# Inference and forecasting for continuous-time integer-valued trawl processes

Mikkel Bennedsen, Asger Lunde, Neil Shephard, Almut E. D. Veraart  Department
of Economics and Business Economics and CREATES, Aarhus University, Fuglesangs
Allé 4, 8210 Aarhus V, Denmark. E-mail<EMAIL_ADDRESS>Copenhagen
Economics, Langebrogade 1B, 1411 Copenhagen K, Denmark, and CREATES, Aarhus
University, Fuglesangs Allé 4, 8210 Aarhus V, Denmark. E-mail:
<EMAIL_ADDRESS>Department of Economics and Department of
Statistics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA.
E-mail<EMAIL_ADDRESS>Department of Mathematics, Imperial College
London, South Kensington Campus, London SW7 2AZ, UK and CREATES, Aarhus
University. E-mail<EMAIL_ADDRESS>
###### Abstract

This paper develops likelihood-based methods for estimation, inference, model
selection, and forecasting of continuous-time integer-valued trawl processes.
The full likelihood of integer-valued trawl processes is, in general, highly
intractable, motivating the use of composite likelihood methods, where we
consider the pairwise likelihood in lieu of the full likelihood. Maximizing
the pairwise likelihood of the data yields an estimator of the parameter
vector of the model, and we prove consistency and, in the short memory case,
asymptotic normality of this estimator. When the underlying trawl process has
long memory, the asymptotic behaviour of the estimator is more involved; we
present some partial results for this case. The pairwise approach further
allows us to develop probabilistic forecasting methods, which can be used to
construct the predictive distribution of integer-valued time series. In a
simulation study, we document the good finite sample performance of the
likelihood-based estimator and the associated model selection procedure.
Lastly, the methods are illustrated in an application to modelling and
forecasting financial bid-ask spread data, where we find that it is beneficial
to carefully model both the marginal distribution and the autocorrelation
structure of the data.

Keywords: Count data; Lévy basis; pairwise likelihood; estimation; model
selection; forecasting.

JEL Codes: C13; C51; C52; C53; C58.

##  1 Introduction

In this paper, we develop likelihood-based methods for estimation, inference,
model selection, and forecasting of continuous-time integer-valued trawl (IVT)
processes. IVT processes, introduced in Barndorff-Nielsen et al. (2014), are a
flexible class of integer-valued, serially correlated, stationary, and
infinitely divisible continuous-time stochastic processes. In general,
however, IVT processes are not Markovian, which implies that the structure of
the full likelihood of an IVT process is highly intractable (Shephard & Yang,
2016). This is the impetus of the present paper, where we propose to use
composite likelihood (CL, Lindsay, 1988) methods for estimation and inference.
Specifically, we propose to estimate the parameters of an IVT model by
maximizing the pairwise likelihood of the data. CL methods in general, and the
pairwise likelihood approach in particular, have been successfully used in
many applications, such as statistical genetics (Larribe & Fearnhead, 2011),
geostatistics (Hjort & Omre, 1994), and finance (Engle et al., 2020). See
Varin et al. (2011) for an excellent overview of CL methods. Although the
theory behind CL estimation is quite well understood in the case of iid
observations (e.g. Cox & Reid, 2004; Varin & Vidoni, 2005), the time series
case, which is what we consider here, generally requires separate treatment
(Varin et al., 2011, p. 11). For instance, Davis & Yau (2011) develops the
theory of CL estimators in the setting of linear Gaussian time series models,
while Chen et al. (2016) and Ng et al. (2011) consider CL methods for a hidden
Markov model and a time series model with a latent autoregressive process,
respectively. Also, Sørensen (2019) develops a two-step CL estimation method
for parameter-driven count time series models with covariates. Our paper adds
to the literature on CL methods for time series models by deriving the
theoretical properties (consistency, asymptotic normality) of a pairwise CL
estimator applied to IVT models.

A central feature of IVT processes is that they allow for specifying the
correlation structure of the model separately from the marginal distribution
of the model, making them flexible and well-suited for modelling count- or
integer-valued data. In particular, the marginal distribution of an IVT
process can be any integer-valued infinitely divisible distribution, while the
correlation structure can be specified independently using a so-called trawl
function. This setup allows for both short- and long-memory of the IVT
process. So far, IVT processes have been applied to financial data (Barndorff-
Nielsen et al., 2014; Shephard & Yang, 2017; Veraart, 2019) and real-valued
trawl processes to the modelling of extreme events in environmental time
series (Noven et al., 2018). IVT processes are, under weak conditions,
stationary and ergodic, which motivated Barndorff-Nielsen et al. (2014) to
suggest a method-of-moments-based estimator for the parameters of the IVT
model. This method-of-moments-based estimator has been used in most applied
work using IVT processes (e.g. Barndorff-Nielsen et al., 2014; Shephard &
Yang, 2017; Veraart, 2019). Exceptions are Shephard & Yang (2016) and Noven et
al. (2018). In Noven et al. (2018), a pairwise likelihood was used for a
hierarchical model involving a latent (Gamma-distributed) trawl process and
the corresponding asymptotic theory was derived in Courgeau & Veraart (2021).
However, the asymptotic theory for inference for integer-valued trawl
processes which are observed directly is not covered by these earlier papers.
In Shephard & Yang (2016), the authors derive a prediction decomposition of
the likelihood function of a particularly simple IVT process, the so-called
Poisson-Exponential IVT process, allowing them to conduct likelihood-based
estimation and inference. Although the likelihood estimation method developed
in Shephard & Yang (2016) theoretically applies to more general IVT processes,
the computational burden quickly becomes overwhelming in these scenarios,
making estimation by classical maximum likelihood methods infeasible in
practice.

The contributions of this paper can be summarized as follows. First, we derive
the theoretical mixing properties of IVT processes. Using these, we prove
consistency and, in the short memory case, asymptotic normality of the maximum
composite likelihood (MCL) estimator of the parameter vector of an IVT model.
We discuss the long memory case and, based on a result about the asymptotic
behaviour of partial sums of IVT processes, conjecture that the MCL estimator
has an $\alpha$-stable limit with infinite variance in this case. For the
purpose of conducting feasible inference and model selection, we propose two
alternative estimators of the asymptotic variance of the MCL estimator in the
short memory case: a kernel-based estimator, inspired by the heteroskedastic
and autocorrelation consistent (HAC) estimator of Newey & West (1987), and a
simulation-based estimator. Second, we use the same principle of considering
the pairwise likelihood in lieu of the full likelihood, to derive the
predictive distribution of an IVT model, conditional on the current value of
the process; this allows us to use the IVT framework for forecasting integer-
valued data. In a simulation study, we compare the MCL estimator to the
standard method-of-moments-based estimator suggested in Barndorff-Nielsen et
al. (2014) and find that the MCL estimator provides substantial improvements
in most cases. Indeed, in a realistic simulation setup, we find that the MCL
estimator can improve on the method-of-moments-based estimator by more than
$50\%$, in terms of finite sample root median squared error. Since the
asymptotic theory for (G)MM estimation of trawl processes has not been worked
out elsewhere, we also derive the asymptotic theory for GMM estimation and
present the results for comparison purposes in the Supplementary Material, see
Section S12.

We apply the methods developed in the paper to a time series of the bid-ask
spread of a financial asset. The time series behaviour of the bid-ask spread
has been extensively studied in the literature on the theory of the
microstructure of financial markets (e.g. Huang & Stoll, 1997; Bollen et al.,
2004). The model selection procedure developed in the paper indicates that a
model with Negative Binomial marginal distribution and slowly decaying
autocorrelations most adequately describe the data. These findings are in line
with those of Groß-KlußMann & Hautsch (2013), who also found strong
persistence in bid-ask spread time series. Then, in a pseudo-out-of-sample
forecast exercise, we find that it is important to carefully model both the
marginal distribution and the autocorrelation structure to get accurate
forecasts of the future bid-ask spread. These findings highlight the strength
of modelling using a framework where the choice of marginal distribution can
be made independently of the choice of autocorrelation structure.

The rest of the paper is structured as follows. Section 2 outlines the
mathematical setup of IVT processes, while Section 3 contains details on the
estimation and model selection procedures. Section 4 presents the theory
behind our proposed forecasting approach. Section 5 summarises the results
from our simulation study, investigating the finite sample properties of the
estimation and model selection procedures. Section 6 illustrates the use of
the new methodology in an empirical application to financial bid-ask spread
data. Section 7 concludes. The proofs of the main mathematical results are
given in an Appendix. Practical details on the implementation of the
asymptotic theory and additional derivations are given in the Supplementary
Material, which also contains further simulation results and extensive details
on various calculations used in the implementation of the methods. A software
package for the implementation of simulation, estimation, inference, model
selection, and forecasting of IVT processes is freely available in the MATLAB
programming language.111The software package can be found at
https://github.com/mbennedsen/Likelihood-based-IVT.

##  2 Integer-valued trawl processes

Let $(\Omega,\mathcal{F},\mathbb{P})$ denote a probability space, supporting a
Poisson random measure $N$, defined on
$\mathbb{Z}\times[0,1]\times\mathbb{R}$, with mean (intensity) measure
$\eta\otimes Leb\otimes Leb$. Throughout $Leb$ denotes the Lebesgue measure
and $\eta$ is a Lévy measure. A _Lévy basis_ $L$ is a homogeneous and
independently scattered random measure on $[0,1]\times\mathbb{R}$, defined as

$\displaystyle
L(dx,ds):=\int_{-\infty}^{\infty}yN(dy,dx,ds),\quad(x,s)\in[0,1]\times\mathbb{R}.$
(2.1)

See, e.g., Rajput & Rosinski (1989) and Barndorff-Nielsen (2011) for further
details on Lévy bases. Since we are only interested in integer-valued Lévy
bases, we will work under the following assumption.

######  Assumption 2.1.

The Lévy basis $L$ is given by (2.1) with Lévy measure $\eta$, concentrated on
the integers ($y\in\mathbb{Z}$), such that
$\|\eta\|:=\sum_{y=-\infty}^{\infty}y^{2}\eta(y)<\infty$.

The Lévy basis $L$ is an infinitely divisible random measure with cumulant
(log-characteristic) function

$C_{L(dx,ds)}(\theta):=\log\mathbb{E}[\exp(i\theta
L(dx,ds))]=\int_{-\infty}^{\infty}\left(e^{i\theta
y}-1\right)\eta(dy)dxds,\quad(x,s)\in[0,1]\times\mathbb{R}.$

An important random variable associated with the Lévy basis $L,$ is the so-
called _Lévy seed_ , $L^{\prime}$, which we define as the random variable
$L^{\prime}$ satisfying $\mathbb{E}[\exp(i\theta
L^{\prime})]=\exp(C_{L^{\prime}}(\theta)),$ with
$C_{L^{\prime}}(\theta)=\sum_{y=-\infty}^{\infty}\left(e^{i\theta
y}-1\right)\eta(y)$.

######  Remark 2.1.

Because the distribution of a Lévy process is entirely determined by its
distribution at a particular time point, we can specify a Lévy process
$L_{t}^{\prime}$ from a Lévy seed $L^{\prime}$, by requiring that
$L_{1}^{\prime}\sim L^{\prime}$.

Using the Lévy seed, we can rewrite the cumulant function of the Lévy basis as
$C_{L(dx,ds)}(\theta)=C_{L^{\prime}}(\theta)dxds$, or, for a Borel set
$B\in\mathcal{B}([0,1]\times\mathbb{R}),$

$\displaystyle C_{L(B)}(\theta)=C_{L^{\prime}}(\theta)Leb(B).$ (2.2)

From (2.2) we have that $\kappa_{j}(L(B))=\kappa_{j}(L^{\prime})Leb(B)$,
$j\geq 0$, where $\kappa_{j}(Z)$ denotes the $j$th cumulant of the random
variable $Z$, when it exists.222Recall that the cumulants $\kappa_{j}(Z)$ of
the random variable $Z$ are defined implicitly through the power series
expansion of the cumulant function of $Z$, i.e.,
$C_{Z}(\theta)=\log\mathbb{E}[\exp(i\theta
Z)]=\sum_{j=1}^{\infty}\kappa_{j}(Z)(i\theta)^{j}/j!$. In particular
$\mathbb{E}[L(B)]=\mathbb{E}[L^{\prime}]Leb(B)$, and
$Var(L(B))=Var(L^{\prime})Leb(B)$. The relationship (2.2) implies that the
distribution of the random variable $L(B)$ is entirely specified by the Lévy
seed $L^{\prime}$ and the Lebesgue measure of the set $B$. In Section 2.1
below, we illustrate how this can be used to construct trawl processes with a
given marginal distribution.

The Lévy basis $L$ acts on sets in $\mathcal{B}([0,1]\times\mathbb{R})$. We
restrict attention to _trawl sets_ of the form

$\displaystyle A_{t}=A+(0,t),\quad A=\\{(x,s):s\leq 0,0\leq x<d(s)\\},\quad
t\geq 0,$ (2.3)

where $d$ is a _trawl function_ , determining the shape of the trawl set
$A_{t}$. Section 2.2 contains several parametric examples for the trawl
function $d$. We will impose the following assumption.

######  Assumption 2.2.

The trawl set $A_{t}$ is given by (2.3), where the trawl function
$d:\mathbb{R}_{-}\to[0,1]$ is continuous and monotonically increasing such
that $Leb(A)=\int_{-\infty}^{0}d(s)<\infty$.

Intuitively, $A_{t}$ is obtained from the set $A$ by “dragging” it along in
time. Note in particular that $Leb(A_{t})=Leb(A)$ for all $t$. Finally, define
the IVT process $X=(X_{t})_{t\geq 0}$ as the Lévy basis evaluated over the
trawl set:

$\displaystyle X_{t}:=L(A_{t}),\quad t\geq 0.$ (2.4)

###  2.1 Modelling the marginal distribution

For an IVT process $X$ as defined in (2.4), we have
$C_{X_{t}}(\theta)=C_{L(A_{t})}(\theta)=Leb(A)C_{L^{\prime}}(\theta)=C_{L^{\prime}_{Leb(A)}}(\theta)$,
where $L^{\prime}_{t}$ is a Lévy process with $L^{\prime}_{1}\sim L^{\prime}$.
Hence we observe that the marginal distribution of the IVT process $X_{t}$ is
entirely decided by the Lebesgue measure of the trawl set $A$ and the Lévy
seed $L^{\prime}$ of the underlying Lévy basis $L$. Indeed, by specifying a
distribution for $L^{\prime}$, we can build IVT processes with the
corresponding marginal distribution. The following two examples illustrate how
to do this; additional details can be found in the Supplementary Material.

