poisson point process is new.

  2. 2.

Assume $h$ to be predictable in the sense of (2.16). It follows that
$h(\eta-\delta_{(y,s)},y,s)=h(\eta,y,s)$ for all
$(y,s)\in\mathbb{X}\times[0,1]$. For $\phi\in\mathcal{C}^{2}(\mathbb{R})$,
formula (4.2) is now roughly equivalent to the Itô formula given in [IW81,
Theorem II.5.1] in the special case where the semi-martingale in the statement
of [IW81, Theorem II.5.1] has the following properties:

     * •

the point process in question is a Poisson point process;

     * •

the only non-zero part is the one with respect to the compensated Poisson
measure;

     * •

the integrand $h$ is both in $L^{2}$ and in $L^{1}$.

  3. 3.

Our setting is thus both more general ($\phi\in\mathcal{C}^{1}(\mathbb{R})$
and $h$ anticipative) and more restrictive ($h\in L^{1}\cap L^{2}$ instead of
$h\in L^{2,\text{loc}}$ and the Gaussian, finite variation and non-compensated
Poisson terms are zero) than the one given by Ikeda and Watanabe. The proof of
our result relies however on the same ideas as the proof of [IW81, Theorem
II.5.1].

###  4.2 Moment Inequalities

In this section, we present a number of functional inequalities that are of
independent interest and also crucial to the improved bounds on Wasserstein
and Kolmogorov distances presented in earlier sections.

To the best of our knowledge, Theorem 4.2 is the first bound of its kind on
functionals of general Poisson-Skorohod integrals. Partial results are known
in the particular case where $h$ is predictable, see Corollary 4.3 and the
discussion thereafter. In particular, Theorem 4.2 below contains the first
general estimate in terms of add-one costs for $p$-moments of the Skorohod
integral, where $p\in[1,2]$, the cases $p=1$ and $p=2$ being the only ones
known. See also [LMS22].

In the special case $\phi(x)=x^{2}$, the theorem below follows immediately
from the isometry relation reported in formula (A.4) of Appendix A.

######  Theorem 4.2.

Let $h\in L^{2}(\mathbf{N}\times\mathbb{X}\times[0,1])$ satisfy (2.12). Let
$\phi:\mathbb{R}\rightarrow\mathbb{R}$ be a differentiable function with
$(p-1)$-Hölder continuous derivative, for some $p\in(1,2]$ and assume that
$\phi(0)=0$. Then

$\displaystyle\left|\mathbb{E}\phi(\delta(h))\right|$
$\displaystyle\leqslant\frac{c_{\phi}}{p}\mathbb{E}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds\
|h(\eta,y,s)|^{p}$
$\displaystyle+c_{\phi}\mathbb{E}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds\int_{\mathbb{X}}\lambda(dx)\int_{0}^{s}dt\
\left|\operatorname{D}_{(y,s)}h(\eta,x,t)\right|\cdot\left|\operatorname{D}_{(x,t)}h(\eta,y,s)\right|^{p-1}$
$\displaystyle+2c_{\phi}\mathbb{E}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds\int_{\mathbb{X}}\lambda(dx)\int_{0}^{s}dt\
\left|\operatorname{D}_{(y,s)}h(\eta,x,t)\right|\cdot\left|h(\eta,y,s)\right|^{p-1},$
(4.3)

where $c_{\phi}$ is the Hölder constant of $\phi^{\prime}$. In particular,
this inequality holds with $\phi(x)=|x|^{p}$ and $c_{\phi}=p2^{2-p}$, for
$p\in[1,2]$.

Our first corollary is a version of the above inequality for predictable
functions $h$ and contains a generalisation of the classical Poincaré
inequality.

######  Corollary 4.3.

Let $h\in L^{2}(\mathbf{N}\times\mathbb{X}\times[0,1])$ be predictable in the
sense of (2.16). Let $\phi:\mathbb{R}\rightarrow\mathbb{R}$ be a
differentiable function with $(p-1)$-Hölder continuous derivative, for some
$p\in(1,2]$. Assume $\phi(0)=0$. Then

$|\mathbb{E}\phi(\delta(h))|\leqslant\frac{c_{\phi}}{p}\mathbb{E}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds|h(\eta,y,s)|^{p}.$
(4.4)

Moreover, for $F\in L^{2}(\mathbb{P}_{\eta})$ and $p\in[1,2]$,

$\mathbb{E}|F|^{p}-|\mathbb{E}F|^{p}\leqslant
2^{2-p}\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}\left|\mathbb{E}[\operatorname{D}_{(x,t)}F|\eta_{\mathbb{X}\times[0,t)}]\right|^{p}\lambda(dx)dt.$
(4.5)

######  Remark 4.4.

  1. 1.

We can extend inequality (4.5) to $F\in L^{1}(\mathbb{P}_{\eta})$ at the cost
of introducing an additional absolute value on the RHS:

$\mathbb{E}|F|^{p}-|\mathbb{E}F|^{p}\leqslant
2^{2-p}\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}\mathbb{E}[|\operatorname{D}_{(x,t)}F||\eta_{\mathbb{X}\times[0,t)}]^{p}\lambda(dx)dt.$
(4.6)

This can be seen easily by approximating $F$ by $F_{n}:=(F\wedge n)\vee(-n)$
and using monotone and dominated convergence.

  2. 2.

When removing the conditional expectation in (4.5) using Jensen’s inequality,
the inequality can be extended to functionals $G\in L^{1}(\mathbb{P}_{\chi})$,
where $\chi$ is a $(\mathbb{X},\lambda)$-Poisson measure without time
component. Indeed, as discussed in Section 2, the marginal
$\eta(.\times[0,1])$ has the same law as $\chi$, which means one can see $G$
as a functional on $\mathbf{N}_{\mathbb{X}\times[0,1]}$. We have then

$\mathbb{E}|G|^{p}-|\mathbb{E}G|^{p}\leqslant
2^{2-p}\mathbb{E}\int_{\mathbb{X}}\left|\operatorname{D}_{x}G\right|^{p}\lambda(dx).$
(4.7)

######  Remark 4.5 (Literature review).

The proof of Theorem 4.2 relies on a combination of the Clark-Ocône type
representation result (2.18) and the version of Itô formula given in Theorem
4.1. This method of combining a Clark-Ocône result with Itô formulae to deduce
functional inequalities has been applied before in various settings, e.g. in
[Wu00] and [Cha04], where it was used to deduce a modified log-Sobolev
inequality and $\Phi$-Sobolev inequalities respectively.

Inequalities (4.4), (4.5) and (4.7) can be seen as part of a larger family of
functional inequalities on the Poisson space. The first to mention is the
classical Poincaré inequality, given e.g. in [Las16, Theorem 10] (see also
[HPA95, Cor. 4.4] for a very early appearance of this inequality). Our
inequality extends the classical one, which is (4.7) in the case $p=2$.
Another well-known inequality is the modified log-Sobolev inequality shown in
[Wu00] (see also [AL00]). It is extended in [Cha04, (5.10)] to the so-called
$\Phi$-Sobolev inequalities, which in the case $\Phi(x)=x^{p}$, imply (4.7)
when $F\geqslant 0$ or $\mathbb{E}F=0$. Similarly, the Beckner type $p$
inequalities discussed in [APS22, Section 4.6] imply (4.7) when $F\geqslant 0$
or $\mathbb{E}F=0$, albeit with a worse constant. [Zhu10, Theorem 3.3.2] gives
a version of (4.4) for $p$-norms in martingale type $p$ Banach spaces.
Although we did not check the details, it is reasonable to assume that one can
deduce (4.4) in the case $\phi(x)=|x|^{p}$ from such a result when applied to
$\mathbb{R}^{d}$.

######  Remark 4.6 (Comparison with the Gaussian case and extensions when
$p\geqslant 2$).

luckluckluckluckluck

  1. 1.

Inequality (4.7) for $p<2$ does not hold for functionals of Gaussian random
measures, as can be seen by taking $G=W_{t}$ and letting $t\rightarrow 0$,
with $W$ a standard Brownian motion. This is in contrast with the classical
Poincaré inequality ($p=2$) which holds in both Gaussian and Poisson settings.

  2. 2.

Inequality (4.7) (and hence also (4.5)) is false in general for $p>2$. Indeed,
consider $(\mathbb{X},\lambda)=(\mathbb{R}^{d},dx)$ and $G=\chi(A)-\lambda(A)$
for some measurable $A\subset\mathbb{R}^{d}$. Then $\mathbb{E}G=0$,
$\mathbb{E}G^{2}=\lambda(A)$ and $\operatorname{D}_{x}G=\mathbbm{1}_{A}(x)$.
On the LHS we have
$\mathbb{E}|G|^{p}\geqslant(\mathbb{E}G^{2})^{p/2}=\lambda(A)^{p/2}$ by
Jensen’s inequality and on the RHS

$\int_{\mathbb{X}}\mathbb{E}|\operatorname{D}_{x}G|^{p}\lambda(dx)=\lambda(A).$
(4.8)

However, since $p>2$, we have $\lambda(A)^{p/2}\gg\lambda(A)$ for $\lambda(A)$
large enough. Hence the inequality fails for any multiplying constant.

  3. 3.

Moment estimates for $p\geqslant 2$ are given in [GST21, Theorem 4.1] and
[APS22, Proposition 4.20]. The RHSs of these inequalities involve related, but
different quantities.

The versatility of Theorem 4.2 can be appreciated when considering the
following corollary, which will be crucial in finding a bound on the
Kolmogorov distance.

######  Corollary 4.7.

Let $h\in L^{1}(\mathbf{N}\times\mathbb{X}\times[0,1])$ and $G\in
L^{0}(\mathbb{P}_{\eta})$ bounded by a constant $c_{G}>0$. Then for any
$p\in[1,2]$,

$\displaystyle\left|\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}h(\eta,x,s)\operatorname{D}_{(x,s)}G\lambda(dx)ds\right|$
$\displaystyle\leqslant
c_{G}\left(2^{2-p}\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}|h(\eta,x,s)|^{p}\lambda(dx)ds\right.$
$\displaystyle+p2^{2-p}\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}\int_{\mathbb{X}}\int_{0}^{1}|\operatorname{D}_{(x,s)}h(\eta,y,u)|^{p}\lambda(dx)ds\lambda(dy)du$
$\displaystyle\left.+p2^{3-p}\int_{\mathbb{X}}\int_{0}^{1}\int_{\mathbb{X}}\int_{0}^{1}\mathbbm{1}_{\\{s<u\\}}\mathbb{E}[|\operatorname{D}_{(y,u)}h(\eta,x,s)|^{p}]^{1/p}\mathbb{E}[|h(\eta,x,s)|^{p}]^{1-1/p}\lambda(dx)ds\lambda(dy)du\right)^{1/p}.$
(4.9)

######  Remark 4.8.

Provided that we upper bound the indicator in the third term on the RHS of
(4.9) by 1, this inequality can be extended to a space $\mathbb{X}$ without
time component.

##  5 Applications

In this section, we look at four types of graphs built on Poisson measures and
assess the speeds of convergence to Normality of $\alpha$-power-weighted edge-
lengths such as (1.1). As was found in previous work [LPS16, ST17], we find
for certain ranges of exponents $\alpha$ that the speed is given by
$t^{-d/2}$, which corresponds to the order of the square root of the variance.
This is the presumably optimal speed corresponding to the one in the classical
Berry-Esseen theorem (see e.g. [Pet75, Theorem 4, p.111]). Beyond a certain
threshold, we find a slower speed of convergence that depends on $\alpha$.
Generally speaking, a $2p$th moment integrability of the first and second
order add-one costs of the functionals leads to a speed of convergence of
$t^{-d(1-1/p)}$. Whether this speed is optimal or not is an open question.

###  5.1 Online Nearest Neighbour Graph

Let $\mu\subset\mathbb{R}^{d}\times[0,1]$ be a finite set such that the
projection of $\mu$ onto $\mathbb{R}^{d}$ is generic and does not contain any
multiplicities and the projections onto $[0,1]$ are distinct. The ONNG on
$\mu$ is an (undirected) graph in $\mathbb{R}^{d}$ constructed as follows:

  * •

Vertices are given by $\\{x\in\mathbb{R}^{d}:(x,s)\in\mu\\}$

  * •

Let $(x,s)\in\mu$. If $\mu\cap\left(\mathbb{R}^{d}\times[0,s)\right)$ is non-
empty, then the online nearest neighbour of $(x,s)$ is given by the point
$(z,u)\in\mu\cap\left(\mathbb{R}^{d}\times[0,s)\right)$ which minimises
$|x-z|$. In this case there is an edge from $x$ to $z$ and we denote this
event by $\\{(x,s)\rightarrow(z,u)\text{ in }\mu\\}$.

For a point $(x,s)\in\mu$, the coordinate $s$ can be seen as the arrival time
of the point $x\in\mathbb{R}^{d}$, or its mark. Any point $(x,s)\in\mu$ has
exactly one online nearest neighbour, except for the point in $\mu$ whose mark
is minimal, which has none. Even though the graph is undirected, we think of
edges going from a point to its nearest neighbour, as this simplifies the
discussion.

For $(x,s)\in\mu$, let

$e(x,s,\mu):=\begin{cases*}\inf\\{|x-z|:(z,u)\in\mu\cap\left(\mathbb{R}^{d}\times[0,s)\right)\\},&if
$\mu\cap\left(\mathbb{R}^{d}\times[0,s)\right)\neq\emptyset$\\\
0,&otherwise.\end{cases*}$ (5.1)

This is the length of the edge from $x$ to its online nearest neighbour if
there is one, and zero otherwise. Note that one can find a unique online
nearest neighbour in $\mu$ for any point $(x,s)\in\mathbb{R}^{d}\times[0,1]$
such that the time coordinate $s$ and the position $x$ do not occur in $\mu$.
For convenience, we shall extend the above definitions to any such
$(x,s)\in\mathbb{R}^{d}\times[0,1]$ and tacitly adopt the corresponding
notation.

We will be studying the sums of power-weighted edge-lengths defined as
follows: for $\alpha>0$, let

$F^{(\alpha)}(\mu):=\sum_{(x,s)\in\mu}e(x,s,\mu)^{\alpha}.$ (5.2)

Note that here we make use of the convention explained in Section 2 to
identify a set of points $\mu$ with the point measure whose support is given
by $\mu$.

Let $\eta$ be a Poisson measure on $\mathbb{R}^{d}\times[0,1]$ with Lebesgue
intensity. Let $H\subset\mathbb{R}^{d}$ be a convex body. For $t\geqslant 1$,
define

$F^{(\alpha)}_{t}:=F^{(\alpha)}(\eta_{|tH\times[0,1]}).$ (5.3)

Figure 1: Realisation of the Online Nearest Neighbour graph

The ONNG is a relatively simple model for networks growing in time. Already
mentioned in [Ste89], the Online Nearest Neighbour graph came to general
attention in [BBB+07], where is was presented as a simplified version of the
FKP model developed in [FKP02], used to model the internet graph. The name of
the graph was coined in [Pen05], where the martingale method is used to show
central limit theorems for stabilising random systems satisfying a 4th moment
condition. In particular, it is shown in the case of the ONNG:

######  Theorem 5.1 ([Pen05, Theorem 3.6]).

For $0\leqslant\alpha<\frac{d}{4}$, there is a constant $\sigma_{\alpha,d}>0$
such that as $t\rightarrow\infty$,

$t^{-d}\operatorname{Var}\left(F^{(\alpha)}_{t}\right)\longrightarrow\sigma_{\alpha,d}\quad\text{and}\quad\frac{F^{(\alpha)}_{t}-\mathbb{E}F^{(\alpha)}_{t}}{t^{d/2}}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{N}(0,\sigma_{\alpha,d}).$ (5.4)

A quantitative counterpart to this result is shown in [LRPY20]. Our Theorem
5.3 provides a speed of convergence that is faster than the one given in
[LRPY20].