######  Example 2.1 (Poissonian Lévy seed).

Let $L^{\prime}\sim\textnormal{Poisson}(\nu)$, i.e. $L^{\prime}$ is
distributed as a Poisson random variable with intensity $\nu>0$. It follows
from standard properties of the Poisson distribution that
$X_{t}\sim\textnormal{Poisson}(\nu Leb(A))$. In other words, for all $t\geq
0$, $P\left(X_{t}=x\right)=(\nu Leb(A))^{x}e^{-\nu Leb(A)}/x!$,
$x=0,1,2,\ldots$.

######  Example 2.2 (Negative Binomial Lévy seed).

Let $L^{\prime}\sim\textnormal{NB}(m,p)$, i.e. $L^{\prime}$ is distributed as
a Negative Binomial random variable with parameters $m>0$ and $p\in[0,1]$. It
follows from standard properties of the Negative Binomial distribution that
$X_{t}\sim\textnormal{NB}(mLeb(A),p)$. In other words, for all $t\geq 0$,
$P(X_{t}=x)=\frac{\Gamma(Leb(A)m+x)}{x!\Gamma(Leb(A)m)}(1-p)^{Leb(A)m}p^{x}$,
$x=0,1,2,\ldots$, where $\Gamma(z)=\int_{0}^{\infty}y^{z-1}e^{-y}dy$ for $z>0$
is the $\Gamma$-function.

###  2.2 Modelling the correlation structure

Recall that the shape of the trawl set $A_{t}$ is determined by the trawl
function $d$, see Equation (2.3). A particularly tractable and flexible class
of parametrically specified trawl functions are the so-called _superposition
trawls_ (Barndorff-Nielsen et al., 2014; Shephard & Yang, 2017). They are
defined as $d(s):=\int_{0}^{\infty}e^{\lambda s}\pi(d\lambda)$, for $s\leq 0$,
where $\pi$ is a probability measure on $\mathbb{R}_{+}.$ This construction
essentially randomizes the decay parameter $\lambda$ in an otherwise
exponential function.

The IVT process with a superposition trawl function is stationary. Hence, we
get the autocorrelation function (Barndorff-Nielsen et al., 2014)

$\displaystyle\rho(h):=Corr(L(A_{t+h}),L(A_{t}))=\frac{Leb(A_{h}\cap
A)}{Leb(A)}=\frac{\int_{h}^{\infty}d(-s)ds}{\int_{0}^{\infty}d(-s)ds},\quad
h>0.$ (2.5)

######  Example 2.3 (Exponential trawl function).

For the case where the measure $\pi$ has an atom at $\lambda>0,$ i.e.
$\pi(dx)=\delta_{\lambda}(dx),$ where $\delta_{x}(\cdot)$ is the Dirac delta
function at $x\in\mathbb{R}_{+},$ we get $d(s)=e^{\lambda s}$ for $s\leq 0$.
Consequently, $\rho(h)=\exp(-\lambda h)$, for $h\geq 0$.

######  Example 2.4 (Inverse Gaussian trawl function).

Letting $\pi$ be given by the inverse Gaussian distribution
$\pi(dx)=\frac{(\gamma/\delta)^{1/2}}{2K_{1/2}(\delta\gamma)}x^{-1/2}\exp\left(-\frac{1}{2}(\delta^{2}x^{-1}+\gamma^{2}x)\right)dx$,
where $K_{\nu}(\cdot)$ is the modified Bessel function of the third kind and
$\gamma,\delta\geq 0$ with both not zero simultaneously. It can be shown that
the resulting trawl function is given by
$d(s)=\left(1-\frac{2s}{\gamma^{2}}\right)^{-1/2}\exp\left(\delta\gamma\left(1-\sqrt{1-\frac{2s}{\gamma^{2}}}\right)\right)$,
for $s\leq 0$, and hence that the correlation function of the IVT process with
inverse Gaussian trawl function becomes
$\rho(h)=Corr(X_{t+h},X_{t})=\exp\left(-\delta\gamma(\sqrt{1+2h/\gamma^{2}}-1)\right)$,
for $h\geq 0$. The details on these calculations can be found in the
Supplementary Material.

######  Example 2.5 (Gamma trawl function).

Let $\pi$ have the $\Gamma(1+H,\alpha)$ density,
$\pi(dx)=\frac{1}{\Gamma(1+H)}\alpha^{1+H}\lambda^{H}e^{-\lambda\alpha}dx$,
where $\alpha>0$ and $H>0.$ We can show that
$d(s)=\left(1-\frac{s}{\alpha}\right)^{-(H+1)}$, $s\leq 0$, which implies the
correlation function $\rho(h)=Corr(X_{t+h},X_{t})=\frac{Leb(A_{h}\cap
A)}{Leb(A)}=\left(1+\frac{h}{\alpha}\right)^{-H}$. Note that in this case
$\int_{0}^{\infty}\rho(h)dh=\infty$ for $H\in(0,1]$ and
$\int_{0}^{\infty}\rho(h)dh=\alpha(H-1)^{-1}$ for $H>1$, from which we see
that an IVT process with a Gamma trawl function enjoys the long memory
property, in the sense of a non-integrable autocorrelation function, when
$H\in(0,1].$ The details on these calculations can be found in the
Supplementary Material.

###  2.3 Modelling IVT processes

Using the above methods, we can build flexible continuous-time integer-valued
processes with a marginal distribution determined by the underlying Lévy
basis, and independently specified correlation structure determined by the
trawl function. In our main examples given above, we considered a Lévy basis
with Poisson or Negative Binomial marginals, and various trawl functions,
namely the Exponential trawl function, the IG trawl function, and the Gamma
trawl function. Other specifications for the underlying Lévy basis and trawl
function than those given here could of course be considered. In practice,
these choices should be guided by the properties of the data being modelled.

The simplest IVT process we can construct in this way is the Poisson-
Exponential IVT process, i.e., the case where
$L^{\prime}\sim\textnormal{Poisson}(\nu)$ and $d(s)=\exp(\lambda s)$, $s\leq
0$, see Examples 2.1 and 2.3. This special case results in a Markovian
process, which is not in general true of IVT processes (Barndorff-Nielsen et
al., 2014). In fact, the model is similar to the popular Poissonian INAR(1)
model, introduced in McKenzie (1985) and Al-Osh & Alzaid (1987). An
illustration of the exponential trawl set, $A_{t},$ dragged through time,
together with a simulation of the resulting Poisson-Exponential IVT trawl
process $X_{t}=L(A_{t})$, is seen in Figure 1. The parameters used are
$\lambda=1$ and $\nu=5$. At each time point $t$, the value of $X_{t}$ (bottom
plot) is the number of points inside the trawl set $A_{t}$ (top plot).

Figure 1: Top: Simulation of a Poisson Lévy basis on $\mathbb{R}\times[0,1]$
(black dots) with an exponential trawl set $A_{t}$ (shaded) superimposed at
three periods in time, $t\in\\{0,7,8\\}$. Bottom: The associated trawl process
$X_{t}=L(A_{t})$, given by the number of ‘points’ inside the trawl set $A_{t}$
at time $t$. The intensity of the Poisson random measure is
$\|\eta\|=\eta(1)=\nu=5$ and the parameter controlling the exponential trawl
function, $d(s)=\exp(\lambda s)$, is $\lambda=1.$

##  3 Estimation of integer-valued trawl processes

Barndorff-Nielsen et al. (2014) showed that the parameter vector $\theta$ of
an IVT process can be consistently estimated using a generalized method of
moments (GMM) procedure. In Section 3.1, we propose a likelihood-based
approach instead. Both estimation procedures rely on the fact that the IVT
process is stationary and mixing. The mixing property of IVT processes is
obtained from results given in Fuchs & Stelzer (2013), see Barndorff-Nielsen
et al. (2014, p. 699). Although mixing, in general, is sufficient for the
consistency of the estimators, the central limit theorem for the likelihood-
based estimator (Theorem 3.3 below) relies on the stronger mixing concept of
$\alpha$-mixing, where the size (or rate) of mixing can also be established.
Let us recall the definition of $\alpha$-mixing for a stationary process. Let
$\mathcal{F}_{-\infty}^{0}=\sigma(X_{t};t\leq 0)$ and, for $m>0$,
$\mathcal{F}_{m}^{\infty}=\sigma(X_{t};t\geq m)$, and define the numbers
$\alpha_{m}:=\sup_{G\in\mathcal{F}_{-\infty}^{0},H\in\mathcal{F}_{m}^{\infty}}|\mathbb{P}(H\cap
G)-\mathbb{P}(H)\mathbb{P}(G)|$, for $m>0$. The process
$X=(X_{t})_{t\in\mathbb{R}}$ is $\alpha$-mixing if $\alpha_{m}\rightarrow 0$
as $m\rightarrow\infty$. It is $\alpha$-mixing of size $-\phi_{0}$ if
$\alpha_{m}=O(m^{-\phi})$, as $m\rightarrow\infty$, for some $\phi>\phi_{0}$.

We obtain the following important property for IVT processes.

######  Theorem 3.1.

Let the IVT process $X$ be given by (2.4) and let Assumptions 2.1 and 2.2
hold. Now, $X$ is $\alpha$-mixing with $\alpha_{m}=O(\rho(m))$ as
$m\rightarrow\infty$, where $\rho(m)$ is the autocorrelation function of $X$.

######  Remark 3.1.

The autocorrelation functions of the Exponential (Example 2.3) and IG (Example
2.4) IVT models imply that these models are in fact $\alpha$-mixing with an
exponential decay rate. The autocorrelation function of the Gamma (Example
2.5) IVT model implies that it is $\alpha$-mixing of size $-(H-\epsilon)$ for
all $\epsilon>0$.

######  Remark 3.2.

As an alternative to the proof of Theorem 3.1 provided in Appendix A, we could
first show that trawl processes are $\theta$-weakly dependent, which we do in
the Supplementary Material, see Section S12. Then, as pointed out in Curato &
Stelzer (2019, p. 324) and shown in the discrete-time case in Doukhan et al.
(2012), for integer-valued trawl processes, the fact that they are
$\theta$-weakly dependent, implies that they are strongly mixing with the
coefficient as stated in Theorem 3.1.

###  3.1 Estimation by composite likelihoods

Due to the non-Markovianity of the IVT process, we face computational
difficulties when attempting to estimate the model by maximizing the full
likelihood, hence we propose to use the CL method instead. The main idea
behind the CL approach is to consider a, possibly misspecified, likelihood
function which captures the salient features of the data at hand; here this
means capturing the features of the Lévy basis, controlling the marginal
distribution, and those of the trawl function, controlling the dependence
structure. We focus on pairwise CLs.

####  3.1.1 Pairwise composite likelihood

Suppose we have $n\in\mathbb{N}$ observations of the IVT process $X,$
$x_{1},\ldots,x_{n},$ on an equidistant grid of size $\Delta=T/n$, for some
$T>0$. Define the following likelihood function using pairs of observations
$k$ periods apart,

$\displaystyle
CL^{(k)}(\theta;x):=\prod_{i=1}^{n-k}f(x_{i+k},x_{i};\theta),\quad k\geq 1,$
(3.1)

where $f(x_{i+k},x_{i};\theta)$ is the joint probability mass function (PMF)
of the observations $x_{i}$ and $x_{i+k}$, parametrized by the vector
$\theta$. From (3.1), we construct the composite likelihood function

$\displaystyle\mathcal{L}_{CL}(\theta;x):=\mathcal{L}_{CL}^{(K)}(\theta;x):=\prod_{k=1}^{K}CL^{(k)}(\theta;x)=\prod_{k=1}^{K}\prod_{i=1}^{n-k}f(x_{i+k},x_{i};\theta),$
(3.2)

where $K\in\mathbb{N}$ denotes the number of pairwise likelihoods to include
in the calculation of the composite likelihood function.

The maximum composite likelihood (MCL) estimator of $\theta$ is defined as

$\displaystyle\hat{\theta}^{CL}:=\arg\max_{\theta\in\Theta}l_{CL}(\theta;x),$
(3.3)

where $\Theta$ is the parameter space and
$l_{CL}(\theta;x):=\log\mathcal{L}_{CL}(\theta;x)$ is the log composite
likelihood function. To apply this estimator in practice, we need to be able
to calculate the PMFs $f(x_{i+k},x_{i})$. Section S5 in the Supplementary
Material contains a discussion on how to do this in the general integer-valued
case. In the count-valued case, $f(x_{i+k},x_{i})$ takes a particularly simple
form which is convenient in implementations. Indeed, letting
$\mathbb{P}_{\theta}(B)$ denote the probability of the event $B$ given
parameters $\theta$, we have the following.

######  Proposition 3.1.

Let the IVT process $X$ be given by (2.4) and let Assumptions 2.1 and 2.2
hold. Suppose further, that the Lévy basis $L$ is non-negative, i.e.
$\eta(y)=0$ for $y<0$. The joint PMF of two observations $x_{i+k}$ and $x_{i}$
is

$\displaystyle f(x_{i+k},x_{i};\theta)=\sum_{c=0}^{\min\\{x_{i+k},x_{i}\\}}$
$\displaystyle\mathbb{P}_{\theta}\left(L(A_{(i+k)\Delta}\setminus
A_{i\Delta})=x_{i+k}-c\right)\mathbb{P}_{\theta}\left(L(A_{i\Delta}\setminus
A_{(i+k)\Delta})=x_{i}-c\right)$
$\displaystyle\cdot\mathbb{P}_{\theta}\left(L(A_{(i+k)\Delta}\cap
A_{i\Delta})=c\right).$ (3.4)

The probabilities $\mathbb{P}_{\theta}(\cdot)$ in (3.4) can be expressed as a
function of the parameters of the Lévy seed and the trawl function. Indeed,
for a Borel set $B\in\mathcal{B}([0,1]\times\mathbb{R})$ we have
$\mathbb{P}_{\theta}(L(B)=x)=\mathbb{P}_{\theta}(L^{\prime}_{Leb(B)}=x)$,
where $L^{\prime}_{t}$ is a Lévy process with $L_{1}^{\prime}\sim L^{\prime}$,
and $L^{\prime}$ being the Lévy seed associated to $X$, see Remark 2.1. Also,
$Leb(A_{(i+k)\Delta}\cap A_{i\Delta})=\int_{-\infty}^{-k\Delta}d(s)ds$, and
$Leb(A_{(i+k)\Delta}\setminus A_{i\Delta})=Leb(A_{i\Delta}\setminus
A_{(i+k)\Delta})=Leb(A)-Leb(A_{(i+k)\Delta}\cap
A_{i\Delta})=\int_{-k\Delta}^{0}d(s)ds$. Plugging these into (3.4) we obtain
the pairwise likelihoods, $f(x_{i+k},x_{i};\theta)$, and thus the CL function,
$\mathcal{L}_{CL}(\theta;x)$, as a function of $\theta$.

######  Example 3.1 (Poisson-Exponential IVT process).

Let $L^{\prime}\sim\textnormal{Poisson}(\nu)$ and $d(s)=\exp(\lambda s)$,
$s\leq 0$, for some $\nu,\lambda>0$. Since
$L^{\prime}\sim\textnormal{Poisson}(\nu)$ we have
$L(B)\sim\textnormal{Poisson}(Leb(B)\nu)$ for Borel sets $B$ and hence
$\mathbb{P}_{\theta}\left(L(B)=x\right)=(\nu Leb(B))^{x}e^{-\nu Leb(B)}/x!$,
for $x\geq 0$. Further, it is not difficult to show that
$Leb(A_{(i+k)\Delta}\cap A_{i\Delta})=\lambda^{-1}e^{-\lambda k\Delta}$ and
$Leb(A_{(i+k)\Delta}\setminus A_{i\Delta})=\lambda^{-1}(1-e^{-\lambda
k\Delta})$. Using this, the probabilities in (3.4) can be expressed as
functions of $\nu$ and $\lambda$ and hence the maximization (3.3) can be
carried out using standard numerical methods.

####  3.1.2 Asymptotic theory

Because we are only considering dependencies across pairs of observations and
not their dependence with the remaining observations, the pairwise composite
likelihood function (3.2) can be viewed as a misspecified likelihood.
Nonetheless, since the individual PMFs $f(x_{i+k},x_{i};\theta)$ in (3.2) are
proper bivariate PMFs, the _composite score function_ $\partial
l_{CL}(\theta;x)/\partial\theta$ provides unbiased estimating equations and,
under certain regularity assumptions, the usual asymptotic results will apply
(Cox & Reid, 2004). However, as pointed out in Varin et al. (2011), formally
proving the results in the time series case requires more rigorous treatment.
The following two theorems provide the details on the asymptotic theory in the
setup of this paper. We will work under the following identification
assumption.

######  Assumption 3.1.

For all $\theta\in\Theta$, it holds that

$\displaystyle\theta\neq\theta_{0}\Rightarrow\sum_{k=1}^{K}f(x_{1},x_{2};\theta)\neq\sum_{k=1}^{K}f(x_{1},x_{2};\theta_{0})$
(3.5)

for some $x_{1},x_{2}\in\mathbb{Z}$.

First, we have a Law of Large Numbers.

######  Theorem 3.2.

Fix $K\in\mathbb{N}$, let the IVT process $X$ be given by (2.4), and let
Assumptions 2.1–2.2 and 3.1 hold. Then
$\hat{\theta}_{CL}\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}\theta_{0}$,
as $n\rightarrow\infty$.

######  Remark 3.3.

As is often the case, the identification condition in Assumption 3.1 can be
difficult to check in practice. For the IVT processes considered in this paper
and presented in the examples above, our numerical experiments indicate that
requiring $K\geq\textnormal{dim}(\theta_{d})$, where
$\textnormal{dim}(\theta_{d})$ denotes the dimension of the parameters
controlling the trawl function $d$, results in $\theta_{0}$ being identified.
A similar requirement was suggested in Davis & Yau (2011).

We also impose a standard assumption on the parameter space.

######  Assumption 3.2.

The set $\Theta$ is compact such that the true parameter vector, $\theta_{0}$,
lies in the interior of $\Theta$.

It turns out that the asymptotic behaviour of the MCL estimator differs in the
short- and long-memory cases. The former is captured by the following
assumption.

######  Assumption 3.3 (Short memory).

The autocorrelation function of the IVT process satisfies
$\lim_{n\rightarrow\infty}\rho(n)n=0$.

######  Remark 3.4.

Assumption 3.3 is satisfied by IVT processes with the Exponential trawl
(Example 2.3), the Inverse Gaussian trawl (Example 2.4), and the Gamma trawl
(Example 2.5) with $H>1$.

Under this assumption, the mixing property of IVT processes, presented in
Theorem 3.1, implies that we can invoke a Central Limit Theorem for triangular
arrays of mixing processes (Davidson, 1994, Corollary 24.7) to get the
following result.

######  Theorem 3.3.

Let the conditions from Theorem 3.2 hold, together with Assumptions 3.2–3.3.
Then,

$\displaystyle\sqrt{n}(\hat{\theta}^{CL}-\theta_{0})\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,G(\theta_{0})^{-1}\right),\quad
n\rightarrow\infty,$

where $G(\theta_{0})$ is the _Godambe information matrix_ (Godambe, 1960)
matrix with inverse
$G(\theta_{0})^{-1}=H(\theta_{0})^{-1}V(\theta_{0})H(\theta_{0})^{-1}$, where

$\displaystyle H(\theta_{0})=$
$\displaystyle-\sum_{k=1}^{K}\mathbb{E}\left[\frac{\partial^{2}}{\partial\theta^{\prime}\partial\theta}\log
f(X_{k\Delta},X_{0};\theta)|_{\theta=\theta_{0}}\right],\quad\mathrm{and}$
$\displaystyle V(\theta_{0})=$
$\displaystyle\sum_{k=1}^{K}Var\left(\frac{\partial}{\partial\theta}\log
f(X_{k\Delta},X_{0};\theta)|_{\theta=\theta_{0}}\right)$
$\displaystyle+2\sum_{k=1}^{K}\sum_{k^{\prime}=1}^{K}\sum_{i=1}^{\infty}Cov\left(\frac{\partial}{\partial\theta}\log
f(X_{k\Delta},X_{0};\theta)|_{\theta=\theta_{0}},\frac{\partial}{\partial\theta^{\prime}}\log
f(X_{(i+k^{\prime})\Delta},X_{i\Delta};\theta)|_{\theta=\theta_{0}}\right).$

Further, the infinite sum in the expression for $V(\theta_{0})$ converges.

Theorem 3.3 implies that feasible inference can be conducted using an estimate
of the inverse of the Godambe information matrix
$\hat{G}(\hat{\theta}^{CL})^{-1}=\hat{H}(\hat{\theta}^{CL})^{-1}\hat{V}(\hat{\theta}^{CL})\hat{H}(\hat{\theta}^{CL})^{-1}$,
where $\hat{\theta}^{CL}$ is the MCL estimate from (3.3). Note that while the
straight-forward estimator
$\hat{H}(\hat{\theta}^{CL})=-n^{-1}\frac{\partial}{\partial\theta\partial\theta^{\prime}}l_{CL}(\hat{\theta}^{CL};x)$
is consistent for $H(\theta)$ due to the stationarity and ergodicity of the
IVT process, $\hat{V}(\hat{\theta}^{CL})$ is more difficult to obtain, since
the obvious candidate
$n^{-1}\frac{\partial}{\partial\theta}l_{CL}(\theta;x)\frac{\partial}{\partial\theta}l_{CL}(\theta;x)^{\prime}$
vanishes at $\theta=\hat{\theta}^{CL}$, a fact also remarked in Varin & Vidoni
(2005). While it is possible to estimate $V(\theta_{0})$ using a Newey-West-
type kernel estimator (Newey & West, 1987), we obtained more precise results
using a simulation-based approach to estimating $V(\theta_{0})$. The details
of both approaches are provided in the Supplementary Material, Section
S4.333It is also possible to approximate the standard error of
$\hat{\theta}^{CL}$ using a standard parametric bootstrap approach. However,
as we discuss in Section S4.2 of the Supplementary Material, this solution is
more computationally expensive than the two alternative approaches suggested
here.

####  3.1.3 Asymptotic theory in the long memory case

While the consistency result in Theorem 3.2 applies for all IVT processes
satisfying Assumptions 2.1–2.2 and 3.1, Assumption 3.3, required in the
central limit result in Theorem 3.3, excludes IVT processes with long memory,
e.g. those with autocorrelation function adhering to $\rho(h)=O(h^{-H})$ for
$H\in(0,1]$. As mentioned in Remark 3.1, this is for instance the case for the
Gamma trawl function (Example 2.5) with $H\in(0,1]$.

Although a long memory CLT as such eludes us, we can say some things about the
asymptotic behaviour of the MCL estimator $\hat{\theta}^{CL}$ in the long
memory case. For instance, the convergence rate is likely slower than
$\sqrt{n}$, as the following result suggests.

######  Theorem 3.4.

Let the conditions from Theorem 3.2 hold and assume that the autocorrelation
function of the IVT process satisfies $\rho(h)=L_{\infty}(h)h^{-H}$ for some
$H\in(0,1]$, where $L_{\infty}$ is a function which is slowly varying at
infinity, i.e. for all $a>0$ it holds that
$\lim_{x\rightarrow\infty}\frac{L_{\infty}(ax)}{L_{\infty}(x)}=1$. Then,

  1. (i)

For all $\epsilon>0$,
$n^{H/2-\epsilon}(\hat{\theta}^{CL}-\theta_{0})\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}0$,
as $n\rightarrow\infty$.

  2. (ii)

Let $J=\textnormal{dim}(\theta_{0})$ be the dimension of $\theta_{0}$ and
denote by $\hat{\theta}^{CL}_{i}$ and $\theta_{0,i}$ the $i$th component of
the vectors $\hat{\theta}^{CL}$ and $\theta_{0}$, respectively. Then, for
$i=1,2,\ldots,J$, we have that for all $\epsilon>0$,
$Var\left(n^{H/2+\epsilon}(\hat{\theta}^{CL}_{i}-\theta_{0,i})\right)\rightarrow\infty$,
as $n\rightarrow\infty$.

Theorem 3.4(i) implies that the convergence rate of $\hat{\theta}^{CL}$ cannot
be slower than $n^{H/2}$ for $H\in(0,1]$. Further, Theorem 3.4(ii) implies
that if the convergence rate is faster than $n^{H/2}$ it must necessarily be
the case that the limiting random variable has an infinite variance. We
conjecture that
$n^{H/(H+1)}(\hat{\theta}^{CL}-\theta_{0})\stackrel{{\scriptstyle(d)}}{{\rightarrow}}MY_{1+H}$
for a matrix $M$, where $Y_{\alpha}$ is an $\alpha$-stable random vector. Note
that, for $H\in(0,1)$ it is the case $H/(1+H)\in(H/2,1/2)$, meaning that the
conjectured convergence rate is faster than $n^{H/2}$, but slower that
$\sqrt{n}$. Our reason for the conjecture has its roots in Theorem 3.5 below.
First, we introduce a technical assumption on the trawl function $d$, ensuring
that we are in the long memory case.

######  Assumption 3.4 (Long memory).

Assume that $H\in(0,1)$ and

  1. 1.

$d(-x)=g_{1}(x)x^{-H-1}$, $x>0$, where $g_{1}$ is a function that is slowly
varying at infinity.

  2. 2.

$d^{\prime}(-x)=g_{2}(x)x^{-H-2}$, $x>0$, where $g_{2}$ is a function that is
slowly varying at infinity.

######  Remark 3.5.

The key condition in Theorem 3.4, namely $\rho(h)=L_{\infty}(h)h^{-H}$ for
some $H\in(0,1)$, is implied by Assumption 3.4.

######  Remark 3.6.

The Gamma trawl function (Example 2.5) fulfils Assumption 3.4 with
$g_{1}(x)=\left(x^{-1}+\alpha^{-1}\right)^{-H-1}$ and
$g_{2}(x)=\frac{H+1}{\alpha}\left(x^{-1}+\alpha^{-1}\right)^{-H-2}$.

######  Theorem 3.5.

Suppose $L^{\prime}\sim Poi(\nu)$ and that the parameters of the trawl
function $d$ are known. Let the conditions from Theorem 3.2 hold, along with
Assumptions 3.2 and 3.4. Then,

$\displaystyle
n^{H/(1+H)}(\hat{\nu}^{CL}-\nu_{0}-R_{n})\stackrel{{\scriptstyle
d}}{{\rightarrow}}H(\nu_{0})^{-1}\nu_{0}^{-1}Y_{1+H},\quad
n\rightarrow\infty,$

where $H(\nu_{0})$ is given as in Theorem 3.3, $Y_{\alpha}$ is an
$\alpha$-stable random variable with characteristic function

$\displaystyle\phi_{Y_{\alpha}}(u):=\mathbb{E}[\exp(iuY_{\alpha})]=\exp\left(c|u|^{\alpha}\Gamma(2-\alpha)\
\left(\cos\left(\frac{\pi\alpha}{2}\right)-i\cdot\textnormal{sgn}(u)\sin\left(\frac{\pi\alpha}{2}\right)\right)\right),\quad
u\in\mathbb{R},$ (3.6)

and where $R_{n}$ is given by

$\displaystyle R_{n}=H(\nu_{0})^{-1}\nu_{0}^{-1}n^{-1}S_{n}(U),$

with $S_{n}(U):=\sum_{i=1}^{n}\left(U_{i}-\mathbb{E}[U_{i}]\right)$ denoting
the de-meaned partial sum of the sequence $U=\\{U_{i}\\}_{i=1}^{n}$, where
$U_{i}:=\sum_{k=1}^{K}g(X_{(i+k)\Delta},X_{i\Delta})$ and
$g(X_{(i+k)\Delta},X_{i\Delta}):=\mathbb{E}[L(A_{(i+k)\Delta}\setminus
A_{i\Delta})|X_{(i+k)\Delta},X_{i\Delta}]$.

######  Remark 3.7.

The asymptotic behaviour of the remainder term $R_{n}$ in Theorem 3.5 is
decided by a quite general function $g$ of the pairs $(X_{i},X_{i+k})$ and one
can show that similar issues arise in the more general case where $L^{\prime}$
is integer-valued and the parameters in the trawl function are estimated. The
asymptotic behaviour of such general functions of the data could possibly be
studied using mixing conditions for partial sums with $\alpha$-stable limits
(e.g. Jakubowski, 1993) or by deriving Breuer-Major-like theorems (Breuer &
Major, 1983; Nourdin et al., 2011) valid for IVT processes using Malliavin
calculus for Poissonian spaces, see Basse-O’Connor et al. (2020) for a related
approach. We believe that especially this latter route could be fruitful, but
leave it for future work.

The proof of Theorem 3.5 relies on a result about the partial sums of the IVT
process $X$, which might be of independent interest. We, therefore, state it
here.

######  Theorem 3.6.

Suppose the Lévy basis $L$ is non-negative, i.e. $\eta(y)=0$ for $y<0$, and
let the conditions from Theorem 3.2 hold, along with Assumptions 3.2 and 3.4.
Then,

$\displaystyle n^{-\frac{1}{1+H}}S_{n}(X)\stackrel{{\scriptstyle
d}}{{\rightarrow}}Y_{1+H},\quad n\rightarrow\infty,$

where $Y_{\alpha}$ is an $\alpha$-stable random variable with characteristic
function (3.6).