In [Pen05, PW08, Wad09], results similar to Theorem 5.1 were conjectured to
hold for $\alpha\in\left[\frac{d}{4},\frac{d}{2}\right]$. In particular, part
of the Conjectures 2.1 and 2.2. in [Wad09] states

######  Conjecture 5.2 ([Wad09]).

For $\alpha\in\left[\left.\frac{d}{4},\frac{d}{2}\right)\right.$, there is a
constant $\sigma_{\alpha,d}>0$ such that (5.4) holds.

For $\alpha=\frac{d}{2}$, there is a constant $\sigma_{d}>0$ such that

$\log(t)^{-1}t^{-d}\operatorname{Var}\left(F^{(\alpha)}_{t}\right)\longrightarrow\sigma_{d}\quad\text{and}\quad\frac{F^{(\alpha)}_{t}-\mathbb{E}F^{(\alpha)}_{t}}{\log(t)^{1/2}t^{d/2}}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{N}(0,\sigma_{d}).$ (5.5)

Our forthcoming Theorem 5.3 confirms this conjecture by giving quantitative
central limit theorems for $\alpha\in\left(\left.0,\frac{d}{2}\right]\right.$
and upper and lower bounds for the variances that match the conjectured
orders. Upper bounds of the conjectured orders were already given in [Wad09,
Theorem 2.1] for the variances involved. They are shown for an ONNG built on
$n$ uniformly distributed random variables and the corresponding result for
the Poisson version follows by Poissonisation. For the sake of completeness,
we will give purely Poissonian proofs of the upper bounds, following however a
similar strategy as in [Wad09]. A law of large numbers is shown in [Wad07] and
the case $\alpha>\frac{d}{2}$ is discussed in [PW08] (especially for $d=1$)
and in [Wad09], where is is shown that a limit exists in this case, but is
non-Gaussian for $\alpha>d$. For more related results we refer to the survey
[PW09] (for results up to 2010) and to the paper [LM21].

######  Theorem 5.3.

For $0<\alpha<\frac{d}{2}$, and for every $1<p<\frac{d}{2\alpha}$ such that
$p\leqslant 2$, there is a constant $c_{1}>0$ such that for all $t\geqslant 1$
large enough

$\max\left\\{d_{W}\left(\frac{F^{(\alpha)}_{t}-\mathbb{E}F^{(\alpha)}_{t}}{\sqrt{\operatorname{Var}\big{(}F^{(\alpha)}_{t}\big{)}}},N\right),d_{K}\left(\frac{F^{(\alpha)}_{t}-\mathbb{E}F^{(\alpha)}_{t}}{\sqrt{\operatorname{Var}\big{(}F^{(\alpha)}_{t}\big{)}}},N\right)\right\\}\leqslant
c_{1}t^{-d\left(1-\frac{1}{p}\right)},$ (5.6)

where $N$ denotes a standard normal random variable. Moreover, there are
constants $c_{2},C_{2}>0$ such that for all $t\geqslant 1$ large enough

$c_{2}t^{d}<\operatorname{Var}(F^{(\alpha)}_{t})<C_{2}t^{d}.$ (5.7)

For $\alpha=\frac{d}{2}$, there is a constant $c_{3}>0$ such that for all
$t\geqslant 1$ large enough

$d_{W}\left(\frac{F^{(d/2)}_{t}-\mathbb{E}F^{(d/2)}_{t}}{\sqrt{\operatorname{Var}\big{(}F^{(d/2)}_{t}\big{)}}},N\right)\leqslant
c_{3}\log(t)^{-1}.$ (5.8)

Moreover, there are constants $c_{4},C_{4}>0$ such that for all $t\geqslant 1$
large enough

$c_{4}t^{d}\log(t^{d})<\operatorname{Var}(F^{(d/2)}_{t})<C_{4}t^{d}\log(t^{d}).$
(5.9)

The constants $c_{1},c_{2},C_{2},c_{3},c_{4},C_{4}$ may depend on $H,\
\alpha,\ d$ and $p$.

Note that, in the special case $0<\alpha<\frac{d}{4}$, we find a speed of
convergence of $t^{-d/2}$, which corresponds to the square root of the order
of the variance.

###  5.2 Gilbert Graph

For a finite set $\mu\subset\mathbb{R}^{d}$ and a real number $\epsilon>0$,
the Gilbert graph $G(\mu,\epsilon)$ has vertex set $\mu$ and an edge between
$x,y\in\mu$, $x\neq y$ if and only if $|x-y|<\epsilon$. To construct our
functional of interest, we consider

  * •

$W\subset\mathbb{R}^{d}$ a convex body;

  * •

for every $t>0$, we take $\eta^{t}$ a $(W,t\,dx)$-Poisson measure;

  * •

$(\epsilon_{t})_{t>0}$ a sequence of positive real numbers s.t.
$\epsilon_{t}\rightarrow 0$ as $t\rightarrow\infty$.

Then for $\alpha\in\mathbb{R}$, define

$L^{(\alpha)}_{t}:=\sum_{e\in
G(\eta^{t},\epsilon_{t})}|e|^{\alpha}=\frac{1}{2}\sum_{x,y\in\eta^{t},x\neq
y}\mathbbm{1}_{\\{|x-y|\leqslant\epsilon_{t}\\}}|x-y|^{\alpha},$ (5.10)

where $e$ denote the edges of the graph and $|e|$ their length.

Define

$\hat{L}^{(\alpha)}_{t}:=\frac{L^{(\alpha)}_{t}-\mathbb{E}L^{(\alpha)}_{t}}{\sqrt{\operatorname{Var}\big{(}L^{(\alpha)}_{t}\big{)}}}.$
(5.11)

Figure 2: Realisation of the Gilbert graph

######  Remark 5.4.

For the sake of continuity with the article [RST17], we use the convention
that the intensity $t\,dx$ of $\eta$ grows and the observation window $W$
stays constant. In the other applications presented in this article, we keep
the intensity constant and instead let the observation window grow as $tW$.
Note that one can pass from one setting to the other by a simple rescaling.
Indeed, consider $\tilde{\eta}$ a Poisson measure on $\mathbb{R}^{d}$ and
Lebesgue intensity and for $s\geqslant 1$, construct a Gilbert graph on
$\tilde{\eta}_{|sH}$ by connecting two points $x\neq y\in\tilde{\eta}_{|sH}$
if and only if $|x-y|<\tilde{\epsilon}_{s}$. For $\alpha\in\mathbb{R}$, let

$F_{s}^{(\alpha)}:=\frac{1}{2}\sum_{x,y\in\tilde{\eta}_{|sW},x\neq
y}\mathbbm{1}_{\\{|x-y|<\tilde{\epsilon}_{s}\\}}|x-y|^{\alpha}.$ (5.12)

Then $F^{(\alpha)}_{t}$ is equal in law to $s^{\alpha}L_{s^{d}}^{(\alpha)}$
with $\epsilon_{s^{d}}=s^{-1}\tilde{\epsilon}_{s}$. The central limit theorem
for $F^{(\alpha)}_{t}$ can be deduced from the one for $L^{(\alpha)}_{t}$.

The first mention of the Gilbert graph was by Gilbert in [Gil61], in dimension
$d=2$. It has been treated in many works under various names: geometric or
proximity graph, interval graph (when $d=1$) or disk graph (when $d=2$). The
book [Pen03] provides a vast background and literature review and we also
refer to [LRP13a, LRP13b] for central limit theorems of generalisations of the
Gilbert graph and [RS13] for a quantitative CLT on a sum of weighted edge-
lengths. For a comprehensive overview of the Gilbert graph in the context of
$U$-statistics, see [LRR16], especially Section 4.3. See also [McD03, Mü08,
HM09, BP14, DST16, GT20]. In [RST17], the authors give a complete picture of
the asymptotic behaviour of $\hat{L}^{(\alpha)}_{t}$ for
$\alpha\in\mathbb{R}$. In particular, they show that for
$\alpha>-\frac{d}{2}$, the quantity $\hat{L}^{(\alpha)}_{t}$ converges in
distribution to a standard Gaussian as $t\rightarrow\infty$, provided that
$t^{2}\epsilon_{t}^{2}\rightarrow\infty$. They also give a quantitative bound
on the speed of convergence in Kolmogorov distance in the case
$\alpha>-\frac{d}{4}$. As an application of our estimates, we recover this
speed of convergence below and extend to the case
$-\frac{d}{2}<\alpha\leqslant-\frac{d}{4}$. The authors of [RST17] show that
CLTs hold also for $-d<\alpha\leqslant-\frac{d}{2}$ with different rescalings;
however, establishing corresponding speeds of convergence in this range is
still an open problem.

######  Theorem 5.5.

Let $\alpha>-\frac{d}{2}$ and assume that
$t^{2}\epsilon_{t}^{d}\rightarrow\infty$ as $t\rightarrow\infty$. Then for
$t\geqslant 1$ large enough

  * •

if $\alpha>-\frac{d}{4}$, there is a constant $c_{1}>0$ such that

$\max\left\\{d_{W}\left(\hat{L}^{(\alpha)}_{t},N\right),d_{K}\left(\hat{L}^{(\alpha)}_{t},N\right)\right\\}\leqslant
c_{1}\left(t^{-1/2}\vee(t^{2}\epsilon_{t}^{d})^{-1/2}\right).$ (5.13)

  * •

if $-\frac{d}{2}<\alpha\leqslant-\frac{d}{4}$, then for any
$1<p<-\frac{d}{2\alpha}$, there is a constant $c_{2}>0$ such that

$\max\left\\{d_{W}\left(\hat{L}^{(\alpha)}_{t},N\right),d_{K}\left(\hat{L}^{(\alpha)}_{t},N\right)\right\\}\leqslant
c_{2}\left(t^{-1+1/p}\vee(t^{2}\epsilon_{t}^{d})^{-1+1/p}\right).$ (5.14)

######  Remark 5.6.

A careful inspection of the bounds applied to $\gamma_{3}$ in the proof of
Theorem 5.5 reveals that in the sparse regime ($t\epsilon_{t}^{d}\rightarrow
0$) when $-\frac{d}{2}<\alpha\leqslant-\frac{d}{4}$, a slightly improved rate
can be found for the Wasserstein distance. Indeed, for any
$1<p<-\frac{d}{2\alpha}$ and any $0<r<-\frac{d}{\alpha}-2$, there is a
constant $c>0$ such that

$d_{W}(\hat{L}^{(\alpha)}_{t},N)\leqslant
c\left(t^{-1+1/p}\vee(t^{2}\epsilon_{t}^{d})^{-r/2}\right).$ (5.15)

Since $\frac{r}{2}\in\left(0,-\frac{d}{2\alpha}-1\right)$ and
$1-\frac{1}{p}\in\left(0,1+\frac{2\alpha}{d}\right)$ and
$1+\frac{2\alpha}{d}<-\frac{d}{2\alpha}-1$, one can choose
$\frac{r}{2}>1-\frac{1}{p}$. This then gives a slightly faster convergence
rate. As an illustrating example, consider the case where
$\epsilon_{t}^{d}=t^{-\theta}$ with $1<\theta<2$. Then
$t\epsilon_{t}^{d}\rightarrow 0$ and $t^{2}\epsilon_{t}^{d}\rightarrow\infty$.
Theorem 5.5 provides the rate of convergence $t^{(-1+\frac{1}{p})(2-\theta)}$,
and by following this strategy it can be improved to $t^{-1+\frac{1}{p}}\vee
t^{-\frac{r}{2}(2-\theta)}$ with $\frac{r}{2}>1-\frac{1}{p}$.

###  5.3 $k$-Nearest Neighbour graphs

For a finite generic set $\mu\subset\mathbb{R}^{d}$ and a positive integer
$k\in\mathbb{N}$, the $k$-Nearest Neighbour graph has vertex set $\mu$ and an
edge between $x,y\in\mu$ if and only if $y$ is one of the $k$ nearest points
to $x$ or vice-versa.

For our functional of interest, consider the following framework:

  * •

$H\subset\mathbb{R}^{d}$ is a convex body;

  * •

$\eta$ is an $(\mathbb{R}^{d},dx)$-Poisson measure;

  * •

$\phi:(0,\infty)\rightarrow(0,\infty)$ is a decreasing function such that
there is an $r>2$ verifying

$\int_{0}^{1}\phi(s)^{r}s^{d-1}ds<\infty.$ (5.16)

For any finite generic set $\mu\subset\mathbb{R}^{d}$, define

$F(\mu):=\frac{1}{2}\sum_{x,y\in\mu,x\neq y}\mathbbm{1}_{\\{x\in
N(y,\mu)\text{ or }y\in N(x,\mu)\\}}\phi(|x-y|),$ (5.17)

where $N(x,\mu)$ is the set of $k$-nearest neighbours of $x$ in $\mu$. For
$t\geqslant 1$, define $F_{t}:=F(\eta_{|tH})$ and set
$\hat{F}_{t}:=(F_{t}-\mathbb{E}F_{t})\operatorname{Var}(F_{t})^{-1/2}$.

Figure 3: Realisation of a 6-Nearest Neighbour graph

The $k$-nearest neighbour graph is a model frequently used in e.g. social
sciences or geography, see [Wad07] for a discussion of applications.
Quantitative central limit theorems for edge-related quantities were shown in
[AB93, PY05] and subsequently improved in [LPS16]. For a discussion of the
literature, we refer to [LPS16].

In [LPS16], the authors give a quantitative central limit theorem for the sum
of power-weighted edge-lengths with powers $\alpha\geqslant 0$, at a speed of
convergence of $t^{-d/2}$. We complement this result by dealing with the case
$\alpha\in\left(-\frac{d}{2},0\right)$. Note that in this case, the CLT is new
even in its qualitative version. In the regime
$\alpha\in\left(-\frac{d}{4},0\right)$, we also recover the same, presumably
optimal, speed of convergence of $t^{-d/2}$ as in [LPS16], whereas in the case
$\alpha\in\left(\left.-\frac{d}{2},-\frac{d}{4}\right]\right.$, we find a
speed of convergence that decreases as $\alpha$ approaches $-\frac{d}{2}$. It
is natural to ask what happens when $\alpha\leqslant-\frac{d}{2}$. We consider
this a separate issue and leave it open for further research.

######  Theorem 5.7.

Under the conditions stated above, for any $p\in(1,2]$ such that
$p<\frac{r}{2}$, there is a constant $c>0$ such that, for $t\geqslant 1$,

$\max\left\\{d_{W}\big{(}\hat{F}_{t},N\big{)},d_{K}\big{(}\hat{F}_{t},N\big{)}\right\\}\leqslant
ct^{d(1/p-1)}.$ (5.18)

This inequality holds in particular for the function $\phi(x)=x^{-\alpha}$
with $0<\alpha<\frac{d}{2}$, for any $p\in(1,2]$ such that
$p<\frac{d}{2\alpha}$.

###  5.4 Radial Spanning Tree

Let $\mu\subset\mathbb{R}^{d}\setminus\\{0\\}$ be a finite set, generic with
respect to the point $0$. The radial spanning tree on $\mu$, in short
$RST(\mu)$, is constructed as follows:

  * •

The set of vertices is given by $\mu\cup\\{0\\}$;

  * •

for every $x\in\mu$, we add exactly one edge to the point
$z\in\mu\cup\\{0\\}\cap B^{d}(0,|x|)$ which minimises $|x-z|$. We call $z$ the
radial nearest neighbour of $x$ and say ‘$x$ connects to $z$’, denoted by
‘$x\rightarrow z$ in $\mu$’. We denote the length $|x-z|$ by $g(x,\mu)$.

In order to define our functional of interest, consider the following setting:

  * •

$H\subset\mathbb{R}^{d}$ a convex body such that $B^{d}(0,\epsilon)\subset H$
for some $\epsilon>0$;

  * •

$\eta$ is an $(\mathbb{R}^{d},dx)$-Poisson measure;

  * •

$\phi:(0,\infty)\rightarrow(0,\infty)$ is a decreasing function such that
there is an $r>2$ satisfying

$\int_{0}^{1}\phi(s)^{r}s^{d-1}ds<\infty.$ (5.19)

For any finite set $\mu\subset\mathbb{R}^{d}$ generic with respect to $0$,
define

$F(\mu):=\sum_{x\in\mu}\phi(g(x,\mu))$ (5.20)

and for $t\geqslant 1$, define $F_{t}:=F(\eta_{|tH})$. Set
$\hat{F}_{t}:=(F_{t}-\mathbb{E}F_{t})\operatorname{Var}(F_{t})^{-1/2}$.

Figure 4: Realisation of the Radial Spanning Tree

The radial spanning tree was developed in [BB07] as a model related to the
minimal directed spanning tree and to Poisson forests. The paper also
discusses various applications, most notably in communication networks.
Further work on the radial spanning tree has been done in [PW09, BCT13, ST17].
In [ST17], the authors give a quantitative central limit theorem for sums of
power-weighted edge-lengths of the radial spanning tree for powers
$\alpha\geqslant 0$. The framework is one where the intensity of the Poisson
measure increases while the observation window stays constant. After rescaling
to our framework of a constant intensity and a growing window, one obtains by
[ST17, Theorem 1.2] a speed of convergence of $t^{-d/2}$. We add quantitative
central limit theorems for $\alpha\in\left(-\frac{d}{2},0\right)$, recovering
the same speed of $t^{-d/2}$ for $\alpha\in\left(-\frac{d}{4},0\right)$. Note
that this CLT is new even in its qualitative version. As for the $k$-Nearest
Neighbour graph, the case $\alpha\leqslant-\frac{d}{2}$ will be the object of
further research.

######  Theorem 5.8.

Under the conditions stated above, for any $p\in(1,2]$ such that
$p<\frac{r}{2}$, there is a constant $c>0$ such that for $t\geqslant 1$,

$\max\left\\{d_{W}\big{(}\hat{F}_{t},N\big{)},d_{K}\big{(}\hat{F}_{t},N\big{)}\right\\}\leqslant
ct^{d(1/p-1)}.$ (5.21)

This inequality holds in particular for the function $\phi(x)=x^{-\alpha}$
with $0<\alpha<\frac{d}{2}$, for any $p\in(1,2]$ such that
$p<\frac{d}{2\alpha}$.

The rest of the paper is devoted to providing the proofs of the results in
Sections 3 to 5.

##  Appendix A Background on Malliavin Calculus

In this section, we present several useful notions related to Malliavin
calculus. Unless otherwise indicated, these results are explained in [Las16].
We work in the setting of Section 2: in particular, $\chi$ indicates a
$(\mathbb{W},\nu)$-Poisson measure.

We start with three useful isometry relations. Let $f\in
L^{2}(\mathbb{W}^{n},\nu^{(n)})$ and $g\in L^{2}(\mathbb{W}^{m},\nu^{(m)})$.
Then

$\mathbb{E}\operatorname{I}_{n}(f)\operatorname{I}_{m}(g)=\mathbbm{1}_{\\{m=n\\}}n!\int_{\mathbb{W}^{n}}\tilde{f}(x)\tilde{g}(x)\,\nu^{(n)}(dx),$
(A.1)

where $\tilde{f}$ and $\tilde{g}$ are the symmetrisations of $f$ and $g$
defined by

$\tilde{f}(x_{1},...,x_{n})=\frac{1}{n!}\sum_{\sigma\in\Sigma_{n}}f(x_{\sigma(1)},...,x_{\sigma(n)}),$
(A.2)

the set $\Sigma_{n}$ being the set of all permutations of $\\{1,...,n\\}$. See
[Las16, Lemma 4] and the remark thereafter on page 10 for a proof.

Relation (A.1) implies that for $F,G\in L^{2}(\mathbb{P}_{\chi})$ having an
expansion (2.5) with kernels $f_{n}$ and $g_{n}$ respectively,

$\mathbb{E}FG=\mathbb{E}F\mathbb{E}G+\sum_{n=1}^{\infty}n!\int_{\mathbb{W}^{n}}f_{n}g_{n}\,d\nu^{(n)}.$
(A.3)

By [Las16, Theorem 5], if $h\in L^{2}(\mathbf{N}\times\mathbb{W})$ satisfies
(2.12), then

$\mathbb{E}\delta(h)^{2}=\mathbb{E}\int_{\mathbb{W}}h(\chi,w)^{2}\nu(dw)+\mathbb{E}\int_{\mathbb{W}}\int_{\mathbb{W}}\operatorname{D}_{z}h(\chi,w)\operatorname{D}_{w}h(\chi,z)\,\nu(dz)\nu(dw).$
(A.4)

A well-known relation in Malliavin calculus is the so-called integration by
parts formula: for $F\in\operatorname{dom}\operatorname{D}$ and
$h\in\operatorname{dom}\delta$, we have
$\mathbb{E}\int_{\mathbb{W}}h(\chi,w)\operatorname{D}_{w}F\nu(dw)=\mathbb{E}[F\delta(h)]$
(cf. [Las16, Theorem 4]). The condition on $F$ is however suboptimal in our
context, which is why we need a version of integration by parts under slightly
different assumptions.

######  Lemma A.1.

Let $h\in\operatorname{dom}\delta\cap L^{1}(\mathbf{N}\times\mathbb{W})$ and
$F\in L^{0}(\mathbb{P}_{\chi})$ bounded. Then

$\mathbb{E}\int_{\mathbb{W}}h(\chi,w)\operatorname{D}_{w}F\nu(dw)=\mathbb{E}[F\delta(h)].$
(A.5)

###### Proof.

Since $h\in\operatorname{dom}\delta\cap L^{1}(\mathbf{N}\times\mathbb{W})$, it
is easy to check that the expectations appearing in the statement are well-
defined and finite. Note that

$\displaystyle\mathbb{E}\int_{\mathbb{X}}\operatorname{D}_{w}F\,h(\chi,w)\nu(dw)$
$\displaystyle=\mathbb{E}\int_{\mathbb{X}}(F(\chi+\delta_{w})-F(\chi))h(\chi,w)\nu(dw)$
$\displaystyle=\mathbb{E}\int_{\mathbb{X}}F(\chi+\delta_{w})h(\chi,w)\nu(dw)-\mathbb{E}\int_{\mathbb{X}}F(\chi)h(\chi,w)\nu(dw),$
(A.6)

where the last line is justified by the fact that $F$ is bounded and $h\in
L^{1}(\mathbf{N}\times\mathbb{W})$, so both integrals are well-defined. We now
apply Mecke formula (2.6) to deduce that (A.6) equals

$\mathbb{E}\int_{\mathbb{X}}F(\chi)h(\chi-\delta_{w},w)\chi(dw)-\mathbb{E}\int_{\mathbb{X}}F(\chi)h(\chi,w)\nu(dw)\\\
=\mathbb{E}F(\chi)\left(\int_{\mathbb{X}}h(\chi-\delta_{w},w)\chi(dw)-\int_{\mathbb{X}}h(\chi,w)\nu(dw)\right).$
(A.7)

Since $h\in\operatorname{dom}\delta\cap L^{1}(\mathbf{N}\times\mathbb{X})$,

$\int_{\mathbb{X}}h(\chi-\delta_{w},w)\chi(dw)-\int_{\mathbb{X}}h(\chi,w)\nu(dw)=\delta(h)\qquad\mathbb{P}\text{-a.s.}$
(A.8)

The result follows. ∎

Next, we introduce the Ornstein-Uhlenbeck operator $P_{\tau}$. For $F\in
L^{1}(\mathbb{P}_{\chi})$ and $\tau\in[0,1]$, we define

$P_{\tau}F=\int\mathbb{E}\big{[}F(\chi^{\tau}+\xi)\big{|}\chi\big{]}\Pi_{\tau}(d\xi),$
(A.9)

where $\chi^{\tau}$ is a $\tau$-thinning of $\chi$ (see [Las16, p. 24] and the
reference given therein) and $\Pi_{\tau}$ is the law of an independent Poisson
measure with intensity measure $(1-\tau)\nu$. It follows by Jensen’s
inequality that for all $p\geqslant 1$, one has

$\mathbb{E}|P_{\tau}F|^{p}\leqslant\mathbb{E}|F|^{p}.$ (A.10)

By [Las16, Lemma 6], for all $F\in L^{2}(\mathbb{P}_{\chi})$ and all
$\tau\in[0,1]$, for $\nu^{(n)}$-a.e. $w_{1},...,w_{n}\in\mathbb{W}$ it holds
$\mathbb{P}$-a.s. that

$\operatorname{D}^{(n)}_{w_{1},...,w_{n}}(P_{\tau}F)=\tau^{n}P_{\tau}\operatorname{D}^{(n)}_{w_{1},...,w_{n}}F.$
(A.11)

This implies that for $F\in L^{2}(\mathbb{P}_{\chi})$, the following expansion
holds (see also [Las16, (79)]):

$P_{\tau}F=\mathbb{E}F+\sum_{n=1}^{\infty}\tau^{n}\operatorname{I}_{n}(f_{n}).$
(A.12)

The following lemma summarises some useful approximation properties of the
Ornstein-Uhlenbeck operator.

######  Lemma A.2.

Let $h\in L^{2}(\mathbf{N}_{\mathbb{W}}\times\mathbb{W})$ and let
$\tau\in(0,1)$. Then $P_{\tau}h$ satisfies condition (2.12) and
$P_{\tau}h\rightarrow h$ in $L^{2}(\mathbf{N}_{\mathbb{W}}\times\mathbb{W})$
as $\tau\rightarrow 1$. Moreover, for $w,z\in\mathbb{W}$, and all $p\geqslant
1$,

$\displaystyle\mathbb{E}|P_{\tau}h(\chi,w)|^{p}\leqslant\mathbb{E}|h(\chi,w)|^{p}$
(A.13) and
$\displaystyle\mathbb{E}|\operatorname{D}_{z}P_{\tau}h(\chi,w)|^{p}\leqslant\mathbb{E}|\operatorname{D}_{z}h(\chi,w)|^{p}.$
(A.14)

Under the additional assumption that $h\in\operatorname{dom}\delta$, it holds
that $\delta(P_{\tau}h)\rightarrow\delta(h)$ in $L^{2}(\mathbb{P}_{\chi})$ as
$\tau\rightarrow 1$.

###### Proof.

By the isometry property (A.3) and the expansions (A.12) and (2.8), we infer
that

$\mathbb{E}\int_{\mathbb{W}}\int_{\mathbb{W}}\left(\operatorname{D}_{w}P_{\tau}h(\chi,z)\right)^{2}\nu(dw)\nu(dz)=\sum_{n=0}^{\infty}n\cdot
n!\,\tau^{2n}\lVert h_{n}\rVert_{n+1}^{2},$ (A.15)

where $\lVert.\rVert_{n}$ is the norm in $L^{2}(\mathbb{W}^{n},\nu^{(n)})$.
Now note that $\sup_{n\geqslant 1}n\tau^{2n}<\infty$ and

$\sum_{n=0}^{\infty}n!\,\lVert
h_{n}\rVert_{n+1}^{2}=\int_{\mathbb{W}}\mathbb{E}h(w)^{2}\nu(dw)<\infty,$
(A.16)

hence $P_{\tau}h$ satisfies (2.12). Similarly using the expansions, we deduce
that

$\mathbb{E}\int_{\mathbb{W}}\left(P_{\tau}h(\chi,w)-h(\chi,w)\right)^{2}\nu(dw)=\sum_{n=1}^{\infty}n!\,(\tau^{n}-1)^{2}\lVert
h_{n}\rVert_{n+1}^{2}.$ (A.17)

By dominated convergence, this expression tends to $0$ as $\tau\rightarrow 1$.
Properties (A.13) and (A.14) follow immediately from (A.10) and (A.11). For
the last point, note that

$\delta(P_{\tau}h)-\delta(h)=\sum_{n=0}^{\infty}\operatorname{I}_{n+1}((\tau^{n}-1)h_{n})$
(A.18)

and

$\mathbb{E}\left(\delta(P_{\tau}h)-\delta(h)\right)^{2}=\sum_{n=0}^{\infty}(n+1)!(1-\tau^{n})^{2}\lVert\tilde{h}_{n}\rVert_{n+1}^{2},$
(A.19)

which converges to $0$ as $\tau\rightarrow 1$ by dominated convergence since
$h\in\operatorname{dom}\delta$. ∎

The following lemma is used on several occasions:

######  Lemma A.3 ([LP11b, Theorem 1.5]).

Let $\eta$ be a $(\mathbb{W}\times[0,1],\nu\otimes ds)$-Poisson measure and
let $F,G\in L^{2}(\mathbb{P}_{\eta})$. Then

$\mathbb{E}\int_{\mathbb{W}}\int_{0}^{1}\mathbb{E}[\operatorname{D}_{(y,s)}F|\eta_{|\mathbb{W}\times[0,1]}]^{2}\lambda(dy)ds<\infty,$
(A.20)

and an analogous estimate holds for $G$. Moreover,

$\operatorname{Cov}(F,G)=\mathbb{E}\int_{\mathbb{W}}\int_{0}^{1}\mathbb{E}[\operatorname{D}_{(y,s)}F|\eta_{|\mathbb{W}\times[0,1]}]\mathbb{E}[\operatorname{D}_{(y,s)}G|\eta_{|\mathbb{W}\times[0,1]}]\lambda(dy)ds.$
(A.21)

##  Appendix B Proof of Theorem 4.1

Each summand on the RHS of (4.1) is well defined by virtue of Mecke formula
(2.6) and the discussion thereafter. By Mecke formula (2.6) it can be seen
that $h(\eta-\delta_{(y,s)},y,s)$ is almost surely integrable with respect to
the measure $\eta(dy,ds)$. By assumption, $h$ is also integrable with respect
to $\lambda(dy)ds$. It now follows by dominated convergence that the process
$(X_{t})$ is càdlàg.