######  Remark 3.8.

Closely related results about partial sums of trawl processes have previously
been put forth in Doukhan et al. (2019) and Pakkanen et al. (2021) in the
context of a discrete-time trawl process under a standard asymptotic scheme
($n\rightarrow\infty$) and in the context of a continuous-time trawl process
under an infill asymptotic sampling scheme ($\delta\rightarrow 0$),
respectively. In this paper, we consider $n$ observations of the continuous-
time trawl process sampled on an equidistant $\delta$-grid,
$X_{\delta},X_{2\delta},\ldots,X_{n\delta}$, where $\delta>0$ is fixed, and
let $n\rightarrow\infty$. In this sense, our setup is closer to the one in
Doukhan et al. (2019). Indeed, it can be shown that the law of
$(X_{\delta},X_{2\delta},\ldots,X_{n\delta})$ is equal to the law of
$(Y_{1},Y_{2},\ldots,Y_{n})$, where $Y$ is an appropriately specified
discrete-time trawl process in the sense of Doukhan et al. (2019). Although
this highlights a close connection between the long memory results of Doukhan
et al. (2019) and those presented in this section, the underlying assumptions
in the two approaches are different. Firstly, the assumptions on the marginal
distribution made in Doukhan et al. (2019) are different from ours. In
particular, while Doukhan et al. (2019) are not restricting the marginal
distribution of the process to be infinitely divisible, they do impose a
restrictions on the size of the jumps of the Lévy basis, see Equation (3.36)
in Doukhan et al. (2019). Secondly, our assumptions on the correlation
structure of the process are slightly different than the assumptions made in
Doukhan et al. (2019). In particular, besides polynomial decay, we also allow
for a slowly varying function to enter the correlation structure, compare
Assumption 3.4 with Equation (2.12) in Doukhan et al. (2019). For these
reasons, although our setting is closely related to that in Doukhan et al.
(2019), we cannot use their results directly. We can, however, follow similar
lines of arguments as done in Doukhan et al. (2019), and this is what we do in
the proof of Theorem 3.6, given in the Appendix.

###  3.2 Information criteria for model selection

Takeuchi’s Information Criterion (Takeuchi, 1976) is an information criterion,
which can be used for model selection in the case of misspecified likelihoods.
Varin & Vidoni (2005) adapted the ideas of Takeuchi to the composite
likelihood framework and provided arguments for using the composite likelihood
information criterion (CLAIC)

$CLAIC=l_{LC}(\hat{\theta}^{CL};x)+\textnormal{tr}\left\\{\hat{V}(\hat{\theta}^{CL})\hat{H}(\hat{\theta}^{CL})^{-1}\right\\}$

as a basis for model selection, where $\textnormal{tr}\\{M\\}$ is the trace of
the matrix $M$. Specifically, Varin & Vidoni (2005) suggest picking the model
that maximizes $CLAIC$.

Analogous to the usual Bayesian/Schwarz Information Criterion (BIC, Schwarz,
1978), we also suggest the alternative composite likelihood information
criterion (Gao & Song, 2010)

$CLBIC=l_{CL}(\hat{\theta}^{CL};x)+\frac{\log(n)}{2}\textnormal{tr}\left\\{\hat{V}(\hat{\theta}^{CL})\hat{H}(\hat{\theta}^{CL})^{-1}\right\\},$

where $n$ is the number of observations of the data series $x$. Note that the
various models we consider are generally non-nested, whereas most research on
model selection using the composite likelihood approach has considered nested
model (Ng & Joe, 2014). An analysis of the properties of $CLAIC$ and $CLBIC$
in the non-nested case in the spirit of, e.g., Vuong (1989) would be very
valuable but is beyond the scope of the present article.

##  4 Forecasting integer-valued trawl processes

Let $\mathcal{F}_{t}=\sigma((X_{s})_{s\leq t})$ be the sigma-algebra generated
by the history of the IVT process $X$ up until time $t$ and let $h>0$ be a
forecast horizon. We are interested in the predictive distribution of the IVT
process, i.e. the distribution of $X_{t+h}|\mathcal{F}_{t}$. However, since
the IVT process $X$ is in general non-Markovian, the distribution of
$X_{t+h}|\mathcal{F}_{t}$ is highly intractable. This problem is similar to
the one encountered when considering the likelihood of observations of $X$,
cf. Section 3.1. For this reason, we propose to approximate the distribution
of $X_{t+h}|\mathcal{F}_{t}$ by $X_{t+h}|X_{t}$, i.e. instead of conditioning
on the full information set, we only condition on the most recent observation.
Thus, our proposed solution to the forecasting problem is akin to the proposed
solution to the problem of the intractability of the full likelihood. That is,
instead of considering the full distribution of $X_{t+h}|\mathcal{F}_{t}$, we
use the conditional “pairwise” distribution implied by $X_{t+h}|X_{t}$.

To fix ideas, let $t\in\mathbb{R}$ and $h>0$, and consider the random
variables $X_{t}=L(A_{t})=L(A_{t}\cap A_{t+h})+L(A_{t}\setminus A_{t+h})$ and
$X_{t+h}=L(A_{t+h})=L(A_{t}\cap A_{t+h})+L(A_{t+h}\setminus A_{t})$. The goal
is to find the conditional distribution of $X_{t+h}$ given $X_{t}$. Note that
$L(A_{t}\cap A_{t+h})$ and $L(A_{t+h}\setminus A_{t})$ are independent random
variables. Further, since $L(A_{t+h}\setminus A_{t})$ is independent of
$X_{t}$ with a known distribution, we only need to determine the distribution
of $L(A_{t}\cap A_{t+h})$ given $X_{t}$. The following lemma characterises the
conditional distribution of $L(A_{t}\cap A_{t+h})$.

######  Lemma 4.1.

Let $x\in\mathbb{N}\cup\\{0\\}$ and $l\in\\{0,1,\ldots,x\\}$, then

$\displaystyle\mathbb{P}(L(A_{t}\cap
A_{t+h})=l|X_{t}=x)=\frac{\mathbb{P}(L(A_{t}\setminus
A_{t+h})=x-l)\mathbb{P}(L(A_{t}\cap A_{t+h})=l)}{\mathbb{P}(X_{t}=x)}.$

######  Example 4.1.

In the case when $L^{\prime}\sim\mathrm{Poi}(\nu)$, we get the Binomial
distribution:

$\displaystyle L(A_{t}\cap
A_{t+h})|X_{t}\sim\mathrm{Bin}\left(X_{t},\frac{Leb(A_{0}\cap
A_{h})}{Leb(A_{0})}\right),$

which implies that $\mathbb{E}(L(A_{t}\cap A_{t+h})|X_{t})=X_{t}Leb(A_{0}\cap
A_{h})(Leb(A_{0}))^{-1}.$

######  Example 4.2.

In the case when $L^{\prime}\sim\mathrm{NB}(m,p)$, we get the Dirichlet-
multinomial distribution:

$\displaystyle L(A_{t}\cap A_{t+h})|X_{t}\sim\mathrm{Dirichlet-
multinomial}(X_{t},\alpha_{1},\alpha_{2}),$

where $\alpha_{1}=Leb(A_{0}\setminus A_{h})m,\alpha_{2}=Leb(A_{0}\cap
A_{h})m)$ and $\alpha_{1}+\alpha_{2}=Leb(A_{0})m$. For
$x-l,x\in\\{0,\ldots,X_{t}\\}$, the corresponding probability mass function is
given by

$\displaystyle\mathbb{P}(L(A_{t}\cap A_{t+h})=l|X_{t}=x)$
$\displaystyle={x\choose l}\frac{\Gamma(Leb(A_{0}\setminus
A_{h})m+x-l)}{\Gamma(Leb(A_{0}\setminus A_{h})m)}\frac{\Gamma(Leb(A_{0}\cap
A_{h})m+l)}{\Gamma(Leb(A_{0}\cap
A_{h})m)}\frac{\Gamma(Leb(A_{0})m)}{\Gamma(Leb(A_{0})m+x)},\quad x\geq l\geq
0,$

where ${x\choose l}=\frac{x!}{l!(x-l)!}$ is the binomial coefficient. This
implies that, as before, $\mathbb{E}(L(A_{t}\cap
A_{t+h})|X_{t})=X_{t}Leb(A_{0}\cap A_{h})(Leb(A_{0}))^{-1}.$

Using Lemma 4.1, we can derive the distribution of $X_{t+h}|X_{t}$, which can
be used for probabilistic forecasting. The details for non-negative-valued
Lévy bases are given in the following proposition.

######  Proposition 4.1.

Let the IVT process $X$ be given by (2.4) and let Assumptions 2.1 and 2.2
hold. Suppose further, that the Lévy basis $L$ is non-negative, i.e.
$\eta(y)=0$ for $y<0$. Now,

$\displaystyle\mathbb{P}(X_{t+h}=x_{t+h}|X_{t}=x_{t})=\sum_{c=0}^{\min(x_{t},x_{t+h})}\mathbb{P}(L(A_{t+h}\setminus
A_{t})=x_{t+h}-c)\mathbb{P}(L(A_{t}\cap A_{t+h})=c|X_{t}=x_{t}).$

The following corollaries give the specific details for our two main
specifications for the marginal distribution of $X_{t}$, studied in Examples
4.1 and 4.2 above.

######  Corollary 4.1.

If $L^{\prime}\sim\mathrm{Poi}(\nu)$, then

$\mathbb{P}(X_{t+h}=x_{t+h}|X_{t}=x_{t})\\\
=\sum_{c=0}^{\min(x_{t},x_{t+h})}\frac{(\nu{Leb}(A_{h}\setminus
A_{0}))^{x_{t+h}-c}}{(x_{t+h}-c)!}e^{-\nu{Leb}(A_{h}\setminus
A_{0})}{x_{t}\choose c}\left(\frac{{Leb}(A_{h}\cap
A_{0})}{{Leb}(A_{0})}\right)^{c}\left(1-\frac{{Leb}(A_{h}\cap
A_{0})}{{Leb}(A_{0})}\right)^{x_{t}-c}.$

######  Corollary 4.2.

If $L^{\prime}\sim\mathrm{NB}(m,p)$, then

$\displaystyle\mathbb{P}(X_{t+h}=x_{t+h}|X_{t}=x_{t})$
$\displaystyle=\sum_{c=0}^{\min(x_{t},x_{t+h})}(1-p)^{Leb(A_{h}\setminus
A_{0})m}p^{x_{t+h}-c}{x_{t}\choose c}\frac{1}{(x_{t+h}-c)!}$
$\displaystyle\cdot\frac{\Gamma(Leb(A_{h}\setminus
A_{0})m+x_{t+h}-c)}{\Gamma(Leb(A_{h}\setminus
A_{0})m)}\frac{\Gamma(Leb(A_{h}\setminus
A_{0})m+x_{t}-c)}{\Gamma(Leb(A_{h}\setminus
A_{0})m)}\frac{\Gamma(Leb(A_{h}\cap A_{0})m+c)}{\Gamma(Leb(A_{h}\cap
A_{0})m)}\frac{\Gamma(Leb(A_{0})m)}{\Gamma(Leb(A_{0})m+x_{t})}.$

If the parameters of an IVT process $X$ with Poisson or Negative Binomial
marginal distribution are known, we can use Corollary 4.1 or 4.2, and the
calculations for the Lebesgue measures of the trawl sets given in Section 2.2,
for computing the predictive PMFs and thus for forecasting. When the true
parameter values are unknown, they can be estimated using the MCL estimator
suggested above, and plugged into the formulas to arrive at estimates of the
predictive PMFs.

##  5 Monte Carlo simulation experiments

Using simulations, we examine the finite sample properties of the composite
likelihood-based estimation procedure and of the model selection procedure.
Details are available in the Supplementary Material, see Section S6. Here we
summarise our main findings.

We consider six data-generating processes (DGPs): the Poisson-Exponential
(P-Exp), the Poisson-Inverse Gaussian (P-IG), the Poisson-Gamma (P-Gamma), the
Negative Binomial-Exponential (NB-Exp), the Negative Binomial-Inverse Gaussian
(NB-IG), and the Negative Binomial-Gamma (NB-Gamma) IVT models. The parameter
choices used in the simulation study, see Table S1 in the Supplementary
Material, are motivated by the estimates of our empirical study.

We compare the finite sample properties of the MCL estimator with the GMM
estimator, which has been used in the existing literature. Figure 2 plots the
root median squared error (RMSE) of the MCL estimator of a given parameter
divided by the RMSE of the GMM estimator of the same parameter for the six
DGPs. Thus, numbers smaller than one indicate that the MCL estimator has a
lower RMSE than the GMM estimator and vice versa for numbers larger than one.
We see that for most parameters in most of the DGPs, the MCL estimator
outperforms the GMM estimator substantially; indeed, in many cases, the RMSE
of the MCL estimator is around $50\%$ that of the GMM estimator. The exception
seems to be the trawl parameters, i.e. the parameters controlling the
autocorrelation structure, in the case of the Gamma and IG trawls, where the
GMM estimator occasionally performs on par with the MCL estimator. However, in
most cases, it appears that the MCL estimator is able to provide large
improvements over the GMM estimator.

In the Supplementary Material (Section S6.3), we also examine how close the
finite sample distribution of the MCL estimator is to the true (Gaussian)
asymptotic limit, as presented in Theorem 3.3. We find that the Gaussian
approximation is very good for the case of the parameters governing the
marginal distribution, as well as for the parameter $\lambda$ for the case of
IVTs with exponential trawl functions. When there are two parameters in the
trawl function ($\delta$ and $\gamma$ in the case of the IG trawl and $H$ and
$\alpha$ in the case of the Gamma trawl), however, the Gaussian distribution
can be a poor approximation to the finite sample distribution of the MCL
estimator in case of the trawl parameters. This indicates that for
constructing confidence intervals or testing hypotheses on these parameters,
it might be useful to consider bootstrap approaches instead of relying on the
Gaussian distribution.

Figure 2: Root median square error (RMSE) of the MCL estimator (3.3) divided
by the RMSE of the GMM estimator. The underlying IVT process $X_{t}$ is
simulated on the grid $t=\Delta,2\Delta,\ldots,n\Delta$, with $\Delta=0.10$,
see Table S1 for the values of the parameters used in the simulations. For the
Poisson-Exp and NB-Exp models we set $K=1$ in Equations (S3.1) and (3.3); for
the other models we set $K=10$.

##  6 Application to financial bid-ask spread data

In this section, we apply the IVT modelling framework to the bid-ask spread of
equity prices. The bid-ask spread has been extensively studied in the market
microstructure literature, see, e.g., Huang & Stoll (1997) and Bollen et al.
(2004). An application similar to the one studied in this section was
considered in Barndorff-Nielsen et al. (2014). To illustrate the use of the
methods proposed in this paper, we study the time series of the bid-ask
spread, measured in U.S. dollar cents, of the Agilent Technologies Inc. stock
(ticker: A) on a single day, May $4$, $2020$. We cleaned the data and sampled
the data in 5s intervals, leading to $n=3961$ observations, see Section S7 in
the Supplementary Material for details.