As a next step, we show that the integrals on the RHS of (4.2) are well-
defined. For this, note that $(X_{t})_{t\in[0,1]}$ (and
$(X_{t-})_{t\in[0,1]}$) are a.s. bounded on $[0,1]$. Indeed,

$\displaystyle\mathbb{E}\sup_{t\in[0,1]}|X_{t}|$
$\displaystyle\leqslant\mathbb{E}\int_{\mathbb{X}\times[0,1]}|h(\eta-\delta_{(y,s)},y,s)|\eta(dy,ds)+\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}|h(\eta,y,s)|\lambda(dy)ds$
$\displaystyle=2\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}|h(\eta,y,s)|\lambda(dy)ds<\infty,$
(B.1)

where the second inequality follows by Mecke formula (2.6). Now write

$\int_{\mathbb{X}\times[0,t]}\left|\phi(X_{s-}+h(\eta-\delta_{(y,s)},y,s))-\phi(X_{s-})\right|\eta(dy,ds)\\\
\leqslant\int_{\mathbb{X}\times[0,1]}\left|\int_{0}^{1}\phi^{\prime}\left(X_{s-}+uh(\eta-\delta_{(y,s)},y,s)\right)h(\eta-\delta_{(y,s)},y,s)du\right|\eta(dy,ds).$
(B.2)

By the boundedness of $h$, we infer from (B.1) that
$X_{s-}+uh(\eta-\delta_{(y,s)},y,s)$ almost surely takes values in a compact
interval. Since the function $\phi^{\prime}$ is continuous, this entails

$\sup_{s,u\in[0,1],y\in\mathbb{X}}|\phi^{\prime}(X_{s-}+uh(\eta-\delta_{(y,s)},y,s))|<\infty\qquad\mathbb{P}\text{-a.s.}$
(B.3)

Hence the RHS of (B.2) is bounded by

$\left(\sup_{s,u\in[0,1],y\in\mathbb{X}}|\phi^{\prime}(X_{s-}+uh(\eta-\delta_{(y,s)},y,s))|\right)\int_{\mathbb{X}\times[0,1]}|h(\eta-\delta_{(y,s)},y,s)|\eta(dy,ds)<\infty\qquad\mathbb{P}\text{-a.s.}$
(B.4)

which implies that the first integral on the RHS of (4.2) is well-defined.
Similarly,

$\sup_{s\in[0,1]}|\phi^{\prime}(X_{s})|<\infty\qquad\mathbb{P}\text{-a.s.}$
(B.5)

and hence

$\int_{\mathbb{X}}\int_{0}^{1}|\phi^{\prime}(X_{s})h(\eta,x,s)|\lambda(dx)ds<\infty\qquad\mathbb{P}\text{-a.s.}$
(B.6)

This concludes the proof that all terms in (4.2) are well-defined.

To show (4.2), we start by showing it for an approximation of $X_{t}$. Let
$(U_{m})_{m\in\mathbb{N}}\subset\mathbb{X}$ s.t. $\bigcup_{m}U_{m}=\mathbb{X}$
and $\forall\,m\in\mathbb{N}$, $\lambda(U_{m})<\infty$ and $U_{m}\subset
U_{m+1}$. Define

$X_{t}^{(m)}(\eta):=X_{0}+\int_{U_{m}\times[0,t]}h(\eta-\delta_{(y,s)},y,s)\eta(dy,ds)-\int_{U_{m}}\int_{0}^{t}h(\eta,y,s)\lambda(dy)ds.$
(B.7)

Define the event $\Omega_{0}:=\\{\eta(U_{m}\times[0,1])<\infty,m\geqslant
1\\}$. Then $\mathbb{P}(\Omega_{0})=1$ and, since $\eta$ is proper,
$\forall\omega\in\Omega_{0}$ and all $m\geqslant 1$ there exists a finite
collection of points $(y_{1},s_{1}),...,(y_{n_{m}},s_{n_{m}})\in
U_{m}\times[0,1]$ (all depending on $\omega$) s.t.

$\eta_{|U_{m}\times[0,1]}=\sum_{i=1}^{n_{m}}\delta_{(y_{i},s_{i})}.$ (B.8)

W.l.o.g. we can assume that $0<s_{1}<s_{2}<...<s_{n_{m}}<1$ and set $s_{0}:=0$
and $s_{n_{m}+1}:=1$. Now the process $X^{(m)}$ can be written as

$X^{(m)}_{t}=\sum_{i=1}^{n_{m}}\mathbbm{1}_{\\{s_{i}\leqslant
t\\}}h(\eta-\delta_{(y_{i},s_{i})},y_{i},s_{i})-\int_{U_{m}}\int_{0}^{t}h(\eta,y,s)\lambda(dy)ds,$
(B.9)

and one has the telescopic sums:

$\displaystyle\phi\left(X^{(m)}_{t}\right)-\phi(X_{0})$
$\displaystyle=\sum_{i=1}^{n_{m}+1}\left(\phi\left(X^{(m)}_{s_{i}\wedge
t}\right)-\phi\left(X^{(m)}_{s_{i-1}\wedge t}\right)\right)$
$\displaystyle=\sum_{i=1}^{n_{m}}\left(\phi\left(X^{(m)}_{s_{i}\wedge
t}\right)-\phi\left(X^{(m)}_{(s_{i}-)\wedge
t}\right)\right)+\sum_{j=1}^{n_{m}+1}\left(\phi\left(X^{(m)}_{(s_{j}-)\wedge
t}\right)-\phi\left(X^{(m)}_{s_{j-1}\wedge t}\right)\right)$
$\displaystyle=I_{1}(t)+I_{2}(t),$ (B.10)

where

$X^{(m)}_{(s-)\wedge t}=\begin{cases*}X^{(m)}_{s-}&if $s\leqslant t$\\\
X^{(m)}_{t}&if $s>t$.\end{cases*}$ (B.11)

The sum $I_{1}(t)$ represents what is happening at jump times, whereas
$I_{2}(t)$ shows what happens in between jump times. The index $i=n_{m}+1$
does not appear in the sum $I_{1}(t)$ because $\mathbb{P}$-a.s. there is no
jump at time $t=1$.

We first study $I_{1}(t)$.

$\displaystyle I_{1}(t)$
$\displaystyle=\sum_{i=1}^{n_{m}}\left(\phi\left(X^{(m)}_{s_{i}\wedge
t}\right)-\phi\left(X^{(m)}_{(s_{i}-)\wedge t}\right)\right)$
$\displaystyle=\sum_{i=1}^{n_{m}}\mathbbm{1}_{\\{s_{i}\leqslant
t\\}}\left(\phi\left(X^{(m)}_{s_{i}-}+h(\eta-\delta_{(y_{i},s_{i})},y_{i},s_{i})\right)-\phi\left(X^{(m)}_{s_{i}-}\right)\right)$
$\displaystyle=\int_{U_{m}\times[0,t]}\left(\phi\left(X^{(m)}_{s-}+h(\eta-\delta_{(y,s)},y,s)\right)-\phi\left(X^{(m)}_{s-}\right)\right)\eta(dy,ds).$
(B.12)

Now consider $I_{2}(t)$. For $s\in[s_{i-1},s_{i})$,

$X^{(m)}_{s}=\sum_{j=1}^{i-1}h(\eta-\delta_{(y_{j},s_{j})})-\int_{U_{m}}\int_{0}^{s}h(\eta,y,u)\lambda(dy)du$
(B.13)

and so for $s\in(s_{i-1},s_{i})$

$\frac{d}{ds}X^{(m)}_{s}=-\int_{U_{m}}h(\eta,y,s)\lambda(dy).$ (B.14)

This implies that

$\phi\left(X^{(m)}_{(s_{i}-)\wedge t}\right)-\phi\left(X^{(m)}_{s_{i-1}\wedge
t}\right)=-\int_{s_{i-1}\wedge t}^{(s_{i}-)\wedge
t}\phi^{\prime}\left(X^{(m)}_{s}\right)\int_{U_{m}}h(\eta,y,s)\lambda(dy)ds.$
(B.15)

We conclude that

$\displaystyle I_{2}(t)$
$\displaystyle=\sum_{i=1}^{n_{m}+1}\phi\left(X^{(m)}_{(s_{i}-)\wedge
t}\right)-\phi\left(X^{(m)}_{s_{i-1}\wedge t}\right)$
$\displaystyle=-\sum_{i=1}^{n_{m}+1}\int_{s_{i-1}\wedge t}^{(s_{i}-)\wedge
t}\int_{U_{m}}\phi^{\prime}\left(X^{(m)}_{s}\right)h(\eta,y,s)\lambda(dy)ds$
$\displaystyle=-\int_{0}^{t}\int_{U_{m}}\phi^{\prime}\left(X^{(m)}_{s}\right)h(\eta,y,s)\lambda(dy)ds.$
(B.16)

We have shown until now that

$\phi\left(X^{(m)}_{t}\right)=\phi(X_{0})+\int_{U_{m}\times[0,t]}\left(\phi\left(X^{(m)}_{s-}+h(\eta-\delta_{(y,s)},y,s)\right)-\phi\left(X^{(m)}_{s-}\right)\right)\eta(dy,ds)\\\
-\int_{U_{m}}\int_{0}^{t}\phi^{\prime}\left(X^{(m)}_{s}\right)h(\eta,x,s)\lambda(dx)ds\qquad\mathbb{P}\text{-a.s.}$
(B.17)

Our aim is now to let $m\rightarrow\infty$. By dominated convergence,
$X^{(m)}_{t}\rightarrow X_{t}$ a.s. for fixed $t\in[0,1]$. We would like to
use dominated convergence for both the second and third terms on the RHS of
(B.17). Start by noting that $(X_{t}^{(m)})_{t\in[0,1]}$ (as well as
$(X_{t-}^{(m)})_{t\in[0,1]}$ and $(X_{t})_{t\in[0,1]}$) are $\mathbb{P}$-a.s.
uniformly bounded on $[0,1]$ and in $m$. Indeed,

$\sup_{\begin{subarray}{c}m\in\mathbb{N}\\\
t\in[0,1]\end{subarray}}\max\left\\{\left|X_{t}^{(m)}\right|,\left|X_{t-}^{(m)}\right|,\left|X_{t}\right|\right\\}\\\
\leqslant\int_{\mathbb{X}\times[0,1]}|h(\eta-\delta_{(y,s)},y,s)|\eta(dy,ds)+\int_{\mathbb{X}}\int_{0}^{1}|h(\eta,y,s)|\lambda(dy)ds<\infty\qquad\mathbb{P}\text{-a.s.}$
(B.18)

We start with the second term on the RHS of (B.17) and write

$\int_{U_{m}\times[0,t]}\left|\phi\left(X^{(m)}_{s-}+h(\eta-\delta_{(y,s)},y,s)\right)-\phi\left(X^{(m)}_{s-}\right)\right|\eta(dy,ds)\\\
\leqslant\int_{\mathbb{X}\times[0,1]}\left|\int_{0}^{1}\phi^{\prime}\left(X^{(m)}_{s-}+uh(\eta-\delta_{(y,s)},y,s)\right)h(\eta-\delta_{(y,s)},y,s)du\right|\eta(dy,ds).$
(B.19)

By boundedness of $h$, we get that $X^{(m)}_{s-}+uh(\eta-\delta_{(y,s)},y,s)$
almost surely takes values in a compact interval independent of $m$. The
function $\phi^{\prime}$ being continuous, we deduce

$\sup_{\begin{subarray}{c}m\in\mathbb{N}\\\ s,u\in[0,1]\\\
y\in\mathbb{X}\end{subarray}}\left|\phi^{\prime}\left(X^{(m)}_{s-}+uh(\eta-\delta_{(y,s)},y,s)\right)\right|<\infty\qquad\mathbb{P}\text{-a.s.}$
(B.20)

Hence the RHS of (B.19) is bounded by

$\bigg{(}\sup_{\begin{subarray}{c}m\in\mathbb{N}\\\ s,u\in[0,1]\\\
y\in\mathbb{X}\end{subarray}}\left|\phi^{\prime}\left(X^{(m)}_{s-}+uh(\eta-\delta_{(y,s)},y,s)\right)\right|\bigg{)}\int_{\mathbb{X}\times[0,1]}|h(\eta-\delta_{(y,s)},y,s)|\eta(dy,ds)<\infty\qquad\mathbb{P}\text{-a.s.}$
(B.21)

We can thus apply dominated convergence and deduce that

$\int_{U_{m}\times[0,t]}\left(\phi\left(X^{(m)}_{s-}+h(\eta-\delta_{(y,s)},y,s)\right)-\phi\left(X^{(m)}_{s-}\right)\right)\eta(dy,ds)\\\
\stackrel{{\scriptstyle
m\rightarrow\infty}}{{\longrightarrow}}\int_{\mathbb{X}\times[0,t]}\left(\phi\left(X_{s-}+h(\eta-\delta_{(y,s)},y,s)\right)-\phi\left(X_{s-}\right)\right)\eta(dy,ds)\qquad\mathbb{P}\text{-a.s.}$
(B.22)

Now, similarly,

$\int_{U_{m}}\int_{0}^{t}\left|\phi^{\prime}\left(X^{(m)}_{s}\right)h(\eta,x,s)\right|\lambda(dx)ds\\\
\leqslant\left(\sup_{\begin{subarray}{c}m\in\mathbb{N}\\\
s\in[0,1]\end{subarray}}\left|\phi^{\prime}\left(X^{(m)}_{s}\right)\right|\right)\int_{\mathbb{X}}\int_{0}^{t}\left|h(\eta,x,s)\right|\lambda(dx)ds<\infty\qquad\mathbb{P}\text{-a.s.}$
(B.23)

and by dominated convergence

$\int_{U_{m}}\int_{0}^{t}\phi^{\prime}\left(X^{(m)}_{s}\right)h(\eta,x,s)\lambda(dx)ds\\\
\stackrel{{\scriptstyle
m\rightarrow\infty}}{{\longrightarrow}}\int_{\mathbb{X}}\int_{0}^{t}\phi^{\prime}\left(X_{s}\right)h(\eta,x,s)\lambda(dx)ds\qquad\mathbb{P}\text{-a.s.}$
(B.24)

The fact that $\phi\left(X^{(m)}_{t}\right)\rightarrow\phi\left(X_{t}\right)$
a.s. follows by continuity of $\phi$. This concludes the proof. ∎

##  Appendix C Proofs of Theorem 4.2 and Corollaries 4.3 and 4.7

######  Lemma C.1.

For $p\in(1,2]$, the function
$\phi:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto|x|^{p}$ is continuously
differentiable and its derivative $\phi^{\prime}$ is $(p-1)$-Hölder continuous
with Hölder constant $c_{\phi}=p2^{2-p}$.