Figure 3: Analysis of the A spread level on May 4, 2020. Top: The data (spread
level in cents) from $10$:$30$AM to $4$:$00$PM sampled every $5$ seconds. The
vertical red line separates the initial in-sample period (left) from the out-
of-sample period (right), used in the forecasting exercise in Section 6.1. The
bottom six rows show the empirical autocorrelation (left; blue bars) and the
empirical marginal distribution of the spread level (right; blue bars)
together with the fits from the six IVT models (red lines). The parameters of
the models have been estimated using the MCL estimator (3.3) with $K=10$, see
Table 2.

Let $s_{t}$ be the bid-ask spread level at time $t$, the time series of which
is displayed in the top panel of Figure 3. Since the minimum spread level in
the data is one tick (one dollar cent), we work on this time series minus one,
i.e on $x_{t}=s_{t}-1$. We can now apply our model selection method. We first
inspect the empirical autocorrelation of the data (shown in the right panels
of Figure 3) which shows evidence of a very persistent process; we, therefore,
set $K$ to a moderately large value to accurately capture the dependence
structure of the data. Here, we choose $K=10$ but the results are robust to
other choices.

Table 1: Model selection results: bid-ask spread data

Model: | Poisson-Exponential | Poisson-IG | Poisson-Gamma | NB-Exponential | NB-IG | NB-Gamma
---|---|---|---|---|---|---
$CL$ | $-244125.5$ | $-242885.2$ | $-242835.8$ | $-216363.9$ | $-216318.1$ | ${\bf-216313.5}$
$CLAIC$ | $-244125.7$ | $-242899.1$ | $-242855.8$ | $-216364.0$ | $-216318.5$ | ${\bf-216313.9}$
$CLBIC$ | $-244126.4$ | $-242942.8$ | $-242918.8$ | $-216364.1$ | $-216319.7$ | ${\bf-216315.1}$

Composite likelihood and information criteria values for fitting the A bid-ask
spread data on May 4, 2020, shown in the top plot of Figure 3, calculated
using six different models as given in the top row of the table using $K=10$.
The maximum value for a given criteria (i.e. row-wise) is given in bold. The
parameter estimates corresponding to the fits are given in Table 2.

Table 2: Estimation results: bid-ask spread data

DGP | $\nu$ | $m$ | $p$ | $\lambda$ | $\delta$ | $\gamma$ | $H$ | $\alpha$
---|---|---|---|---|---|---|---|---
P-Exp | $28.9319$ |  |  | $4.0399$ |  |  |  |
| $(0.6644)$ |  |  | $(0.0904)$ |  |  |  |
P-IG | $294.9102$ |  |  |  | $1.5293$ | $0.0371$ |  |
| $(28.2715)$ |  |  |  | $(0.0424)$ | $(0.0040)$ |  |
P-Gamma | $83.8197$ |  |  |  |  |  | $0.6123$ | $0.0523$
| $(-)$ |  |  |  |  |  | $(-)$ | $(-)$
NB-Exp |  | $6.4273$ | $0.6665$ | $1.7835$ |  |  |  |
|  | $(0.9324)$ | $(0.0215)$ | $(0.1349)$ |  |  |  |
NB-IG |  | $7.7104$ | $0.6675$ |  | $1.7816$ | $0.8292$ |  |
|  | $(1.0111)$ | $(0.0238)$ |  | $(0.4436)$ | $(0.2094)$ |  |
NB-Gamma |  | $7.7336$ | $0.6675$ |  |  |  | $1.7020$ | $0.7897$
|  | $(1.1316)$ | $(0.0260)$ |  |  |  | $(0.7365)$ | $(0.3363)$

Parameter estimates (standard errors in parentheses) from the six different
DGPs when applied to the bid-ask spread data of A on May 4, 2020 using the MCL
estimator with $K=10$. The standard errors have been obtained using the
simulation-based approach to estimating the asymptotic covariance matrix of
the MCL estimator, see Section S4 in the Supplementary Material. Since our
asymptotic theory does not cover the long memory case, no standard deviations
are reported for the P-Gamma model. See Figure 3 for the resulting fits of the
models to the empirical distribution and autocorrelation.

Using this setting, we calculated the maximized composite likelihood value,
CL, and the two information criteria, $CLAIC$ and $CLBIC$, obtained for these
data using the six models considered in Section 5. The results are shown in
Table 1. The table shows that the NB-Gamma model is the preferred model on all
three criteria, while the second-best model is the NB-IG model.

To further examine the fit of the various models, the bottom six rows of
Figure 3 contain the empirical autocorrelation (left; blue bars) and the
empirical marginal distribution of the spread level (right; blue bars). Each
respective row also shows the fit of one of the six models considered in Table
1; the parameter estimates corresponding to the models are given in Table 2.
The fit of the models shown in the bottom six panels of Figure 3 and the
selection criteria of Table 1 indicate that the models based on the Negative
Binomial distribution are preferred to the models based on the Poisson
distribution. We conclude that, for this data series, the Poisson distribution
is unable to accurately describe the marginal distribution of the spread level
sampled every $5$ seconds. That the Gamma and IG trawl functions are preferred
to the Exponential trawl function indicates that the Exponential
autocorrelation function is not flexible enough to capture the correlation
structure of the data. By both visual inspection of the autocorrelations in
Figure 3 and the selection criteria of Table 1, we conclude that the NB-Gamma
model is the preferred model for these data. As shown in Table 2, this model
has $\hat{H}=1.70$ (s.e. $0.74$), implying that the model possesses a slowly
decaying autocorrelation structure, albeit not the long memory property.

###  6.1 Forecasting the spread level

This section illustrates the use of IVT models for forecasting, as outlined in
Section 4. The aim is to forecast the future spread level of the A stock on
May $4$, i.e. the data studied above and plotted in the top panel of Figure 3.
We set aside the first $n_{1}=3221$ observations as an “in-sample period” for
initial estimation of the parameters of the models, see the vertical red line
in the top panel of Figure 3 for the placement of this split. We then forecast
the spread level from $5$ seconds until $100$ seconds into the future, using
the approach presented in Section 4. That is, we forecast
$x_{n_{1}+1},x_{n_{1}+2},\ldots,x_{n_{1}+20}$ given the current value
$x_{n_{1}}$. After this, we update the in-sample data set with one additional
observation so that this sample now contains $n_{2}=n_{1}+1=3222$
observations. Then we again forecast the next $20$ observations,
$x_{n_{2}+1},x_{n_{2}+2},\ldots,x_{n_{2}+20}$, given $x_{n_{2}}$. We repeat
this procedure until the end of the sample, which yields
$n_{oos}=n-n_{1}-20=720$ out-of-sample forecasts for each forecast horizon. To
ease the computational burden, we only re-estimate the model every $24$
periods (i.e. every $2$ minutes).

To evaluate the forecasts, we consider four different loss metrics. The first
two, the mean absolute error (MAE) and the mean squared error (MSE), are often
used in econometric forecasting studies of real-valued data (e.g. Elliott &
Timmermann, 2016). For a forecast horizon $h=1,2,\ldots,20$, define the mean
absolute forecast error,

$MAE(h)=\frac{1}{n_{oos}}\sum_{i=n_{1}+h}^{n-(20-h)}|x_{i}-\hat{x}_{i|i-h}|,$

where $\hat{x}_{i|i-h}$ is the $h$-step ahead forecast of $x_{i}$, constructed
using the information available up to observation $i-h$. That is,
$\hat{x}_{i|i-h}$ is the point forecast of $x_{i}$ coming from a particular
IVT model, such as the conditional mean, median, or mode. In what follows, we
set $\hat{x}_{i|i-h}$ equal to the estimated conditional mean, i.e. we set
$\hat{x}_{i|i-h}=\sum_{k=0}^{M}\widehat{\mathbb{P}}(X_{i|i-h}=k)k$, where
$M\geq 1$ is a large (cut-off) number and $\widehat{\mathbb{P}}$ is the
estimated predictive density of the IVT model.444Letting $\hat{x}_{i|i-h}$ be
the conditional mode, instead of the conditional mean, produces results
similar to those reported here. These results are reported in the
Supplementary Material, Section S7.2. Here, we set $M=60$ but the results are
very robust to other choices. Define also the mean squared forecast error

$MSE(h)=\frac{1}{n_{oos}}\sum_{i=n_{1}+h}^{n-(20-h)}(x_{i}-\hat{x}_{i|i-h})^{2}.$

We consider two additional loss metrics, designed to directly evaluate the
accuracy of the estimated predictive PMF $\widehat{\mathbb{P}}$, which is
arguably more relevant to the problem at hand than MAE and MSE. The first is
the logarithmic score (Elliott & Timmermann, 2016, p. 30),

$logS(h)=\frac{1}{n_{oos}}\sum_{i=n_{1}+h}^{n-(20-h)}-\log\widehat{\mathbb{P}}(X_{i|i-h}=x_{i}),$

where $x_{i}$ is the realized outcome. The second is the ranked probability
score (RPS; Epstein, 1969),

$RPS(h)=\frac{1}{n_{oos}}\sum_{i=n_{1}+h}^{n-(20-h)}\sum_{k=0}^{M}(\widehat{F}_{i|i-h}(k)-\mathbb{I}_{\\{x_{i}\leq
k\\}})^{2},$

where $\widehat{F}_{i|i-h}(k)=\sum_{j=0}^{k}\widehat{\mathbb{P}}(X_{i|i-h}=j)$
is the estimated cumulative distribution function of $X_{i}|x_{i-h}$ coming
from a given model and $\mathbb{I}_{\\{x_{i}\leq k\\}}$ is the indicator
function of the event $\\{x_{i}\leq k\\}$.

Figure 4 shows the four different forecast loss metrics for the preferred NB-
Gamma IVT model as a ratio of the forecasting loss of a given benchmark model
in the out-of-sample forecasting exercise described above. The numbers plotted
in the figure are $Loss(h)_{NB-Gamma}/Loss(h)_{benchmark}$, where “$Loss$”
denotes one of the four loss metrics given above and $h=1,2,\ldots,20$ denotes
the forecasting horizon. Thus, numbers less than one favour the NB-Gamma model
compared to the benchmark model and vice versa for numbers greater than one.
Initially, we choose the Poisson-Exponential IVT process as the benchmark
model (Figure 4, first column); as remarked above, this process is identical
to the Poissonian INAR(1) model, which is often used for forecasting count-
valued data (e.g. Freeland & McCabe, 2004; McCabe & Martin, 2005; Silva et
al., 2009). It is evident from the figure that losses from the NB-Gamma model
are smaller than those from the Poisson-Exponential model for practically all
forecast horizons and loss metrics. In the case of the two most relevant loss
functions for evaluating the predictive distribution, the logS and RPS, the
reduction in losses are substantial for all forecast horizon, on the order of
$20\%$.

Figure 4: Forecasting the spread level of the A stock on May 4, 2020. Four
different loss metrics (row-wise, as indicated above each plot) and twenty
forecast horizons, $h=1,2,\ldots,20$. The numbers plotted are relative average
losses of the NB-Gamma forecasting model, compared with the Poisson-
Exponential model (first column), the Poisson-Gamma model (second column), and
the NB-Exponential model (third column), over $n_{oos}=720$ out-of-sample
forecasts. A circle above the bars indicates rejection null of equal
forecasting performance between the two models, against the alternative that
the NB-Gamma model provides superior forecasts, using the Diebold-Mariano
(Diebold and Mariano, 1995) test at a $5\%$ level; an asterisk denotes
rejection at a $1\%$ level.

To assess whether these loss differences are also statistically significant,
we perform the Diebold-Mariano test of superior predictive ability (Diebold
and Mariano, 1995). The null hypothesis of the statistical test is that the
two models have equal predictive power, while the alternative hypothesis is
that the NB-Gamma model provides superior forecasts compared to the benchmark
model. In Figure 4, a circle (asterisk) denotes rejection of the Diebold-
Mariano test at a $5\%$ ($1\%$) level. The test rejects the null hypothesis of
equal predictive ability for almost all forecast horizons and loss metrics at
a $1\%$ level.

To investigate whether the increased forecast performance of the NB-Gamma
model comes from having a more flexible marginal distribution than the
Poisson-Exponential benchmark model (Negative Binomial vs. Poisson marginals)
or from having a more flexible correlation structure (polynomial decay vs.
exponential decay) or both, we compare the forecasts from NB-Gamma model to
those coming from a Poisson-Gamma model and from an NB-Exp model. These
results are given in the second and third columns of Figure 4, respectively.
From the second column, we see that the NB-Gamma model outperforms the
Poisson-Gamma model considerably, especially for the shorter forecast
horizons, indicating that it is important to use a model with Negative
Binomial marginals when forecasting these data. From the third column of the
figure, we see that for the shorter forecast horizons, the NB-Exp model
performs on par with the NB-Gamma model, but for the longer forecast horizons,
the NB-Gamma model is superior. Hence, when forecasting, it also appears to be
important to specify a model with an accurate autocorrelation structure,
especially for longer forecasting horizons.

##  7 Conclusion

This paper has developed likelihood-based methods for the estimation,
inference, model selection, and forecasting of IVT processes. We proved the
consistency and asymptotic normality of the MCL estimator and provided the
details on how to conduct feasible inference and model selection. We also
developed a pairwise approach to approximating the conditional predictive PMF
of the IVT process, which can be used for forecasting integer-valued data. All
these methods are implemented in a freely available software package written
in the MATLAB programming language.

In a simulation exercise, we demonstrated the good properties of the MCL
estimator compared to the often-used method-of-moments-based estimator.
Indeed, the reduction in root median squared error of the MCL estimator was in
many cases more than $50\%$ compared to the corresponding GMM estimator.

In an empirical application to financial bid-ask spread data, we illustrated
the model selection procedure and found that the Negative Binomial-Gamma IVT
model provided the best fit for the data. Using the forecast tools developed
in the paper, we saw that this model outperformed the simpler Poisson-
Exponential IVT model considerably, resulting in a reduction in forecast loss
on the order of $20\%$ for most forecast horizons. We demonstrated that most
of the superior forecasting performance came from accurate modelling of the
marginal distribution of the data; however, we also found that it was
beneficial to carefully model the autocorrelation structure, especially for
longer forecasting horizons. These findings highlight the strengths of the IVT
modelling framework, where the marginal distribution and autocorrelation
structure can be modelled independently in a flexible fashion.