###### Proof.

We want to show that

$c_{\phi}:=\sup_{\begin{subarray}{c}a\neq b\\\
a,b\in\mathbb{R}\end{subarray}}\frac{|\phi^{\prime}(a)-\phi^{\prime}(b)|}{|a-b|^{p-1}}=p2^{2-p}.$
(C.1)

First we observe that $\phi^{\prime}(x)=p\operatorname{sgn}(x)|x|^{p-1}$, with
the convention that $sgn(0)=1$. Let $a\neq b$ and assume without loss of
generality that $|a|\geqslant|b|\geqslant 0$. Then $a\neq 0$ and

$\frac{\left|\operatorname{sgn}(a)|a|^{p-1}-\operatorname{sgn}(b)|b|^{p-1}\right|}{|a-b|^{p-1}}=\frac{\left|1-\operatorname{sgn}\left(\frac{b}{a}\right)\left|\frac{b}{a}\right|^{p-1}\right|}{\left|1-\frac{b}{a}\right|^{p-1}}.$
(C.2)

It follows that

$c_{\phi}=p\sup_{x\in[-1,1)}\frac{1-\operatorname{sgn}(x)\left|x\right|^{p-1}}{(1-x)^{p-1}}=:p\sup_{x\in[-1,1)}f(x)$
(C.3)

The function $f$ is differentiable on $[-1,1)$ with derivative
$f^{\prime}(x)=(p-1)(1-x)^{-p}(1-|x|^{p-2})\leqslant 0$, so $f$ is decreasing
and therefore

$f(x)\leqslant f(-1)=2^{2-p}.$ (C.4)

We conclude that $c_{\phi}=p2^{2-p}$. ∎

###### Proof of Theorem 4.2.

We start by showing the result for an approximation of $h$. Let
$(U_{m})_{m\in\mathbb{N}}\subset\mathbb{X}$ s.t. $\bigcup_{m}U_{m}=\mathbb{X}$
and $\forall\,m\in\mathbb{N}$, $\lambda(U_{m})<\infty$ and $U_{m}\subset
U_{m+1}$. Define for $\mu\in\mathbf{N}$, $(y,s)\in\mathbb{X}\times[0,1]$:

$h_{m}(\mu,y,s):=\left[(h(\mu,y,s)\wedge m)\vee(-m)\right]\mathbbm{1}_{\\{y\in
U_{m}\\}}.$ (C.5)

Now $h_{m}\in L^{1}(\mathbf{N}\times\mathbb{X}\times[0,1])\cap
L^{2}(\mathbf{N}\times\mathbb{X}\times[0,1])$ and $h_{m}\rightarrow h$ in
$L^{2}$ as $m\rightarrow\infty$. Moreover, $|h_{m}|\leqslant|h|$ and
$|\operatorname{D}_{(x,t)}h_{m}(\eta,y,s)|\leqslant|\operatorname{D}_{(x,t)}h(\eta,y,s)|$
for all $(x,t),(y,s)\in\mathbb{X}\times[0,1]$. This implies that $h_{m}$
satisfies (2.12) and hence $h_{m}\in\operatorname{dom}\delta$.

Finally, $h_{m}$ is also bounded. In particular, for any $\mu\in\mathbf{N}$,
$(y,s)\in\mathbb{X}\times[0,1]$,

$|h_{m}(\mu,y,s)|\leqslant m\mathbbm{1}_{\\{y\in U_{m}\\}}.$ (C.6)

We conclude that $\delta(h_{m})$ is well-defined and has by (2.13) the
pathwise expression

$\delta(h_{m})=\int_{\mathbb{X}\times[0,1]}h_{m}(\eta-\delta_{(y,s)},y,s)\eta(dy,ds)-\int_{\mathbb{X}}\int_{0}^{1}h_{m}(\eta,y,s)\lambda(dy)ds.$
(C.7)

Define $X_{t}$ as in Theorem 4.1 with $h=h_{m}$ and $X_{0}=0$. Then
$\delta(h_{m})=X_{1}$ and we infer from (4.2) that

$\phi(\delta(h_{m}))-\phi(0)=\int_{\mathbb{X}\times[0,1]}\left(\phi(X_{s-}+h_{m}(\eta-\delta_{(y,s)},y,s))-\phi(X_{s-})\right)\eta(dy,ds)\\\
-\int_{\mathbb{X}}\int_{0}^{1}\phi^{\prime}(X_{s})h_{m}(\eta,y,s)\lambda(dy)ds\qquad\mathbb{P}\text{-a.s.}$
(C.8)

We would now like to take expectations on both sides of (C.8), but in order to
do so, we must first show that the terms in question are in
$L^{1}(\mathbb{P}_{\eta})$. Start with a simple estimate for $\phi$ that uses
Hölder continuity of $\phi^{\prime}$. Let $a,b\in\mathbb{R}$. Then

$\displaystyle|\phi(a+b)-\phi(a)|$
$\displaystyle=\left|\int_{0}^{1}\phi^{\prime}(a+ub)b\,du\right|$
$\displaystyle\leqslant|\phi^{\prime}(a)b|+\int_{0}^{1}|b|\cdot\left|\phi^{\prime}(a+ub)-\phi^{\prime}(a)\right|du$
$\displaystyle\leqslant|\phi^{\prime}(0)b|+|\phi^{\prime}(a)-\phi^{\prime}(0)|\cdot|b|+|b|\int_{0}^{1}c_{\phi}|ub|^{p-1}du$
$\displaystyle\leqslant|\phi^{\prime}(0)|\cdot|b|+c_{\phi}|a|^{p-1}\cdot|b|+\frac{c_{\phi}}{p}|b|^{p}.$
(C.9)

We are also going to need a bound on
$\sup_{s\in[0,1]}\left\\{\max{\left\\{|X_{s}|,|X_{s-}(\eta+\delta_{(y,s)})|\right\\}}\right\\}$.
Using (C.6), we find the following:

$\displaystyle\max{\left\\{|X_{s}|,|X_{s-}(\eta+\delta_{(y,s)})|\right\\}}$
$\displaystyle\leqslant\int_{\mathbb{X}\times[0,1]}m\mathbbm{1}_{\\{x\in
U_{m}\\}}\eta(dx,dt)+\int_{\mathbb{X}}\int_{0}^{1}m\mathbbm{1}_{\\{x\in
U_{m}\\}}\lambda(dx)dt$ $\displaystyle\leqslant
m\left(\eta(U_{m}\times[0,1])+\lambda(U_{m})\right)$ (C.10)

Using (C.9) with $a=0$ and $b=\delta(h_{m})$, we get for the LHS in (C.8)

$\mathbb{E}\left|\phi(\delta(h_{m}))\right|\leqslant|\phi^{\prime}(0)|\cdot\mathbb{E}|\delta(h_{m})|+\frac{c_{\phi}}{p}\mathbb{E}|\delta(h_{m})|^{p}<\infty$
(C.11)

which is finite since $\delta(h_{m})\in L^{2}(\mathbb{P}_{\eta})$. To show
that the first term on the RHS in (C.8) is integrable, we first apply Mecke
formula (2.6) to get

$\mathbb{E}\left|\int_{\mathbb{X}\times[0,1]}\left(\phi(X_{s-}+h_{m}(\eta-\delta_{(y,s)},y,s))-\phi(X_{s-})\right)\eta(dy,ds)\right|\\\
\leqslant\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}\left|\phi(X_{s-}(\eta+\delta_{(y,s)})+h_{m}(\eta,y,s))-\phi(X_{s-}(\eta+\delta_{(y,s)}))\right|\lambda(dy)ds.$
(C.12)

Now we combine (C.9), (C.10) and (C.6) to get that the RHS of (C.12) is
bounded by

$\displaystyle\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}\bigg{(}|\phi^{\prime}(0)|\cdot
m\mathbbm{1}_{\\{y\in
U_{m}\\}}+c_{\phi}|m\left(\eta(U_{m}\times[0,1])+\lambda(U_{m})\right)|^{p-1}\cdot
m\mathbbm{1}_{\\{y\in U_{m}\\}}$ $\displaystyle\hskip
260.17464pt+\frac{c_{\phi}}{p}m^{p}\mathbbm{1}_{\\{y\in
U_{m}\\}}\bigg{)}\lambda(dy)ds$ $\displaystyle\leqslant
m\lambda(U_{m})\left(|\phi^{\prime}(0)|+c_{\phi}m^{p-1}\mathbb{E}\left[\left(\eta(U_{m}\times[0,1])+\lambda(U_{m})\right)^{p-1}\right]+\frac{c_{\phi}m^{p-1}}{p}\right)<\infty$
(C.13)

which is finite since $\eta(U_{m}\times[0,1])$ is a Poisson random variable
with parameter $\lambda(U_{m})$ and thus all its moments are finite. This also
shows that
$\phi(X_{s-}(\eta+\delta_{(y,s)})+h_{m}(\eta,y,s))-\phi(X_{s-}(\eta+\delta_{(y,s)}))\in
L^{1}(\mathbf{N}\times\mathbb{X}\times[0,1])$. The second term on the RHS can
be treated by the same method.

We can now take expectations on both sides of (C.8) and apply Mecke formula
(2.6) to get:

$\mathbb{E}\phi(\delta(h_{m}))=\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}\phi(X_{s-}(\eta+\delta_{(y,s)})+h_{m}(\eta,y,s))-\phi(X_{s-}(\eta+\delta_{(y,s)}))\lambda(dy)ds\\\
-\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}\phi^{\prime}(X_{s}(\eta))h_{m}(\eta,y,s)\lambda(dy)ds$
(C.14)

Now add and subtract the integral of
$\phi(X_{s-}(\eta)+h_{m}(\eta,y,s))-\phi(X_{s-}(\eta))$ (which can be shown to
be in $L^{1}(\mathbf{N}\times\mathbb{X}\times[0,1])$ by the same methods as
above). We obtain:

$\displaystyle\mathbb{E}\phi(\delta(h_{m}))$
$\displaystyle=\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}\phi(X_{s-}(\eta+\delta_{(y,s)})+h_{m}(\eta,y,s))-\phi(X_{s-}(\eta+\delta_{(y,s)}))$
$\displaystyle\hskip
130.08731pt-\phi(X_{s-}(\eta)+h_{m}(\eta,y,s))+\phi(X_{s-}(\eta))\lambda(dy)ds$
$\displaystyle+\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}\phi(X_{s-}(\eta)+h_{m}(\eta,y,s))-\phi(X_{s-}(\eta))-\phi^{\prime}(X_{s}(\eta))h_{m}(\eta,y,s)\lambda(dy)ds$
$\displaystyle=:I_{1}+I_{2}.$ (C.15)

To deal with $I_{1}$, note that for $a,b,c\in\mathbb{R}$

$\displaystyle|\phi(a+c)-\phi(a)-\phi(b+c)+\phi(b)|$
$\displaystyle=\left|\int_{a}^{b}\phi^{\prime}(u+c)-\phi^{\prime}(u)du\right|$
(C.16) $\displaystyle\leqslant\int_{a\wedge b}^{a\vee b}c_{\phi}|c|^{p-1}du$
(C.17) $\displaystyle=c_{\phi}|a-b|\cdot|c|^{p-1}.$ (C.18)

Hence

$|I_{1}|\leqslant
c_{\phi}\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}\left|X_{s-}(\eta+\delta_{(y,s)})-X_{s-}(\eta)\right|\cdot|h_{m}(\eta,y,s)|^{p-1}\lambda(dy)ds.$
(C.19)

Now we bound and rewrite part of the integrand on the RHS of (C.19) to find

$\displaystyle\left|X_{s-}(\eta+\delta_{(y,s)})-X_{s-}(\eta)\right|$
$\displaystyle=\left|\int_{\mathbb{X}\times[0,s)}h_{m}(\eta+\delta_{(y,s)}-\delta_{(x,t)},x,t)\eta(dx,dt)-\int_{\mathbb{X}}\int_{0}^{s}h_{m}(\eta+\delta_{(y,s)},x,t)\lambda(dx)dt\right.$
$\displaystyle\phantom{=}\left.-\int_{\mathbb{X}\times[0,s)}h_{m}(\eta-\delta_{(x,t)},x,t)\eta(dx,dt)+\int_{\mathbb{X}}\int_{0}^{s}h_{m}(\eta,x,t)\lambda(dx)dt\right|$
$\displaystyle=\left|\int_{\mathbb{X}\times[0,s)}\operatorname{D}_{(y,s)}h_{m}(\eta-\delta_{(x,t)},x,t)\eta(dx,dt)-\int_{\mathbb{X}}\int_{0}^{s}\operatorname{D}_{(y,s)}h_{m}(\eta,x,t)\lambda(dx)dt\right|$
$\displaystyle\leqslant\int_{\mathbb{X}\times[0,s)}\left|\operatorname{D}_{(y,s)}h_{m}(\eta-\delta_{(x,t)},x,t)\right|\eta(dx,dt)+\int_{\mathbb{X}}\int_{0}^{s}\left|\operatorname{D}_{(y,s)}h_{m}(\eta,x,t)\right|\lambda(dx)dt$
(C.20)

Multiplying this by $|h_{m}(\eta,x,t)|^{p-1}$ and taking expectations, after
an application of Mecke formula (2.6) we deduce that

$|I_{1}|\leqslant
c_{\phi}\mathbb{E}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds\int_{\mathbb{X}}\lambda(dx)\int_{0}^{s}dt\,\left|\operatorname{D}_{(y,s)}h_{m}(\eta,x,t)\right|\\\
\left(|h_{m}(\eta+\delta_{(x,t)},y,s)|^{p-1}+|h_{m}(\eta,y,s)|^{p-1}\right)\lambda(dx)dt\lambda(dy)ds.$
(C.21)

Since $|a-b|^{p-1}\leqslant|a|^{p-1}+|b|^{p-1}$ for all $a,b\in\mathbb{R}$,
one has that

$|h_{m}(\eta+\delta_{(x,t)},y,s)|^{p-1}\leqslant|\operatorname{D}_{(x,t)}h_{m}(\eta,y,s)|^{p-1}+|h_{m}(\eta,y,s)|^{p-1}$
(C.22)

This implies that

$\displaystyle|I_{1}|$ $\displaystyle\leqslant
c_{\phi}\mathbb{E}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds\int_{\mathbb{X}}\lambda(dx)\int_{0}^{s}dt\left|\operatorname{D}_{(y,s)}h_{m}(\eta,x,t)\right|\cdot|\operatorname{D}_{(x,t)}h_{m}(\eta,y,s)|^{p-1}$
$\displaystyle\phantom{\leqslant}+2c_{\phi}\mathbb{E}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds\int_{\mathbb{X}}\lambda(dx)\int_{0}^{s}dt\,\left|\operatorname{D}_{(y,s)}h_{m}(\eta,x,t)\right|\cdot|h_{m}(\eta,y,s)|^{p-1}.$
(C.23)

Now we consider $|I_{2}|$. For this, note that for $a,b\in\mathbb{R}$,

$\displaystyle|\phi(a+b)-\phi(a)-\phi^{\prime}(a)b|$
$\displaystyle=\left|\int_{0}^{b}\phi^{\prime}(a+u)-\phi^{\prime}(a)du\right|$
$\displaystyle\leqslant c_{\phi}\int_{0}^{|b|}u^{p-1}du$
$\displaystyle=\frac{c_{\phi}}{p}|b|^{p}.$ (C.24)

Applying this to $a=X_{s}(\eta)$ and $h=h_{m}(\eta,y,s)$ yields

$|I_{2}|\leqslant\frac{c_{\phi}}{p}\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}|h_{m}(\eta,y,s)|^{p}\lambda(dy)ds.$
(C.25)

Observe that in the previous computation we implicitly used the fact that,
$\mathbb{P}$-a.s., the set $\left\\{s\in[0,1]:X_{s}(\eta)\neq
X_{s-}(\eta)\right\\}$ has zero Lebesgue measure. We have therefore shown the
following inequality:

$\displaystyle|\mathbb{E}\phi(\delta(h_{m}))|$
$\displaystyle\leqslant\frac{c_{\phi}}{p}\mathbb{E}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds\,|h_{m}(\eta,y,s)|^{p}$
$\displaystyle\phantom{\leqslant}+c_{\phi}\mathbb{E}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds\int_{\mathbb{X}}\lambda(dx)\int_{0}^{s}dt\,\left|\operatorname{D}_{(y,s)}h_{m}(\eta,x,t)\right|\cdot|\operatorname{D}_{(x,t)}h_{m}(\eta,y,s)|^{p-1}$
$\displaystyle\phantom{\leqslant}+2c_{\phi}\mathbb{E}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds\int_{\mathbb{X}}\lambda(dx)\int_{0}^{s}dt\,\left|\operatorname{D}_{(y,s)}h_{m}(\eta,x,t)\right|\cdot|h_{m}(\eta,y,s)|^{p-1}.$
(C.26)