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##  Appendix A Mathematical proofs

We first give an alternative representation of the Lévy basis $L$, underlying
the IVT process X. From the construction of the IVT process in Section 2 in
the main article, it is clear that the distribution of $L$ is representable as
a compound Poisson distribution. That is, for a Borel set $B$, we can write

$\displaystyle\mathbb{P}_{\theta}(L(B)=x)$
$\displaystyle=\sum_{q=0}^{\infty}\mathbb{P}_{\theta}\left(\sum_{i=1}^{q}Y_{i}=x,\tilde{N}(B)=q\right)$
$\displaystyle=\sum_{q=0}^{\infty}\mathbb{P}_{\theta}\left(\sum_{i=1}^{q}Y_{i}=x\right)\mathbb{P}_{\theta}(\tilde{N}(B)=q),\quad
x\in\mathbb{Z},$ (A.1)

where $Y_{i}$ are iid integer-valued random variables with probability mass
function $\tilde{\eta}(y):=\frac{\eta(y)}{\sum_{y=-\infty}^{\infty}\eta(y)}$,
i.e. $\mathbb{P}_{\theta}(Y_{i}=y)=\tilde{\eta}(y)$, where $\eta$ is the Lévy
measure given in the construction of the IVT process in Equation. Likewise,
$\tilde{N}$ is a Poisson random measure, given by

$\displaystyle\tilde{N}(dx,ds)=\int_{-\infty}^{\infty}N(dy,dx,ds),$

with an underlying intensity $\tilde{\nu}:=\sum_{y=-\infty}^{\infty}\eta(y)$.
The random variables $Y_{i}$ are independent of the random measure
$\tilde{N}$. Intuitively, we have decomposed the event that the sum of the
points in the set $B$ equals $x$ (i.e. $\\{L(B)=x\\}$) into the intersection
of the two events that there are $q$ individual points in $B$ (i.e.
$\\{\tilde{N}(B)=q\\}$) and the “sizes” of these $q$ points add up to $x$
(i.e. $\\{\sum_{i=1}^{q}Y_{i}=x\\}$). With this construction, we have

$\displaystyle\mathbb{P}_{\theta}(\tilde{N}(B)=q)=\frac{(\tilde{\nu}Leb(B))^{q}}{q!}e^{-\tilde{\nu}Leb(B)},\quad
q=0,1,2,\ldots.$ (A.2)

We will use this alternative representation of $L$ in our proofs below.

###### Proof of Theorem 3.1.

Let $m>0$ and define $N_{m}=\tilde{N}(A_{m}\cap A)$ as the Poisson random
variable, which counts the number of ‘events’ in the set $A_{m}\cap A_{0}$.
From (A.2), we know that there exists a constant $\tilde{\nu}>0$ such that

$\displaystyle\mathbb{P}(N_{m}=0)=e^{-\tilde{\nu}Leb(A_{m}\cap A_{0})}$

and, therefore,

$\displaystyle\mathbb{P}(N_{m}\neq
0)=1-\mathbb{P}(N_{m}=0)=1-e^{-\tilde{\nu}Leb(A_{m}\cap
A_{0})}=\tilde{\nu}Leb(A_{m}\cap A_{0})+o(Leb(A_{m}\cap A_{0})),$ (A.3)

as $m\rightarrow\infty$.

Let $G\in\mathcal{F}_{-\infty}^{0}$ and $H\in\mathcal{F}_{m}^{\infty}$ be such
that $\mathbb{P}(G),\mathbb{P}(H)>0$, and write, using the law of total
probability,

$\displaystyle|\mathbb{P}(H\cap G)-\mathbb{P}(H)\mathbb{P}(G)|$
$\displaystyle=|\mathbb{P}(H|G)-\mathbb{P}(H)|\cdot\mathbb{P}(G)$
$\displaystyle=|\mathbb{P}(H|G,N_{m}=0)\mathbb{P}(N_{m}=0|G)+\mathbb{P}(H|G,N_{m}\neq
0)\mathbb{P}(N_{m}\neq 0|G)$
$\displaystyle-\mathbb{P}(H|N_{m}=0)\mathbb{P}(N_{m}=0)-\mathbb{P}(H|N_{m}\neq
0)\mathbb{P}(N_{m}\neq 0)|\cdot\mathbb{P}(G)$
$\displaystyle\leq(D_{1,m}+D_{2,m})\cdot\mathbb{P}(G),$

where

$\displaystyle D_{1,m}$
$\displaystyle:=|\mathbb{P}(H|G,N_{m}=0)\mathbb{P}(N_{m}=0|G)-\mathbb{P}(H|N_{m}=0)\mathbb{P}(N_{m}=0)|,$
$\displaystyle D_{2,m}$ $\displaystyle:=|\mathbb{P}(H|G,N_{m}\neq
0)\mathbb{P}(N_{m}\neq 0|G)-\mathbb{P}(H|N_{m}\neq 0)\mathbb{P}(N_{m}\neq
0)|.$

We seek to bound these expressions. For $D_{1,m}$, we use the fact that on the
event $\\{N_{m}=0\\}$ the two events $G$ and $H$ are independent. For both
$D_{1,m}$ and $D_{2,m}$, we will use that the probability of the complementary
event $\\{N_{m}\neq 0\\}$ is “small enough”, cf. Equation (A.3). For the first
of the terms, we get, using conditional independence of $H$ and $G$ and (A.3),

$\displaystyle D_{1,m}$
$\displaystyle=\mathbb{P}(H|N_{m}=0)\cdot|\mathbb{P}(N_{m}=0|G)-\mathbb{P}(N_{m}=0)|$
$\displaystyle\leq|\mathbb{P}(N_{m}=0|G)-\mathbb{P}(N_{m}=0)|$
$\displaystyle\leq|1-\mathbb{P}(N_{m}=0|G)|+|1-\mathbb{P}(N_{m}=0)|$
$\displaystyle=|1-\mathbb{P}(N_{m}=0|G)|+\tilde{\nu}Leb(A_{m}\cap
A_{0})+o(Leb(A_{m}\cap A_{0})),$

and, using the Bayes formula and then the law of total probability,

$\displaystyle|1-\mathbb{P}(N_{m}=0|G)|$
$\displaystyle=|1-\mathbb{P}(G|N_{m}=0)\mathbb{P}(N_{m}=0)\mathbb{P}(G)^{-1}|$
$\displaystyle=\mathbb{P}(G)^{-1}|\mathbb{P}(G)-\mathbb{P}(G|N_{m}=0)\mathbb{P}(N_{m}=0)|$
$\displaystyle=\mathbb{P}(G)^{-1}|\mathbb{P}(G|N_{m}=0)\mathbb{P}(N_{m}=0)+\mathbb{P}(G|N_{m}\neq
0)\mathbb{P}(N_{m}\neq 0)$
$\displaystyle-\mathbb{P}(G|N_{m}=0)\mathbb{P}(N_{m}=0)|$
$\displaystyle=\mathbb{P}(G)^{-1}|\mathbb{P}(G|N_{m}\neq
0)\mathbb{P}(N_{m}\neq 0)|$
$\displaystyle\leq\mathbb{P}(G)^{-1}\mathbb{P}(N_{m}\neq 0)$
$\displaystyle=\mathbb{P}(G)^{-1}(\tilde{\nu}Leb(A_{m}\cap
A_{0})+o(Leb(A_{m}\cap A_{0}))).$

We conclude that

$\displaystyle D_{1,m}\cdot\mathbb{P}(G)\leq 2\tilde{\nu}Leb(A_{m}\cap
A_{0})+o(Leb(A_{m}\cap A_{0})).$

For the second term above, we get

$\displaystyle D_{2,m}$ $\displaystyle=|\mathbb{P}(H|G,N_{m}\neq
0)\mathbb{P}(N_{m}\neq 0|G)-\mathbb{P}(H|N_{m}\neq 0)\mathbb{P}(N_{m}\neq 0)|$
$\displaystyle=|\mathbb{P}(H|G,N_{m}\neq 0)\mathbb{P}(N_{m}\neq
0|G)-\mathbb{P}(H|G,N_{m}\neq 0)\mathbb{P}(N_{m}\neq 0)$
$\displaystyle+\mathbb{P}(H|G,N_{m}\neq 0)\mathbb{P}(N_{m}\neq
0)-\mathbb{P}(H|N_{m}\neq 0)\mathbb{P}(N_{m}\neq 0)|$ $\displaystyle\leq
E_{1,m}+E_{2,m},$

where

$\displaystyle E_{1,m}$ $\displaystyle:=|\mathbb{P}(H|G,N_{m}\neq
0)\mathbb{P}(N_{m}\neq 0|G)-\mathbb{P}(H|G,N_{m}\neq 0)\mathbb{P}(N_{m}\neq
0)|,$ $\displaystyle E_{2,m}$ $\displaystyle:=|\mathbb{P}(H|G,N_{m}\neq
0)\mathbb{P}(N_{m}\neq 0)-\mathbb{P}(H|N_{m}\neq 0)\mathbb{P}(N_{m}\neq 0)|.$

Now, by (A.3),

$\displaystyle E_{2,m}$ $\displaystyle=\mathbb{P}(N_{m}\neq
0)|\mathbb{P}(H|G,N_{m}\neq 0)-\mathbb{P}(H|N_{m}\neq 0)|$
$\displaystyle\leq\mathbb{P}(N_{m}\neq 0)$
$\displaystyle=\tilde{\nu}Leb(A_{m}\cap A_{0})+o(Leb(A_{m}\cap A_{0})).$

Also,

$\displaystyle E_{1,m}$ $\displaystyle=|\mathbb{P}(H|G,N_{m}\neq
0)\mathbb{P}(N_{m}\neq 0|G)-\mathbb{P}(H|G,N_{m}\neq 0)\mathbb{P}(N_{m}\neq
0)|$ $\displaystyle=\mathbb{P}(H|G,N_{m}\neq 0)\cdot|\mathbb{P}(N_{m}\neq
0|G)-\mathbb{P}(N_{m}\neq 0)|$ $\displaystyle\leq|\mathbb{P}(N_{m}\neq
0|G)-\mathbb{P}(N_{m}\neq 0)|$ $\displaystyle\leq\mathbb{P}(N_{m}\neq
0|G)+\mathbb{P}(N_{m}\neq 0).$

Using Bayes formula, we can write

$\displaystyle\mathbb{P}(N_{m}\neq 0|G)$ $\displaystyle=\mathbb{P}(G|N_{m}\neq
0)\mathbb{P}(N_{m}\neq 0)\mathbb{P}(G)^{-1}$
$\displaystyle\leq\mathbb{P}(N_{m}\neq 0)\mathbb{P}(G)^{-1}$

so that, from (A.3),

$\displaystyle E_{1,m}\leq(1+\mathbb{P}(G)^{-1})(\tilde{\nu}Leb(A_{m}\cap
A_{0})+o(Leb(A_{m}\cap A_{0}))).$

We conclude that

$\displaystyle D_{2,m}\cdot\mathbb{P}(G)\leq 3\tilde{\nu}Leb(A_{m}\cap
A_{0})+o(Leb(A_{m}\cap A_{0})).$

Taking it all together, we have that

$\displaystyle|\mathbb{P}(H\cap G)-\mathbb{P}(H)\mathbb{P}(G)|$
$\displaystyle\leq(D_{1,m}+D_{2,m})\cdot\mathbb{P}(G)$ $\displaystyle\leq
5\tilde{\nu}Leb(A_{m}\cap A_{0})+o(Leb(A_{m}\cap A_{0})),$

implying, since $G$ and $H$ were arbitrary, that (taking supremums)

$\displaystyle\alpha_{m}\leq 5\tilde{\nu}Leb(A_{m}\cap A_{0})+o(Leb(A_{m}\cap
A_{0})),$

which implies that $\alpha_{m}\leq O(Leb(A_{m}\cap A_{0}))$.

To finish the proof, we show that we also have $\alpha_{m}\geq O(Leb(A_{m}\cap
A_{0}))$. Letting $x_{1},x_{2}\in\mathbb{Z}$ and defining the events
$\tilde{H}:=\\{X_{0}=x_{1}\\}$ and $\tilde{G}:=\\{X_{m}=x_{2}\\}$, the results
and the arguments in the proof of Lemma A.1 imply that

$\displaystyle|\mathbb{P}(\tilde{H}\cap\tilde{G})-\mathbb{P}(\tilde{H})\mathbb{P}(\tilde{G})|=O(Leb(A_{m}\cap
A_{0})),\qquad m\rightarrow\infty.$

Since, clearly, $\tilde{H}\in\mathcal{F}_{-\infty}^{0}$ and
$\tilde{G}\in\mathcal{F}_{m}^{\infty}$, we conclude that
$\alpha_{m}=\sup_{H\in\mathcal{F}_{-\infty}^{0},G\in\mathcal{F}_{m}^{\infty}}|\mathbb{P}(H\cap
G)-\mathbb{P}(H)\mathbb{P}(G)|\geq|\mathbb{P}(\tilde{H}\cap\tilde{G})-\mathbb{P}(\tilde{H})\mathbb{P}(\tilde{G})|=O(Leb(A_{m}\cap
A_{0}))$.

∎

###### Proof of Proposition 3.1.

When $\nu(y)=0$ for $y<0$ we have $P_{\theta}(L(A_{(i+k)\Delta}\cap
A_{i\Delta})=c)=0$ for $c<0$. Further, since the maximal amount of events in
$L(A_{(i+k)\Delta}\cap A_{i\Delta})$ is bounded by the number of events in
$x_{t+k}$ and $x_{t}$ (no negative values in the trawl sets), we also have
$P_{\theta}(L(A_{(i+k)\Delta}\cap A_{i\Delta})=c)=0$ for
$c>\min\\{x_{t+k},x_{t}\\}$. This, together with the discussion of the
decomposition of trawl sets in Section 3.1.1 (cf. Figure S1 in the
Supplementary Material), and the law of total probability implies that

$\displaystyle f(x_{i+k},$ $\displaystyle
x_{i};\theta):=\mathbb{P}_{\theta}\left(X_{(i+k)\Delta}=x_{i+k},X_{i\Delta}=x_{i}\right)$
$\displaystyle=$
$\displaystyle\sum_{c=-\infty}^{\infty}\mathbb{P}_{\theta}\left(X_{(i+k)\Delta}=x_{i+k},X_{i\Delta}=x_{i}|L(A_{(i+k)\Delta}\cap
A_{i\Delta})=c\right)\cdot\mathbb{P}_{\theta}\left(L(A_{(i+k)\Delta}\cap
A_{i\Delta})=c\right)$ $\displaystyle=$
$\displaystyle\sum_{c=0}^{\max\\{x_{i},x_{i+k}\\}}\mathbb{P}_{\theta}\left(L(A_{(i+k)\Delta}\setminus
A_{i\Delta})=x_{i+k}-c\right)\mathbb{P}_{\theta}\left(L(A_{i\Delta}\setminus
A_{(i+k)\Delta})=x_{i}-c\right)$
$\displaystyle\cdot\mathbb{P}_{\theta}\left(L(A_{(i+k)\Delta}\cap
A_{i\Delta})=c\right),$

as we wanted to show. ∎

###### Proof of Theorem 3.2.

Due to the stationarity and ergodicity of the IVT processes considered in this
paper, the normalized log-composite likelihood function will converge in
probability to its population counterpart, i.e.

$\displaystyle
Q_{n}(\theta):=\frac{1}{n}l_{CL}(\theta;X)=\frac{1}{n}\sum_{k=1}^{K}\sum_{i=1}^{n-k}\log
f(X_{(i+k)\Delta},X_{i\Delta};\theta)\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}\mathbb{E}\left[\sum_{k=1}^{K}\log
f(X_{k\Delta},X_{0};\theta)\right]=:Q(\theta),$

as $n\rightarrow\infty$. By the identifiability condition (3.5), and the fact
that the pairwise likelihoods are indeed proper (bivariate) likelihoods, the
information inequality implies that $Q(\theta)$ is uniquely maximized at
$\theta=\theta_{0}$ (Lemma 2.2 in Newey & McFadden 1994, p. 2124). The result
now follows from Theorem 4.1 and Theorem 4.3 in Wooldridge (1994). ∎

###### Proof of Theorem 3.3.

Let

$\displaystyle
s_{n}(\theta):=\frac{\partial}{\partial\theta}l_{CL}(\theta;X)=\sum_{k=1}^{K}\sum_{i=1}^{n-k}\frac{\partial}{\partial\theta}\log
f(X_{(i+k)\Delta},X_{i\Delta};\theta)$

denote the score function and consider the estimating equation related to the
MCL estimator $\hat{\theta}=\hat{\theta}^{CL}$, namely
$s_{n}(\hat{\theta})=0$. Using this equation, we Taylor expand
$s_{n}(\hat{\theta})$ around the true parameter vector $\theta_{0}$ to get

$\displaystyle
s_{n}(\theta_{0})+\frac{\partial}{\partial\theta^{\prime}}s_{n}(\bar{\theta})(\hat{\theta}-\theta_{0})=0,$

where $\bar{\theta}$ lies on the line segment between $\theta_{0}$ and
$\hat{\theta}$ and
$\frac{\partial}{\partial\theta^{\prime}}s_{n}(\bar{\theta})$ is shorthand for
$\frac{\partial}{\partial\theta^{\prime}}s_{n}(\theta)|_{\theta=\bar{\theta}}$.
Rearranging this equation and multiplying through by $\sqrt{n}$, we get

$\displaystyle\sqrt{n}(\hat{\theta}-\theta_{0})=-\left(\frac{1}{n}\frac{\partial}{\partial\theta^{\prime}}s_{n}(\bar{\theta})\right)^{-1}n^{-1/2}s_{n}(\theta_{0}).$

Stationarity and ergodicity, along with consistency of $\hat{\theta}$ due to
Theorem 3.2, implies that
$-\frac{1}{n}\frac{\partial}{\partial\theta^{\prime}}s_{n}(\bar{\theta})^{\prime}\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}H(\theta_{0})$
as $n\rightarrow\infty$. To prove the result, we thus need to show that
$n^{-1/2}s_{n}(\theta_{0})\stackrel{{\scriptstyle(d)}}{{\rightarrow}}N(0,V(\theta_{0}))$
as $n\rightarrow\infty$. By the mixing properties of the IVT process $X$,
given in Theorem 3.1, it is enough to show that
$\mathbb{E}[s_{n}(\theta_{0})]=0$ and
$Var\left(n^{-1/2}s_{n}(\theta_{0})\right)\rightarrow V(\theta_{0})$ as
$n\rightarrow\infty$ (e.g. Davidson 1994, Corollary 24.7, p. 387).555Note that
the crucial condition (c’) of Corollary 24.7 in Davidson (1994) relies on the
IVT process $X$ being mixing of size $\phi_{0}$ for some $\phi_{0}>1$. This
rules out the long memory processes, as shown in Theorem 3.1.

To show this, we consider, for simplicity, the case where $\theta$ is a
scalar. The vector case is similar, but with slightly more involved notation.
First note that, clearly,

$\displaystyle\mathbb{E}\left[s_{n}(\theta_{0})\right]=\mathbb{E}\left[\frac{\partial}{\partial\theta}l_{CL}(\theta;X)|_{\theta=\theta_{0}}\right]=0.$

Also

$\displaystyle Var\left(s_{n}(\theta_{0})\right)$
$\displaystyle=Var\left(\frac{\partial}{\partial\theta}l_{CL}(\theta;X)|_{\theta=\theta_{0}}\right)$
$\displaystyle=Var\left(\sum_{k=1}^{K}\sum_{i=1}^{n-k}\frac{\partial}{\partial\theta}\log\left.f(X_{i\Delta},X_{(i+k)\Delta};\theta)\right|_{\theta=\theta_{0}}\right)$
$\displaystyle=\sum_{k=1}^{K}\sum_{i=1}^{n-k}Var\left(\frac{\partial}{\partial\theta}\log\left.f(X_{i\Delta},X_{(i+k)\Delta};\theta)\right|_{\theta=\theta_{0}}\right)$
$\displaystyle+\sum_{k=1}^{K}\sum_{k^{\prime}=1}^{K}\sum_{i\neq
j}Cov\left(\frac{\partial}{\partial\theta}\log\left.f(X_{i\Delta},X_{(i+k)\Delta};\theta)\right|_{\theta=\theta_{0}},\frac{\partial}{\partial\theta}\log\left.f(X_{j\Delta},X_{(j+k^{\prime})\Delta};\theta)\right|_{\theta=\theta_{0}}\right).$

Due to stationarity, the first sum is $O(n)$ as $n\rightarrow\infty$. To prove
the proposition, we, therefore, investigate the second sum. With slight abuse
of notation, let
$\frac{\partial}{\partial\theta}\log\left.f(X_{i\Delta},X_{(i+k)\Delta};\theta)\right|_{\theta=\theta_{0}}$
be denoted by $\frac{\partial}{\partial\theta}\log
f(X_{i\Delta},X_{(i+k)\Delta};\theta_{0})$. For $l,k,k^{\prime}\geq 1$, define
also the joint probability mass functions

$\displaystyle
f_{l}(x_{1},x_{2},x_{3},x_{4};k,k^{\prime},\theta):=\mathbb{P}_{\theta}\left(X_{0}=x_{1},X_{k\Delta}=x_{2},X_{l\Delta}=x_{3},X_{(l+k^{\prime})\Delta}=x_{4}\right),\quad
x_{1},x_{2},x_{3},x_{4}\in\mathbb{Z}.$

and

$\displaystyle
f_{k}(x_{1},x_{2}):=\mathbb{P}_{\theta}\left(X_{0}=x_{1},X_{k\Delta}=x_{2}\right),\quad
x_{1},x_{2}\in\mathbb{Z}.$

Now, using that

$\displaystyle\mathbb{E}\left[\frac{\partial}{\partial\theta}\log
f(X_{i\Delta},X_{(i+k)\Delta};\theta_{0})\right]=0,$

we have, for all $i,j,k,k^{\prime}$,

$\displaystyle Cov\left(\frac{\partial}{\partial\theta}\log
f(X_{i\Delta},X_{(i+k)\Delta};\theta_{0}),\frac{\partial}{\partial\theta}\log
f(X_{j\Delta},X_{(j+k^{\prime})\Delta};\theta_{0})\right)$
$\displaystyle=\mathbb{E}\left[\frac{\partial}{\partial\theta}\log
f(X_{i\Delta},X_{(i+k)\Delta};\theta_{0})\frac{\partial}{\partial\theta}\log
f(X_{j\Delta},X_{(j+k^{\prime})\Delta};\theta_{0})\right]$
$\displaystyle=\sum_{x_{1}=-\infty}^{\infty}\sum_{x_{2}=-\infty}^{\infty}\sum_{x_{3}=-\infty}^{\infty}\sum_{x_{4}=-\infty}^{\infty}\frac{\partial}{\partial\theta}f_{k}(x_{1},x_{2};\theta_{0})\frac{\partial}{\partial\theta}f_{k^{\prime}}(x_{3},x_{4};\theta_{0})\frac{f_{|i-j|}(x_{1},x_{2},x_{3},x_{4};k,k^{\prime},\theta_{0})}{f_{k}(x_{1},x_{2};\theta_{0})f_{k^{\prime}}(x_{3},x_{4};\theta_{0})}$
$\displaystyle=\sum_{x_{1}=-\infty}^{\infty}\sum_{x_{2}=-\infty}^{\infty}\sum_{x_{3}=-\infty}^{\infty}\sum_{x_{4}=-\infty}^{\infty}\frac{\partial}{\partial\theta}f_{k}(x_{1},x_{2};\theta_{0})\frac{\partial}{\partial\theta}f_{k^{\prime}}(x_{3},x_{4};\theta_{0})\left(\frac{f_{|i-j|}(x_{1},x_{2},x_{3},x_{4};k,k^{\prime},\theta_{0})}{f_{k}(x_{1},x_{2};\theta_{0})f_{k^{\prime}}(x_{3},x_{4};\theta_{0})}-1\right),$

where the last equality follows because e.g.,

$\displaystyle\sum_{x_{1}=-\infty}^{\infty}\sum_{x_{2}=-\infty}^{\infty}\frac{\partial}{\partial\theta}f_{k}(x_{1},x_{2};\theta_{0})=\frac{\partial}{\partial\theta}\sum_{x_{1}=-\infty}^{\infty}\sum_{x_{2}=-\infty}^{\infty}f_{k}(x_{1},x_{2};\theta_{0})=\frac{\partial}{\partial\theta}1=0.$

Now, Lemma A.1 below shows that

$\displaystyle\left(\frac{f_{n}(x_{1},x_{2},x_{3},x_{4};k,k^{\prime},\theta_{0})}{f_{k}(x_{1},x_{2};\theta_{0})f_{k^{\prime}}(x_{3},x_{4};\theta_{0})}-1\right)=O(Leb(A_{n\Delta}\cap
A_{0})),\quad n\rightarrow\infty,$

from which we conclude, using Equation (2.5), i.e. $Leb(A_{n\Delta}\cap
A_{0})=\rho(n\Delta)Leb(A)$, and the condition on $\rho$ imposed in the
theorem, that the second sum in the expression for
$Var\left(s_{n}(\theta_{0})\right)$ is $O(n)$ as well. Indeed, taking it all
together, we have that, as $n\rightarrow\infty$,

$\displaystyle n^{-1}Var\left(s_{n}(\theta_{0})\right)$
$\displaystyle\rightarrow\sum_{k=1}^{K}Var\left(\frac{\partial}{\partial\theta}\log\left.f(X_{0},X_{k\Delta};\theta)\right|_{\theta=\theta_{0}}\right)$
$\displaystyle+2\sum_{k=1}^{K}\sum_{k^{\prime}=1}^{K}\sum_{i=1}^{\infty}Cov\left(\frac{\partial}{\partial\theta}\log\left.f(X_{0},X_{k\Delta};\theta)\right|_{\theta=\theta_{0}},\frac{\partial}{\partial\theta}\log\left.f(X_{i\Delta},X_{(i+k^{\prime})\Delta};\theta)\right|_{\theta=\theta_{0}}\right)$
$\displaystyle=:V(\theta_{0}),$

where the series converges. This finalizes the proof.

∎

######  Lemma A.1.

Fix $k,k^{\prime}\geq 1$, let $X$ be an IVT process, let
$f_{k}(\cdot,\cdot;\theta)$ be the joint PMF of $(X_{0},X_{k\Delta})$, and let
$f_{n}(\cdot,\cdot,\cdot,\cdot;k,k^{\prime},\theta)$ be the joint PMF of
$(X_{0},X_{k\Delta},X_{n\Delta},X_{(n+k^{\prime})\Delta})$. That is

$\displaystyle
f_{k}(x_{1},x_{2};\theta):=\mathbb{P}_{\theta}\left(X_{0}=x_{1},X_{k\Delta}=x_{2}\right),\quad
x_{1},x_{2}\in\mathbb{Z},$

and

$\displaystyle
f_{n}(x_{1},x_{2},x_{3},x_{4};k,k^{\prime},\theta):=\mathbb{P}_{\theta}\left(X_{0}=x_{1},X_{k\Delta}=x_{2},X_{n\Delta}=x_{3},X_{(n+k^{\prime})\Delta}=x_{4}\right),\quad
x_{1},x_{2},x_{3},x_{4}\in\mathbb{Z}.$

Define the function

$\displaystyle
G_{n}(x_{1},x_{2},x_{3},x_{4};k,k^{\prime},\theta):=\left(\frac{f_{n}(x_{1},x_{2},x_{3},x_{4};k,k^{\prime},\theta)}{f_{k}(x_{1},x_{2};\theta)f_{k^{\prime}}(x_{3},x_{4};\theta)}-1\right).$

The following holds:

$\displaystyle\frac{1}{Leb(A_{k\Delta}\cap
A_{n\Delta})}G_{n}(x_{1},x_{2},x_{3},x_{4};k,k^{\prime},\theta)\rightarrow
G(x_{1},x_{2},x_{3},x_{4};k,k^{\prime},\theta),\quad n\rightarrow\infty,$

where $G$ is a function, given in Equation (A.7) below, that depends on the
Lévy basis and trawl function of $X$. (In Remark A.1 below, we give the
function $G$ in the special case where the Lévy basis is Poissonian and the
trawl function is the Gamma trawl.)

###### Proof of Lemma A.1.

Letting
$f_{n}(x_{1},x_{2}|x_{3},x_{4};k,k^{\prime},\theta):=\mathbb{P}_{\theta}\left(X_{0}=x_{1},X_{k\Delta}=x_{2}|X_{n\Delta}=x_{3},X_{(n+k^{\prime})\Delta}=x_{4}\right)$,
we can write

$\displaystyle G_{n}(x_{1},x_{2},x_{3},x_{4};k,k^{\prime},\theta)$
$\displaystyle=\frac{f_{n}(x_{1},x_{2},x_{3},x_{4};\theta)-f_{k}(x_{1},x_{2};\theta)f_{k^{\prime}}(x_{3},x_{4};\theta)}{f_{k}(x_{1},x_{2};\theta)f_{k^{\prime}}(x_{3},x_{4};\theta)}$
$\displaystyle=\frac{f_{n}(x_{1},x_{2}|x_{3},x_{4};k,k^{\prime},\theta)f_{k^{\prime}}(x_{3},x_{4};\theta)-f_{k}(x_{1},x_{2};\theta)f_{k^{\prime}}(x_{3},x_{4};\theta)}{f_{k}(x_{1},x_{2};\theta)f_{k^{\prime}}(x_{3},x_{4};\theta)}$
$\displaystyle=(f_{n}(x_{1},x_{2}|x_{3},x_{4};k,k^{\prime},\theta)-f_{k}(x_{1},x_{2};\theta))f_{k}(x_{1},x_{2};\theta)^{-1}.$

To prove the lemma, we, therefore, study the asymptotic behaviour of
$f_{n}(x_{1},x_{2}|x_{3},x_{4};k,k^{\prime},\theta)-f_{k}(x_{1},x_{2};\theta)$
as $n\rightarrow\infty$.