By the construction of $h_{m}$, the RHS of this inequality is upper bounded by

$\frac{c_{\phi}}{p}\mathbb{E}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds\,|h(\eta,y,s)|^{p}\\\
+c_{\phi}\mathbb{E}\int_{\mathbb{X}}\lambda(dx)\int_{0}^{1}ds\int_{\mathbb{X}}\lambda(dx)\int_{0}^{s}dt\left|\operatorname{D}_{(y,s)}h(\eta,x,t)\right|\cdot|\operatorname{D}_{(x,t)}h(\eta,y,s)|^{p-1}\\\
+2c_{\phi}\mathbb{E}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds\int_{\mathbb{X}}\lambda(dx)\int_{0}^{s}dt\,\left|\operatorname{D}_{(y,s)}h(\eta,x,t)\right|\cdot|h(\eta,y,s)|^{p-1}.$
(C.27)

In order to conclude the proof, it remains to show that
$\mathbb{E}\phi(\delta(h_{m}))\rightarrow\mathbb{E}\phi(\delta(h))$, as
$m\rightarrow\infty$. For this, we use (C.9) with $a=\delta(h)$ and
$b=\delta(h_{m})-\delta(h)$ to get

$\displaystyle\mathbb{E}|\phi(\delta(h_{m}))-\phi(\delta(h))|$
$\displaystyle\leqslant\mathbb{E}|\delta(h_{m})-\delta(h)|\left(|\phi^{\prime}(0)|+c_{\phi}|\delta(h)|^{p-1}+\frac{c_{\phi}}{p}|\delta(h_{m})-\delta(h)|^{p-1}\right).$
(C.28)

Using, in order, the Cauchy-Schwarz, Minkowski and Jensen inequalities, we
obtain

$\displaystyle\mathbb{E}|\phi(\delta(h_{m}))-\phi(\delta(h))|$
$\displaystyle\hskip
43.36464pt\leqslant\mathbb{E}\left[\left(\delta(h_{m})-\delta(h)\right)^{2}\right]^{1/2}\mathbb{E}\left[\left(|\phi^{\prime}(0)|+c_{\phi}|\delta(h)|^{p-1}+\frac{c_{\phi}}{p}|\delta(h_{m})-\delta(h)|^{p-1}\right)^{2}\right]^{1/2}$
$\displaystyle\hskip
43.36464pt\leqslant\mathbb{E}\left[\delta(h_{m}-h)^{2}\right]^{1/2}\left(|\phi^{\prime}(0)|+c_{\phi}\mathbb{E}\left[|\delta(h)|^{2}\right]^{(p-1)/2}+\frac{c_{\phi}}{p}\mathbb{E}\left[|\delta(h_{m}-h)|^{2}\right]^{(p-1)/2}\right).$
(C.29)

By the isometry property (A.4) and the Cauchy-Schwarz inequality, we infer
that

$\displaystyle\mathbb{E}\left[|\delta(h)|^{2}\right]\leqslant\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}h(\eta,y,s)^{2}\lambda(dy)ds+\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}\int_{\mathbb{X}}\int_{0}^{1}\left(\operatorname{D}_{(x,t)}h(\eta,y,s)\right)^{2}\lambda(dy)ds\lambda(dx)ds$
(C.30) and
$\displaystyle\mathbb{E}\left[|\delta(h_{m}-h)|^{2}\right]\leqslant\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}(h_{m}(\eta,y,s)-h(\eta,y,s))^{2}\lambda(dy)ds$
$\displaystyle\hskip
86.72267pt+\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}\int_{\mathbb{X}}\int_{0}^{1}\left(\operatorname{D}_{(x,t)}h_{m}(\eta,y,s)-\operatorname{D}_{(x,t)}h(\eta,y,s)\right)^{2}\lambda(dy)ds\lambda(dx)ds.$
(C.31)

The RHS of (C.30) is finite because $h$ satisfies (2.12), and the RHS of
(C.31) tends to $0$ as $m\rightarrow\infty$ by dominated convergence and the
assumption (2.12). This implies convergence to $0$ on the RHS of (C.29).

To show that the inequality holds in particular for $\phi(x)=|x|^{p}$ with
$p\in(1,2]$, it suffices to note that by Lemma C.1, this function satisfies
the conditions of the theorem. For $\phi(x)=|x|$, we define $h_{m}$ as in
(C.5). Then $h_{m}\in L^{1}(\mathbf{N}\times\mathbb{X}\times[0,1])\cap
L^{2}(\mathbf{N}\times\mathbb{X}\times[0,1])\cap\operatorname{dom}\delta$ and
the pathwise representation (2.13) holds. By the triangle inequality, we have

$\mathbb{E}|\delta(h_{m})|\leqslant\mathbb{E}\int_{\mathbb{X}\times[0,1]}|h_{m}(\eta-\delta_{y,s},y,s)|\eta(dy,ds)+\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}|h_{m}(\eta,y,s)|\lambda(dy)ds.$
(C.32)

By Mecke formula (2.6) and the fact that $|h_{m}|\leqslant|h|$, inequality
(4.3) follows with $h_{m}$ on the LHS instead of $h$, and the second and third
term on the RHS being zero. As we have shown before,
$\delta(h_{m})\rightarrow\delta(h)$ in $L^{2}(\mathbb{P}_{\eta})$ as
$m\rightarrow\infty$, hence the inequality follows for $h$. ∎

###### Proof of Corollary 4.3.

Step 1. We first prove the following slightly modified version of (4.3) which
holds under the weaker assumption $h\in\operatorname{dom}\delta$:

$\displaystyle\left|\mathbb{E}\phi(\delta(h))\right|$
$\displaystyle\leqslant\frac{c_{\phi}}{p}\mathbb{E}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds|h(\eta,y,s)|^{p}$
$\displaystyle+c_{\phi}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds\int_{\mathbb{X}}\lambda(dx)\int_{0}^{s}dt\left(\mathbb{E}\left|\operatorname{D}_{(y,s)}h(\eta,x,t)\right|^{p}\right)^{1/p}\cdot\left(\mathbb{E}\left|\operatorname{D}_{(x,t)}h(\eta,y,s)\right|^{p}\right)^{1-1/p}$
$\displaystyle+2c_{\phi}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds\int_{\mathbb{X}}\lambda(dx)\int_{0}^{s}dt\left(\mathbb{E}\left|\operatorname{D}_{(y,s)}h(\eta,x,t)\right|^{p}\right)^{1/p}\cdot\left(\mathbb{E}\left|h(\eta,y,s)\right|^{p}\right)^{1-1/p}.$
(C.33)

For $h$ satisfying (2.12), this is an immediate consequence of Theorem 4.2 by
applying Hölder inequality.

Now let $h\in L^{2}(\mathbf{N}\times\mathbb{X}\times[0,1])$ and for
$\tau\in(0,1)$ define $P_{\tau}h$. By Lemma A.2, we have that $P_{\tau}h$
satisfies (2.12), and so inequality (C.33) holds when $h$ is replaced by
$P_{\tau}h$. Using (A.13) and (A.14), we deduce

$\displaystyle\left|\mathbb{E}\phi(\delta(P_{\tau}h))-\phi(0)\right|$ (C.34)
$\displaystyle\leqslant\frac{c_{\phi}}{p}\mathbb{E}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds|h(\eta,y,s)|^{p}$
$\displaystyle+c_{\phi}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds\int_{\mathbb{X}}\lambda(dx)\int_{0}^{s}dt\left(\mathbb{E}\left|\operatorname{D}_{(y,s)}h(\eta,x,t)\right|^{p}\right)^{1/p}\cdot\left(\mathbb{E}\left|\operatorname{D}_{(x,t)}h(\eta,y,s)\right|^{p}\right)^{1-1/p}$
$\displaystyle+2c_{\phi}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds\int_{\mathbb{X}}\lambda(dx)\int_{0}^{s}dt\left(\mathbb{E}\left|\operatorname{D}_{(y,s)}h(\eta,x,t)\right|^{p}\right)^{1/p}\cdot\left(\mathbb{E}\left|h(\eta,y,s)\right|^{p}\right)^{1-1/p}.$
(C.35)

It remains to show that $\phi(\delta(P_{\tau}h))\rightarrow\phi(\delta(h))$ in
$L^{1}(\eta)$ as $\tau\rightarrow 1$. Using arguments similar to the ones in
the proof of Theorem 4.2, one shows that

$\mathbb{E}|\phi(\delta(P_{\tau}h))-\phi(\delta(h))|\\\
\leqslant\lVert\delta(P_{\tau}h-h)\rVert_{L^{2}(\mathbb{P}_{\eta})}\left(|\phi^{\prime}(0)|+c_{\phi}\lVert\delta(h)\rVert_{L^{2}(\mathbb{P}_{\eta})}^{p-1}+\frac{c_{\phi}}{p}\lVert\delta(P_{\tau}h-h)\rVert_{L^{2}(\mathbb{P}_{\eta})}^{p-1}\right).$
(C.36)

By the isometry property (A.1), the second moment $\mathbb{E}\delta(h)^{2}$ is
finite since $h\in\operatorname{dom}\delta$ and Lemma A.2 implies that
$\lVert\delta(P_{\tau}h-h)\rVert_{L^{2}(\mathbb{P}_{\eta})}\rightarrow 0$ as
$\tau\rightarrow 1$.

Step 2. Let $\in L^{2}(\mathbf{N}\times\mathbb{X}\times[0,1])$ be predictable.
Then $h\in\operatorname{dom}\delta$ and (C.33) holds. If $t<s$, then by
predictability of $h$ one has $h(\eta+\delta_{(y,s)},x,t)=h(\eta,x,t)$ and
hence $\operatorname{D}_{(y,s)}h(\eta,x,t)=0$. It follows that the second and
third terms on the RHS of (C.33) are 0. Inequality (4.4) ensues.

Step 3. To prove (4.5) for $p\in(1,2]$, let
$h(\eta,x,s):=\mathbb{E}[\operatorname{D}_{(x,s)}F|\eta_{\mathbb{X}\times[0,s)}]$.
This is predictable in the sense of (2.16) (see Section 2) and by the Clark-
Ocône representation (2.18), one has that

$F=\mathbb{E}F+\delta(h)\qquad\mathbb{P}\text{-a.s.}$ (C.37)

Define $\phi(x):=|x+\mathbb{E}F|^{p}-|\mathbb{E}F|^{p}$. Then $\phi(0)=0$ and
the derivative $\phi^{\prime}$ is $(p-1)$-Hölder continuous with Hölder
constant $2^{2-p}$ by Lemma C.1. Moreover, it holds that

$\phi(\delta(h))=|F|^{p}-|\mathbb{E}F|^{p}\qquad\mathbb{P}\text{-a.s.}$ (C.38)

We can now apply (4.4) for this choice of $\phi$ and $h$ and inequality (4.5)
follows.

For $p=1$, assume the RHS of the inequality (4.5) to be finite (else there is
nothing to prove). Then
$h(\eta,x,s)=\mathbb{E}[\operatorname{D}_{(x,s)}F|\eta_{\mathbb{X}\times[0,s)}]\in\operatorname{dom}\delta\cap
L^{1}(\mathbb{N}\times\mathbb{X}\times[0,1])$ and hence

$\delta(h)=\int_{\mathbb{X}\times[0,1]}h(\eta-\delta_{(x,s)},x,s)\eta(dx,ds)-\int_{\mathbb{X}}\int_{0}^{1}h(\eta,x,s)\lambda(dx)ds.$
(C.39)

Using triangle inequality and Mecke formula, we deduce the chain of
inequalities

$\displaystyle\mathbb{E}|F|-|\mathbb{E}F|$
$\displaystyle\leqslant\mathbb{E}|F-\mathbb{E}F|$
$\displaystyle=\mathbb{E}|\delta(h)|$ (C.40)
$\displaystyle\leqslant\mathbb{E}\int_{\mathbb{X}\times[0,1]}|h(\eta-\delta_{(x,t)},x,t)|\eta(dx,dt)+\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}|h(\eta,x,t)|\lambda(dx)dt$
$\displaystyle\leqslant
2\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}|h(\eta,x,t)|\lambda(dx)dt,$ (C.41)

which concludes the proof. ∎

###### Proof of Corollary 4.7.

Let $(U_{n})_{n\geqslant 1}\subset\mathbb{X}$ be an increasing sequence of
subsets such that $\bigcup_{n\geqslant 1}U_{n}=\mathbb{X}$ and
$\lambda(U_{n})<\infty$. Define for $n\in\mathbb{N}$

$h_{n}(\eta,y,s):=\mathbbm{1}_{\\{y\in U_{n}\\}}[(h(\eta,y,s)\wedge
n)\vee(-n)].$ (C.42)

Clearly $h_{n}\in L^{1}(\mathbf{N}\times\mathbb{X}\times[0,1])\cap
L^{2}(\mathbf{N}\times\mathbb{X}\times[0,1])$ and we can define
$P_{\tau}h_{n}$ for $\tau\in(0,1)$. We will start by showing (4.9) for
$P_{\tau}h_{n}$ instead of $h$. Using the definition (A.9), one easily checks
that $P_{\tau}h_{n}\in L^{1}(\mathbf{N}\times\mathbb{X}\times[0,1])$. By Lemma
A.2, it also follows that $P_{\tau}h_{n}\in\operatorname{dom}\delta$ and hence
we can apply Lemma A.1 and deduce that

$\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}P_{\tau}h_{n}(\eta,x,s)\operatorname{D}_{(x,s)}G\lambda(dx)ds=\mathbb{E}\left[G\delta(P_{\tau}h_{n})\right].$
(C.43)

This implies by Jensen’s inequality that for any $p\in[1,2]$

$\left|\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}P_{\tau}h_{n}(\eta,x,s)\operatorname{D}_{(x,s)}G\lambda(dx)ds\right|\leqslant
c_{G}\left(\mathbb{E}|\delta(P_{\tau}h_{n})|^{p}\right)^{1/p}.$ (C.44)

As by Lemma A.2 the quantity $P_{\tau}h_{n}$ also satisfies (2.12), we can
apply Theorem 4.2 to $\mathbb{E}|\delta(P_{\tau}h_{n})|^{p}$ with
$\phi(x)=|x|^{p}$ (which satisfies the required conditions by Lemma C.1).
After a further application of Hölder inequality, this yields for $p\in(1,2]$
that

$\displaystyle\mathbb{E}|\delta(P_{\tau}h_{n})|^{p}$ $\displaystyle\leqslant
2^{2-p}\mathbb{E}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds|P_{\tau}h_{n}(\eta,y,s)|^{p}$
$\displaystyle+p2^{2-p}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds\int_{\mathbb{X}}\lambda(dx)\int_{0}^{s}dt\mathbb{E}\left[\left|\operatorname{D}_{(y,s)}P_{\tau}h_{n}(\eta,x,t)\right|^{p}\right]^{1/p}\cdot\mathbb{E}\left[\left|\operatorname{D}_{(x,t)}P_{\tau}h_{n}(\eta,y,s)\right|^{p}\right]^{1-1/p}$
$\displaystyle+p2^{3-p}\int_{\mathbb{X}}\lambda(dy)\int_{0}^{1}ds\int_{\mathbb{X}}\lambda(dx)\int_{0}^{s}dt\mathbb{E}\left[\left|\operatorname{D}_{(y,s)}P_{\tau}h_{n}(\eta,x,t)\right|^{p}\right]^{1/p}\cdot\mathbb{E}\left[\left|P_{\tau}h_{n}(\eta,y,s)\right|^{p}\right]^{1-1/p}.$
(C.45)

Using Lemma A.2 and the definition of $h_{n}$, we find that

$\displaystyle\mathbb{E}|\operatorname{D}_{(x,s)}P_{\tau}h_{n}(\eta,y,u)|^{p}\leqslant\mathbb{E}|\operatorname{D}_{(x,s)}h(\eta,y,u)|^{p}$
(C.46) and
$\displaystyle\mathbb{E}|P_{\tau}h_{n}(\eta,y,u)|^{p}\leqslant\mathbb{E}|h(\eta,y,u)|^{p}$
(C.47)

hence (C.45) is upper bounded by the RHS of (4.9). For $p=1$, we reason as in
the proof of Corollary 4.3 to deduce that

$\mathbb{E}|\delta(P_{\tau}h_{n})|\leqslant
2\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}|P_{\tau}h_{n}(\eta,x,s)|\lambda(dx)ds,$
(C.48)

which is again upper bounded by the RHS of (4.9).

It remains to take $\tau\rightarrow 1$ and $n\rightarrow\infty$ in the LHS of
(C.44). By (A.9), one sees that $P_{\tau}h_{n}(\eta,x,s)=\mathbbm{1}_{\\{x\in
U_{n}\\}}P_{\tau}h_{n}(\eta,x,s)$, and hence by the Cauchy-Schwarz inequality

$\mathbb{E}\int_{\mathbb{X}}\int_{0}^{1}\left|P_{\tau}h_{n}(\eta,x,s)-h_{n}(\eta,x,s)\right|\cdot|\operatorname{D}_{(x,s)}G|\lambda(dx)ds\leqslant
2c_{G}\lambda(U_{n})\lVert
P_{\tau}h_{n}-h_{n}\rVert_{L^{2}(\mathbf{N}\times\mathbb{X}\times[0,1])},$
(C.49)

which converges to zero as $\tau\rightarrow 1$, by Lemma A.2. Therefore
inequality (4.9) holds with $h_{n}$ instead of $h$ on the LHS. By dominated
convergence, $h_{n}\operatorname{D}G\rightarrow h\operatorname{D}G$ in
$L^{1}(\mathbf{N}\times\mathbb{X}\times[0,1])$, thus inequality (4.9) holds
for $h$. ∎

##  Appendix D Proofs of Theorems 3.2, 3.3 and 3.4

Throughout this section, we work with the simplified notation adopted in
Remark 3.1; recall also the definitions of the distances $d_{W}$ and $d_{K}$
given in (3.1) and (3.2), and write $\mathcal{H}$ to denote the set of
Lipschitz-continuous functions $h:\mathbb{R}\rightarrow\mathbb{R}$ with
Lipschitz constant $\lVert h\rVert_{L}\leqslant 1$. Let
$N\sim\mathcal{N}(0,1)$.

Given $h\in\mathcal{H}$ and $\phi(z)=\mathbb{P}(N\leqslant z)$, Stein’s
equation

$f^{\prime}(x)=xf(x)+h(x)-\int_{\mathbb{R}}h(y)\phi^{\prime}(y)dy,\qquad
x\in\mathbb{R},$ (D.1)

admits a canonical solution $f_{h}$ satisfying the two properties: (a)
$f_{h}\in\mathcal{C}^{1}(\mathbb{R})$ and $\lVert
f_{h}^{\prime}\rVert_{\infty}\leqslant\sqrt{\frac{2}{\pi}}$ and (b)
$f_{h}^{\prime}$ is Lipschitz-continuous with $\lVert
f_{h}^{\prime}\rVert_{L}\leqslant 2$, see e.g. [PR16, Theorem 3, Chapter 6]
and the references therein. In particular, one has that

$d_{W}(F,N)=\sup_{h\in\mathcal{H}}|f_{h}^{\prime}(F)-Ff_{h}(F)|.$ (D.2)

Similarly, for fixed $z\in\mathbb{R}$, the canonical solution $f_{z}$ to
Stein’s equation

$f_{z}^{\prime}(x)=xf_{z}(x)+\mathbbm{1}_{(-\infty,z]}(x)-\phi(z),\qquad
x\in\mathbb{R},$ (D.3)

is differentiable everywhere except at $z$, where it is customary to define
$f_{z}^{\prime}(z)=zf_{z}(z)+1-\phi(z)$. One has that

  * •

$\lVert f_{z}\rVert_{\infty}\leqslant\frac{\sqrt{2\pi}}{4}$ and $\lVert
f_{z}^{\prime}\rVert_{\infty}\leqslant 1$

  * •

$|xf_{z}(x)|\leqslant 1$ for all $x\in\mathbb{R}$ and the function $x\mapsto
xf_{z}(x)$ is non-decreasing for all $z\in\mathbb{R}$,

(see e.g. [CGS11, Lemma 2.3]). We consequently have that

$d_{K}(F,N)=\sup_{z\in\mathbb{R}}|f_{z}^{\prime}(F)-Ff_{z}(F)|.$ (D.4)

###### Proof of Thm. 3.2.

We will show the upper bounds in (3.3) and (3.6) by exploiting the
representations (D.2) and (D.4). The proof is divided into three steps. First,
we are going to derive the first terms on the RHSs of (3.3) and (3.6) for both
Wasserstein and Kolmogorov distances at the same time. Second, we deduce the
second term on the RHS of (3.3) and, as a last step, we find the second term
on the RHS of (3.6). Throughout the proof, we fix $h\in\mathcal{H}$ and
$z\in\mathbb{R}$ and consider the corresponding canonical solutions $f_{h}$
and $f_{z}$.

Step 1. Write $f$ for either $f_{h}$ or $f_{z}$. Then, $f$ is Lipschitz and
there is a version of $f^{\prime}$ which is bounded. Since
$F\in\operatorname{dom}\operatorname{D}$ and $f^{\prime}$ is bounded, the
expression

$\mathbb{E}f^{\prime}(F)\int_{\mathbb{Y}}\operatorname{D}_{y}F\
\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\bar{\lambda}(dy)$ (D.5)

is well-defined. Add and subtract this term to
$\mathbb{E}f^{\prime}(F)-\mathbb{E}Ff(F)$ and bound the resulting first term
as follows:

$\displaystyle\left|\mathbb{E}f^{\prime}(F)-\mathbb{E}f^{\prime}(F)\int_{\mathbb{Y}}\operatorname{D}_{y}F\
\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\bar{\lambda}(dy)\right|$
$\displaystyle\leqslant\mathbb{E}|f^{\prime}(F)|\cdot\left|1-\int_{\mathbb{Y}}\operatorname{D}_{y}F\
\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\bar{\lambda}(dy)\right|$
$\displaystyle\leqslant\lVert
f^{\prime}\rVert_{\infty}\,\mathbb{E}\left|1-\int_{\mathbb{Y}}\operatorname{D}_{y}F\
\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\bar{\lambda}(dy)\right|.$ (D.6)

As $\lVert f_{h}^{\prime}\rVert_{\infty}\leqslant\sqrt{\frac{2}{\pi}}$ and
$\lVert f_{z}^{\prime}\rVert_{\infty}\leqslant 1$, the bounds follow.

We are left to deal with

$\left|\mathbb{E}Ff(F)-\mathbb{E}f^{\prime}(F)\int_{\mathbb{Y}}\operatorname{D}_{y}F\
\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\bar{\lambda}(dy)\right|.$ (D.7)

Since $f$ is Lipschitz and $F\in L^{2}(\mathbb{P}_{\eta})$, it follows that
$f(F)\in L^{2}(\mathbb{P}_{\eta})$. Hence by Lemma A.3

$\mathbb{E}Ff(F)=\operatorname{Cov}(F,f(F))=\mathbb{E}\int_{\mathbb{Y}}\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\
\mathbb{E}[\operatorname{D}_{y}f(F)|\eta_{y}]\bar{\lambda}(dy).$ (D.8)

Again by Lipschitzianity of $f$, it follows that
$|\operatorname{D}_{y}f(F)|\leqslant|\operatorname{D}_{y}F|$ and hence an
application of Cauchy-Schwarz inequality justifies that

$\mathbb{E}Ff(F)=\mathbb{E}\int_{\mathbb{Y}}\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\operatorname{D}_{y}f(F)\bar{\lambda}(dy).$
(D.9)

Therefore we are left to bound

$\mathbb{E}\int_{\mathbb{Y}}\left|\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\right|\cdot\left|f^{\prime}(F)\operatorname{D}_{y}F-\operatorname{D}_{y}f(F)\right|\bar{\lambda}(dy).$
(D.10)

Step 2. To bound (D.10) for the Wasserstein distance, we use an argument
borrowed from the proof of [BOPT20, Theorem 3.1] to upper bound
$|f_{h}(b)-f_{h}(a)-f_{h}^{\prime}(a)(b-a)|$. Let $a,b\in\mathbb{R}$. Then by
the properties stated above, $f_{h}$ is Lipschitz and hence

$|f_{h}(b)-f_{h}(a)-f_{h}^{\prime}(a)(b-a)|\leqslant|f_{h}(b)-f_{h}(a)|+|f_{h}^{\prime}(a)(b-a)|\leqslant
2\sqrt{\tfrac{2}{\pi}}|b-a|.$ (D.11)

But at the same time by Lipschitzianity of $f_{h}^{\prime}$,

$|f_{h}(b)-f_{h}(a)-f_{h}^{\prime}(a)(b-a)|=\left|\int_{a}^{b}f_{h}^{\prime}(x)-f_{h}^{\prime}(a)dx\right|\leqslant
2(b-a)^{2}.$ (D.12)

We deduce that for any $q\in[1,2]$

$|f_{h}(b)-f_{h}(a)-f_{h}^{\prime}(a)(b-a)|\leqslant
2\min\\{|b-a|,(b-a)^{2}\\}\leqslant 2|b-a|^{q}.$ (D.13)

It follows that

$\displaystyle\left|f_{h}^{\prime}(F)\operatorname{D}_{y}F-\operatorname{D}_{y}f_{h}(F)\right|$
$\displaystyle=|f_{h}(F(\eta+\delta_{y}))-f_{h}(F(\eta))-f_{h}^{\prime}(F(\eta))(F(\eta+\delta_{y})-F(\eta))|$
$\displaystyle\leqslant 2|\operatorname{D}_{y}F|^{q}.$ (D.14)

and therefore (D.10) is bounded by

$2\,\mathbb{E}\int_{\mathbb{Y}}\left|\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\right||\operatorname{D}_{y}F|^{q}\bar{\lambda}(dy).$
(D.15)

The required bound in the Wasserstein distance follows suit.

Step 3. We reason as in the proof of [LRPY20, Theorem 1.12] to conclude

$\left|f_{z}^{\prime}(F)\operatorname{D}_{y}F-\operatorname{D}_{y}f_{z}(F)\right|\leqslant\operatorname{D}_{y}F\cdot\operatorname{D}_{y}(Ff_{z}(F)+\mathbbm{1}_{\\{F>z\\}})$
(D.16)

Thus (D.10) is upper bounded by

$\mathbb{E}\int_{\mathbb{Y}}\left|\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\right|\operatorname{D}_{y}F\cdot\operatorname{D}_{y}\left(Ff_{z}(F)+\mathbbm{1}_{\\{F>z\\}}\right)\bar{\lambda}(dy).$
(D.17)

The desired bound now follows by taking the supremum over all
$z\in\mathbb{R}$. ∎

###### Proof of Theorem 3.3.

The proof will be split into two steps.

Step 1. We start by showing that under the conditions of the theorem,

$\mathbb{E}\left|1-\int_{\mathbb{Y}}\operatorname{D}_{y}F\,\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\bar{\lambda}(dy)\right|\leqslant\beta_{1}+\beta_{2}.$
(D.18)

We can assume that $\mathbb{E}F=0$ and $\sigma=1$ (indeed, the result then
follows since $\operatorname{D}\hat{F}=\sigma^{-1}\operatorname{D}F$).

For ease of notation, define

$G:=\int_{\mathbb{Y}}\operatorname{D}_{y}F\,\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\bar{\lambda}(dy)-1.$
(D.19)

As a first step, we show that $\mathbb{E}G=0$. As
$F\in\operatorname{dom}\operatorname{D}$, we can use Fubini’s theorem and
Lemma A.3 to deduce

$\mathbb{E}\int_{\mathbb{Y}}\operatorname{D}_{y}F\,\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\bar{\lambda}(dy)=\mathbb{E}\int_{\mathbb{Y}}\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]^{2}\bar{\lambda}(dy)=\operatorname{Var}(F)=1.$
(D.20)

Now by the modification (4.6) of Corollary 4.3 given in Remark 4.4 and since
$G\in L^{1}(\mathbb{P}_{\eta})$, we have for $p\in[1,2]$,

$\mathbb{E}|G|\leqslant\left(\mathbb{E}|G|^{p}\right)^{1/p}\leqslant\left(2^{2-p}\mathbb{E}\int_{\mathbb{Y}}\mathbb{E}[|\operatorname{D}_{x}G||\eta_{x}]^{p}\bar{\lambda}(dx)\right)^{1/p}.$
(D.21)

Let us now study the term

$\operatorname{D}_{x}G=\operatorname{D}_{x}\int_{\mathbb{Y}}\operatorname{D}_{y}F\,\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\bar{\lambda}(dy)$
(D.22)

and define

$h(\eta,y):=\operatorname{D}_{y}F\,\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}].$
(D.23)

We want to show that since $h\in L^{1}(\mathbf{N}\times\mathbb{Y})$,

$\left|\operatorname{D}_{x}\int_{\mathbb{Y}}h(\eta,y)\bar{\lambda}(dy)\right|\leqslant\int_{\mathbb{Y}}\left|\operatorname{D}_{x}h(\eta,y)\right|\bar{\lambda}(dy).$
(D.24)

and to do so we need an argument taken from the proof of [LPS16, Proposition
4.1]: Assume the RHS of (D.24) to be finite (if it is not, then the inequality
(D.24) is trivially true). Then

$\int_{\mathbb{Y}}|h(\eta+\delta_{x},y)|\bar{\lambda}(dy)\leqslant\int_{\mathbb{Y}}\left|\operatorname{D}_{x}h(\eta,y)\right|+|h(\eta,y)|\bar{\lambda}(dy)<\infty$
(D.25)

and hence

$\operatorname{D}_{x}\int_{\mathbb{Y}}h(\eta,y)\bar{\lambda}(dy)=\int_{\mathbb{Y}}\operatorname{D}_{x}h(\eta,y)\bar{\lambda}(dy).$
(D.26)

The inequality (D.24) follows. We have therefore shown

$\displaystyle\mathbb{E}|G|$
$\displaystyle\leqslant\left(2^{2-p}\mathbb{E}\int_{\mathbb{Y}}\mathbb{E}\left[\left.\int_{\mathbb{Y}}|\operatorname{D}_{x}h(\eta,y)|\bar{\lambda}(dy)\right|\eta_{x}\right]^{p}\bar{\lambda}(dx)\right)^{1/p}$
$\displaystyle=\left(2^{2-p}\int_{\mathbb{Y}}\mathbb{E}\left(\int_{\mathbb{Y}}\mathbb{E}\big{[}|\operatorname{D}_{x}h(\eta,y)|\big{|}\eta_{x}\big{]}\bar{\lambda}(dy)\right)^{p}\bar{\lambda}(dx)\right)^{1/p},$
(D.27)

where the second line follows from Tonelli’s theorem. By Minkowski’s integral
inequality,

$\mathbb{E}\left(\int_{\mathbb{Y}}\mathbb{E}\big{[}|\operatorname{D}_{x}h(\eta,y)|\big{|}\eta_{x}\big{]}\bar{\lambda}(dy)\right)^{p}\leqslant\left(\int_{\mathbb{Y}}\mathbb{E}\big{[}\mathbb{E}\big{[}|\operatorname{D}_{x}h(\eta,y)|\big{|}\eta_{x}\big{]}^{p}\big{]}^{1/p}\bar{\lambda}(dy)\right)^{p}.$
(D.28)

By the formula (2.4) for products,

$\displaystyle\operatorname{D}_{x}h(\eta,y)$
$\displaystyle=\operatorname{D}_{x}\left(\operatorname{D}_{y}F\,\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\right)$
$\displaystyle=\operatorname{D}^{(2)}_{x,y}F\cdot\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]+\operatorname{D}_{y}F\cdot\operatorname{D}_{x}\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]+\operatorname{D}^{(2)}_{x,y}F\cdot\operatorname{D}_{x}\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}].$
(D.29)

Since $\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]$ depends on $\eta$ only
through $\eta_{y}$, it follows that
$\operatorname{D}_{x}\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]=0$ whenever
$x\geqslant y$. Moreover, since $F\in L^{2}(\mathbb{P}_{\eta})$ implies that
$\operatorname{D}_{y}F,\operatorname{D}^{(2)}_{x,y}F\in
L^{1}(\mathbb{P}_{\eta})$ by [LP11b, Theorem 1.1], if $x<y$, we can put the
add-one cost operator inside the conditional expectation. Therefore

$\operatorname{D}_{x}\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]=\mathbbm{1}_{\\{x<y\\}}\mathbb{E}[\operatorname{D}^{(2)}_{x,y}F|\eta_{y}].$
(D.30)

By Minkowski’s norm inequality, it follows that

$\displaystyle\mathbb{E}\left[\mathbb{E}\big{[}|\operatorname{D}_{x}h(\eta,y)|\big{|}\eta_{x}\big{]}^{p}\right]^{1/p}$
$\displaystyle\leqslant\mathbb{E}\left[\mathbb{E}\big{[}|\operatorname{D}^{(2)}_{x,y}F\cdot\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]|\big{|}\eta_{x}\big{]}^{p}\right]^{1/p}$
$\displaystyle\phantom{\leqslant}+\mathbb{E}\left[\mathbb{E}\big{[}|\operatorname{D}_{y}F\cdot\mathbbm{1}_{\\{x<y\\}}\mathbb{E}[\operatorname{D}^{(2)}_{x,y}F|\eta_{y}]|\big{|}\eta_{x}\big{]}^{p}\right]^{1/p}$
$\displaystyle\phantom{\leqslant}+\mathbb{E}\left[\mathbb{E}\big{[}|\operatorname{D}^{(2)}_{x,y}F\cdot\mathbbm{1}_{\\{x<y\\}}\mathbb{E}[\operatorname{D}^{(2)}_{x,y}F|\eta_{y}]|\big{|}\eta_{x}\big{]}^{p}\right]^{1/p}.$
(D.31)

By the properties of conditional expectations and splitting the first term
into two parts, (D.31) is bounded by

$\displaystyle\mathbbm{1}_{\\{y\leqslant
x\\}}\mathbb{E}\left[|\mathbb{E}[|\operatorname{D}^{(2)}_{x,y}F||\eta_{x}]^{p}\cdot\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]|^{p}\right]^{1/p}$
$\displaystyle+\mathbbm{1}_{\\{x<y\\}}\mathbb{E}\left[\mathbb{E}\big{[}\mathbb{E}[|\operatorname{D}^{(2)}_{x,y}F||\eta_{y}]\cdot|\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]|\big{|}\eta_{x}\big{]}^{p}\right]^{1/p}$
$\displaystyle+\mathbbm{1}_{\\{x<y\\}}\mathbb{E}\left[\mathbb{E}\big{[}\mathbb{E}[|\operatorname{D}_{y}F||\eta_{y}]\cdot|\mathbb{E}[\operatorname{D}^{(2)}_{x,y}F|\eta_{y}]|\big{|}\eta_{x}\big{]}^{p}\right]^{1/p}$
$\displaystyle+\mathbbm{1}_{\\{x<y\\}}\mathbb{E}\left[\mathbb{E}\big{[}\mathbb{E}[|\operatorname{D}^{(2)}_{x,y}F||\eta_{y}]\cdot|\mathbb{E}[\operatorname{D}^{(2)}_{x,y}F|\eta_{y}]|\big{|}\eta_{x}\big{]}^{p}\right]^{1/p}.$
(D.32)

By an application of Jensen’s inequality (D.32) is now bounded by

$\displaystyle\mathbb{E}\left[|\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]|^{p}\cdot\mathbb{E}\big{[}|\operatorname{D}^{(2)}_{x,y}F|\big{|}\eta_{x\vee
y}\big{]}^{p}\right]^{1/p}$
$\displaystyle+\mathbbm{1}_{\\{x<y\\}}\mathbb{E}\left[\mathbb{E}\big{[}|\operatorname{D}_{y}F|\big{|}\eta_{y}\big{]}^{p}\cdot\big{|}\mathbb{E}\big{[}\operatorname{D}^{(2)}_{x,y}F\big{|}\eta_{y}\big{]}\big{|}^{p}\right]^{1/p}$
$\displaystyle+\mathbbm{1}_{\\{x<y\\}}\mathbb{E}\left[\mathbb{E}\big{[}|\operatorname{D}^{(2)}_{x,y}F|\big{|}\eta_{y}\big{]}^{2p}\right]^{1/p}.$
(D.33)

Plugging the conclusion from (D.28) - (D.33) into (D.27) and applying
Minkowski’s norm inequality again yields

$\displaystyle\mathbb{E}|G|$ $\displaystyle\leqslant
2^{2/p-1}\left(\int_{\mathbb{Y}}\left(\int_{\mathbb{Y}}\mathbb{E}\left[\big{|}\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\big{|}^{p}\cdot\mathbb{E}\big{[}|\operatorname{D}^{(2)}_{x,y}F|\big{|}\eta_{x\vee
y}\big{]}^{p}\right]^{1/p}\bar{\lambda}(dy)\right)^{p}\bar{\lambda}(dx)\right)^{1/p}$
$\displaystyle\phantom{\leqslant}+2^{2/p-1}\left(\int_{\mathbb{Y}}\left(\int_{\mathbb{Y}}\mathbbm{1}_{\\{x<y\\}}\mathbb{E}\left[\mathbb{E}\big{[}|\operatorname{D}_{y}F|\big{|}\eta_{y}\big{]}^{p}\cdot\big{|}\mathbb{E}[\operatorname{D}^{(2)}_{x,y}F|\eta_{y}]\big{|}^{p}\right]^{1/p}\bar{\lambda}(dy)\right)^{p}\bar{\lambda}(dx)\right)^{1/p}$
$\displaystyle\phantom{\leqslant}+2^{2/p-1}\left(\int_{\mathbb{Y}}\left(\int_{\mathbb{Y}}\mathbbm{1}_{\\{x<y\\}}\mathbb{E}\left[\mathbb{E}\big{[}|\operatorname{D}^{(2)}_{x,y}F|\big{|}\eta_{y}\big{]}^{2p}\right]^{1/p}\bar{\lambda}(dy)\right)^{p}\bar{\lambda}(dx)\right)^{1/p}$
(D.34)

which is in turn bounded by

$\displaystyle
2^{2/p}\left(\int_{\mathbb{Y}}\left(\int_{\mathbb{Y}}\mathbb{E}\left[\mathbb{E}\big{[}|\operatorname{D}_{y}F|\big{|}\eta_{y}\big{]}^{p}\cdot\mathbb{E}\big{[}|\operatorname{D}^{(2)}_{x,y}F|\big{|}\eta_{x\vee
y}\big{]}^{p}\right]^{1/p}\bar{\lambda}(dy)\right)^{p}\bar{\lambda}(dx)\right)^{1/p}$
$\displaystyle+2^{2/p-1}\left(\int_{\mathbb{Y}}\left(\int_{\mathbb{Y}}\mathbbm{1}_{\\{x<y\\}}\mathbb{E}\left[\mathbb{E}\big{[}|\operatorname{D}^{(2)}_{x,y}F|\big{|}\eta_{y}\big{]}^{2p}\right]^{1/p}\bar{\lambda}(dy)\right)^{p}\bar{\lambda}(dx)\right)^{1/p}.$
(D.35)

The result follows by an application of the Cauchy-Schwarz inequality.

Step 2. We want to show that

$\mathbb{E}\int_{\mathbb{Y}}\left|\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\right|\cdot|\operatorname{D}_{y}F|^{q}\bar{\lambda}(dy)\leqslant\beta_{3}+\beta_{4}.$
(D.36)

Again it suffices to show (D.36) for $F$ with $\mathbb{E}F=0$ and
$\mathbb{E}F^{2}=1$. Assume the $\beta_{3},\beta_{4}$ to be finite (otherwise
there is nothing to prove). Then, in particular

$\mathbb{E}\int_{\mathbb{Y}}\left|\mathbb{E}\left[\left.\operatorname{D}_{y}F\right|\eta_{y}\right]\right|^{q+1}\bar{\lambda}(dy)<\infty$
(D.37)

and we can add and subtract this term to the LHS of (D.36), yielding
$\beta_{3}$ and the following rest term:

$\displaystyle\left|\mathbb{E}\int_{\mathbb{Y}}\left|\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\right|\cdot|\operatorname{D}_{y}F|^{q}\bar{\lambda}(dy)-\mathbb{E}\int_{\mathbb{Y}}\left|\mathbb{E}\left[\left.\operatorname{D}_{y}F\right|\eta_{y}\right]\right|^{q+1}\bar{\lambda}(dy)\right|$
$\displaystyle=\left|\mathbb{E}\int_{\mathbb{Y}}\left|\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]\right|\cdot\mathbb{E}[|\operatorname{D}_{y}F|^{q}|\eta_{y}]\bar{\lambda}(dy)-\mathbb{E}\int_{\mathbb{Y}}\left|\mathbb{E}\left[\left.\operatorname{D}_{y}F\right|\eta_{y}\right]\right|^{q+1}\bar{\lambda}(dy)\right|$
$\displaystyle\leqslant\mathbb{E}\int_{\mathbb{Y}}|\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]|\cdot\big{|}\mathbb{E}[|\operatorname{D}_{y}F|^{q}|\eta_{y}]-\left|\mathbb{E}\left[\left.\operatorname{D}_{y}F\right|\eta_{y}\right]\right|^{q}\big{|}\bar{\lambda}(dy),$
(D.38)

where the equality is justified by the fact that $\mathbb{E}[G|\eta_{y}]$ is
defined for all $G\in L^{0}(\mathbb{P}_{\eta})$ which are integrable or non-
negative (see the definition (2.17)).

The term
$\mathbb{E}[|\operatorname{D}_{y}F|^{q}|\eta_{y}]-\left|\mathbb{E}\left[\left.\operatorname{D}_{y}F\right|\eta_{y}\right]\right|^{q}$
can be rewritten as

$\tilde{\mathbb{E}}|g_{y}(\chi)|^{q}-|\tilde{\mathbb{E}}g_{y}(\chi)|^{q},$
(D.39)

where $g_{y}(\chi)=\operatorname{D}_{y}F(\eta_{y}+\chi)$ and
$\tilde{\mathbb{E}}$ is the expectation with respect to $\chi$, a Poisson
measure on the space $\\{x\in\mathbb{Y}:x\geqslant y\\}$ with intensity
$\lambda\otimes ds$, independent of $\eta$. As
$F\in\operatorname{dom}\operatorname{D}$, it can be shown that $g_{y}\in
L^{2}(\mathbb{P}_{\chi})$ for a.e. $y\in\mathbb{Y}$.

By (4.5) in Corollary 4.3,

$\tilde{\mathbb{E}}|g_{y}(\chi)|^{q}-|\tilde{\mathbb{E}}g_{y}(\chi)|^{q}\leqslant
2^{2-q}\tilde{\mathbb{E}}\int_{\mathbb{Y}}\mathbbm{1}_{\\{y\leqslant
x\\}}\big{|}\tilde{\mathbb{E}}[\operatorname{D}_{x}g_{y}(\chi)|\chi_{x}]\big{|}^{q}\bar{\lambda}(dx).$
(D.40)

Plugging (D.40) into (D.38), we deduce that the RHS of (D.38) is upper bounded
by

$2^{2-q}\mathbb{E}\int_{\mathbb{Y}}|\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]|\int_{\mathbb{Y}}\mathbbm{1}_{\\{y\leqslant
x\\}}\tilde{\mathbb{E}}\big{|}\tilde{\mathbb{E}}[\operatorname{D}_{x}g_{y}(\chi)|\chi_{x}]\big{|}^{q}\bar{\lambda}(dx)\bar{\lambda}(dy).$
(D.41)

Since $\chi$ and $\eta$ are independent and have the same intensity measure,

$\mathbbm{1}_{\\{y\leqslant
x\\}}|\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]|\cdot\tilde{\mathbb{E}}\big{|}\tilde{\mathbb{E}}[\operatorname{D}_{x}g_{y}(\chi)|\chi_{x}]\big{|}^{q}$
(D.42)

has the same law as

$\mathbbm{1}_{\\{y\leqslant
x\\}}|\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]|\cdot\mathbb{E}\big{[}|\mathbb{E}[\operatorname{D}^{(2)}_{x,y}F|\eta_{x}]|^{p}\big{|}\eta_{y}\big{]}.$
(D.43)

This implies finally that the RHS of (D.38) is upper bounded by

$2^{2-q}\mathbb{E}\int_{\mathbb{Y}}\int_{\mathbb{Y}}\mathbbm{1}_{\\{y\leqslant
x\\}}|\mathbb{E}[\operatorname{D}_{y}F|\eta_{y}]|\cdot|\mathbb{E}[\operatorname{D}^{(2)}_{x,y}F|\eta_{x}]|^{q}\bar{\lambda}(dx)\bar{\lambda}(dy)$
(D.44)

and the result follows by the Cauchy-Schwarz inequality. ∎

###### Proof of Theorem 3.4.

It suffices to prove this result for $F$ such that $\mathbb{E}F=0$ and
$\sigma=1$. If Theorem 3.4 holds for all $\sigma$-finite spaces $\mathbb{X}$,
it holds in particular for the space $\mathbb{X}\times[0,1]$. In fact, it
suffices to prove Theorem 3.4 for functionals $F\in
L^{2}(\mathbb{P}_{\eta})\cap\operatorname{dom}\operatorname{D}$, where $\eta$
is a $(\mathbb{X}\times[0,1],\lambda\otimes ds)$-Poisson measure. Indeed, as
discussed in Section 2, we can regard any functional $F\in
L^{2}(\mathbb{P}_{\chi})$, with the $(\mathbb{X},\lambda)$-Poisson measure
$\chi$, as a functional of $\eta$ without changing the law of $F$ or its add-
one costs. If we replace $\mathbb{X}$ by $\mathbb{X}\times[0,1]$ and $\chi$ by