Recall first the decomposition of the trawl sets into three disjoint sets
which led to Proposition 3.1, see Figure S1. In a similar manner, we can
decompose the four trawl sets associated to $X_{0}=L(A_{0})$,
$X_{k\Delta}=L(A_{k\Delta})$, $X_{n\Delta}=L(A_{n\Delta})$, and
$X_{(n+k^{\prime})\Delta}=L(A_{(n+k^{\prime})\Delta})$, into $10$ disjoint
sets as illustrated in Figure 5 below. For ease of notation, we ignore the
dependence on $n$, $k$, and $k^{\prime}$ for a moment and write

$\displaystyle A_{0}$ $\displaystyle=C_{3}\cup C_{4}\cup C_{6}\cup C_{7},\quad
A_{k\Delta}=C_{1}\cup C_{2}\cup C_{3}\cup C_{4}\cup C_{5}\cup C_{6},$
$\displaystyle A_{n\Delta}$ $\displaystyle=C_{1}\cup C_{2}\cup C_{3}\cup
C_{4}\cup D_{2}\cup D_{3},\quad A_{(n+k^{\prime})\Delta}=C_{1}\cup C_{3}\cup
D_{1}\cup D_{2},$

where the sets $C_{1},C_{2},\ldots,C_{7},D_{1},D_{2},D_{3}$ are disjoint. We
will use below that $\lim_{n\rightarrow\infty}Leb(C_{j})=0$ for $j=1,2,3,4$,
$\lim_{n\rightarrow\infty}Leb(C_{5})=Leb(A_{k\Delta}\setminus A_{0})$,
$\lim_{n\rightarrow\infty}Leb(C_{6})=Leb(A_{0}\cap A_{k\Delta})$,
$\lim_{n\rightarrow\infty}Leb(C_{7})=Leb(A_{0}\setminus A_{k\Delta})$,
$\lim_{n\rightarrow\infty}Leb(D_{1})=Leb(A_{k^{\prime}\Delta}\setminus
A_{0})$, $\lim_{n\rightarrow\infty}Leb(D_{2})=Leb(A_{0}\cap
A_{k^{\prime}\Delta})$, and
$\lim_{n\rightarrow\infty}Leb(D_{3})=Leb(A_{0}\setminus
A_{k^{\prime}\Delta})$, cf. Figure 5.

Using this decomposition and the law of total probability, we may write

$\displaystyle f(x_{1},x_{2};\theta)$
$\displaystyle=\sum_{c_{1},c_{2},c_{3},c_{4}=-\infty}^{\infty}\mathbb{P}_{\theta}\left(L(C_{6})+L(C_{7})=x_{1}-c_{3}-c_{4},L(C_{5})+L(C_{6})=x_{2}-c_{1}-c_{2}-c_{3}-c_{4}\right)$
$\displaystyle\cdot\prod_{j=1}^{4}\mathbb{P}_{\theta}\left(L(C_{j})=c_{j}\right)$

and

$\displaystyle f(x_{1},x_{2}|x_{3},x_{4};\theta)$
$\displaystyle=\sum_{c_{1},c_{2},c_{3},c_{4}=-\infty}^{\infty}\mathbb{P}_{\theta}\left(L(C_{6})+L(C_{7})=x_{1}-c_{3}-c_{4},L(C_{5})+L(C_{6})=x_{2}-c_{1}-c_{2}-c_{3}-c_{4}\right)$
$\displaystyle\cdot\mathbb{P}_{\theta}\left(L(C_{1})=c_{1},L(C_{2})=c_{2},L(C_{3})=c_{3},L(C_{4})=c_{4}|X_{n\Delta}=x_{3},X_{(n+k^{\prime})\Delta}=x_{4}\right).$

Taking these together, we get

$\displaystyle f_{n}(x_{1},x_{2}|x_{3},x_{4};\theta)-f(x_{1},x_{2};\theta)$
$\displaystyle=\sum_{c_{1},c_{2},c_{3},c_{4}=-\infty}^{\infty}\mathbb{P}_{\theta}\left(L(C_{6})+L(C_{7})=x_{1}-c_{3}-c_{4},L(C_{5})+L(C_{6})=x_{2}-c_{1}-c_{2}-c_{3}-c_{4}\right)$
$\displaystyle\cdot\left(\mathbb{P}_{\theta}\left(L(C_{1})=c_{1},L(C_{2})=c_{2},L(C_{3})=c_{3},L(C_{4})=c_{4}|X_{n\Delta}=x_{3},X_{(n+k^{\prime})\Delta}=x_{4}\right)-\prod_{j=1}^{4}\mathbb{P}_{\theta}\left(L(C_{j})=c_{j}\right)\right).$

Note that, for the first term in the parenthesis, the following holds

$\displaystyle\mathbb{P}_{\theta}\left(L(C_{1})=c_{1},L(C_{2})=c_{2},L(C_{3})=c_{3},L(C_{4})=c_{4}|X_{n\Delta}=x_{3},X_{(n+k^{\prime})\Delta}=x_{4}\right)$
$\displaystyle=f(x_{3},x_{4};\theta)^{-1}\mathbb{P}_{\theta}\left(L(C_{1})=c_{1},L(C_{2})=c_{2},L(C_{3})=c_{3},L(C_{4})=c_{4},X_{n\Delta}=x_{3},X_{(n+k^{\prime})\Delta}=x_{4}\right)$
$\displaystyle=f(x_{3},x_{4};\theta)^{-1}\mathbb{P}_{\theta}\left(L(D_{2})+L(D_{3})=x_{3}-c_{1}-c_{2}-c_{3}-c_{4},L(D_{1})+L(D_{2})=x_{4}-c_{1}-c_{3}\right)$
$\displaystyle\cdot\prod_{j=1}^{4}\mathbb{P}_{\theta}\left(L(C_{j})=c_{j}\right),$

which allows us to write

$\displaystyle f_{n}(x_{1},x_{2}|x_{3},x_{4};\theta)-f(x_{1},x_{2};\theta)$
$\displaystyle=\sum_{c_{1},c_{2},c_{3},c_{4}=-\infty}^{\infty}\mathbb{P}_{\theta}\left(L(C_{6})+L(C_{7})=x_{1}-c_{3}-c_{4},L(C_{5})+L(C_{6})=x_{2}-c_{1}-c_{2}-c_{3}-c_{4}\right)$
$\displaystyle\cdot\prod_{j=1}^{4}\mathbb{P}_{\theta}\left(L(C_{j})=c_{j}\right)f(x_{3},x_{4};\theta)^{-1}$
$\displaystyle\cdot\left(\mathbb{P}_{\theta}\left(L(D_{2})+L(D_{3})=x_{3}-c_{1}-c_{2}-c_{3}-c_{4},L(D_{1})+L(D_{2})=x_{4}-c_{1}-c_{3}\right)-f(x_{3},x_{4})\right).$

Define the set
$\mathcal{C}_{0}:=\\{(c_{1},c_{2},c_{3},c_{4})\in\mathbb{Z}^{4}:c_{i}\neq
0\textnormal{ for at least one }i=1,2,3,4\\}$. The above calculations imply
that

$\displaystyle
f_{n}(x_{1},x_{2}|x_{3},x_{4};\theta)-f(x_{1},x_{2};\theta)=f(x_{3},x_{4})^{-1}\left(F_{n}^{(1)}+F_{n}^{(2)}\right),$
(A.4)

where

$\displaystyle F_{n}^{(1)}$
$\displaystyle:=\mathbb{P}_{\theta}\left(L(C_{6})+L(C_{7})=x_{1},L(C_{5})+L(C_{6})=x_{2}\right)\prod_{j=1}^{4}\mathbb{P}_{\theta}\left(L(C_{j})=0\right)$
$\displaystyle\cdot\left(\mathbb{P}_{\theta}\left(L(D_{2})+L(D_{3})=x_{3},L(D_{1})+L(D_{2})=x_{4}\right)-f(x_{3},x_{4};\theta)\right)$

and

$\displaystyle F_{n}^{(2)}$
$\displaystyle:=\sum_{(c_{1},c_{2},c_{3},c_{4})\in\mathcal{C}_{0}}\mathbb{P}_{\theta}\left(L(C_{6})+L(C_{7})=x_{1}-c_{3}-c_{4},L(C_{5})+L(C_{6})=x_{2}-c_{1}-c_{2}-c_{3}-c_{4}\right)$
$\displaystyle\cdot\prod_{j=1}^{4}\mathbb{P}_{\theta}\left(L(C_{j})=c_{j}\right)$
$\displaystyle\cdot\left(\mathbb{P}_{\theta}\left(L(D_{2})+L(D_{3})=x_{3}-c_{1}-c_{2}-c_{3}-c_{4},L(D_{1})+L(D_{2})=x_{4}-c_{1}-c_{3}\right)-f(x_{3},x_{4};\theta)\right).$

We can think of $F_{n}^{(1)}$ as the part of
$(f_{n}(x_{1},x_{2}|x_{3},x_{4};\theta)-f(x_{1},x_{2};\theta))f(x_{3},x_{4})$
where $c_{1}=c_{2}=c_{3}=c_{4}=0$, while $F_{n}^{(2)}$ is the remainder.

We study first the behavior of $F_{n}^{(1)}$ as $n\rightarrow\infty$.
Considering the first two factors of this term, the continuity of the
probability measure $\mathbb{P}_{\theta}(\cdot)$ implies that

$\displaystyle\lim_{n\rightarrow\infty}\mathbb{P}_{\theta}\left(L(C_{6})+L(C_{7})=x_{1},L(C_{5})+L(C_{6})=x_{2}\right)\prod_{j=1}^{4}\mathbb{P}_{\theta}\left(L(C_{j})=0\right)$
$\displaystyle=\mathbb{P}_{\theta}\left(L(A_{0}\cap
A_{k\Delta})+L(A_{0}\setminus A_{k\Delta})=x_{1},L(A_{0}\cap
A_{k\Delta})+L(A_{k\Delta}\setminus A_{0})=x_{2}\right)$
$\displaystyle=f(x_{1},x_{2};\theta).$

The third term in $F_{n}^{(1)}$, i.e.

$\displaystyle\mathbb{P}_{\theta}\left(L(D_{2})+L(D_{3})=x_{3},L(D_{1})+L(D_{2})=x_{4}\right)-f(x_{3},x_{4};\theta)$

will, by the same logic as above, converge to zero as $n\rightarrow\infty$. In
fact, by decomposition of the trawl sets of $f(x_{3},x_{4};\theta)$ in the
same manner as above, we get that

$\displaystyle\mathbb{P}_{\theta}\left(L(D_{2})+L(D_{3})=x_{3},L(D_{1})+L(D_{2})=x_{4}\right)-f(x_{3},x_{4};\theta)$
$\displaystyle=\mathbb{P}_{\theta}\left(L(D_{2})+L(D_{3})=x_{3},L(D_{1})+L(D_{2})=x_{4}\right)\left(1-\prod_{j=1}^{4}\mathbb{P}_{\theta}\left(L(C_{j})=0\right)\right)$
$\displaystyle-\sum_{(c_{1},c_{2},c_{3},c_{4})\in\mathcal{C}_{0}}\mathbb{P}_{\theta}\left(L(D_{2})+L(D_{3})=x_{3}-c_{1}-c_{2}-c_{3}-c_{4},L(D_{1})+L(D_{2})=x_{4}-c_{1}-c_{3}\right)$
$\displaystyle\cdot\prod_{j=1}^{4}\mathbb{P}_{\theta}\left(L(C_{j})=c_{j}\right).$

In the first part of Lemma A.2 below, we show that there exists a constant
$\tilde{\nu}>0$, such that

$\displaystyle
1-\prod_{j=1}^{4}\mathbb{P}_{\theta}\left(L(C_{j})=0\right)=\tilde{\nu}Leb(A_{k\Delta}\cap
A_{n\Delta})+o\left(Leb(A_{k\Delta}\cap A_{n\Delta})\right),$

as $n\rightarrow\infty$. Similarly, in the second part of Lemma A.2, we show
that there exists a non-negative function $\tilde{\eta}$ concentrated on the
integers, such that, for $c_{j}\neq 0$,

$\displaystyle\mathbb{P}_{\theta}\left(L(C_{j})=c_{j}\right)=\tilde{\eta}(c_{j})\tilde{\nu}Leb(C_{j})e^{-\tilde{\nu}Leb(C_{j})}+o\left(Leb(C_{j})\right),$

as $n\rightarrow\infty$, while

$\displaystyle\mathbb{P}_{\theta}\left(L(C_{j})=0\right)=e^{-\tilde{\nu}Leb(C_{j})}+o\left(Leb(C_{j})\right),$

as $n\rightarrow\infty$. This shows that for quadruplets
$(c_{1},c_{2},c_{3},c_{4})$ where $c_{i}\neq 0$ for some $i=1,2,3,4$ and
$c_{j}=0$ for the remaining $j\neq i$ (i.e. quadruplets of the form
$(c_{1},0,0,0)$, $(0,c_{2},0,0)$, $(0,0,c_{3},0)$ or $(0,0,0,c_{4})$), we have

$\displaystyle\prod_{j=1}^{4}\mathbb{P}_{\theta}\left(L(C_{j})=c_{j}\right)$
$\displaystyle=\tilde{\eta}(c_{i})\tilde{\nu}Leb(C_{i})e^{-\tilde{\nu}\sum_{j=1}^{4}Leb(C_{j})}+o\left(\sum_{j=1}^{4}Leb(C_{j})\right)$
$\displaystyle=\tilde{\eta}(c_{i})\tilde{\nu}Leb(C_{i})e^{-\tilde{\nu}Leb(A_{k\Delta}\cap
A_{n\Delta})}+o\left(Leb(A_{k\Delta}\cap A_{n\Delta})\right),$

as $n\rightarrow\infty$. Conversely, for quadruplets
$(c_{1},c_{2},c_{3},c_{4})$ where $c_{i},c_{j}\neq 0$ for at least two
distinct $i,j=1,2,3,4$, we have

$\displaystyle\prod_{j=1}^{4}\mathbb{P}_{\theta}\left(L(C_{j})=c_{j}\right)=o\left(Leb(A_{k\Delta}\cap
A_{n\Delta})\right),$

as $n\rightarrow\infty$.

Define the numbers
$a_{j}:=\lim_{n\rightarrow\infty}\frac{Leb(C_{j})}{Leb(A_{n\Delta}\cap
A_{k\Delta})}\geq 0$, $j=1,2,3,4$, which are such that $\sum_{j=1}^{4}a_{j}=1$
since $C_{1}\cup C_{2}\cup C_{3}\cup C_{4}=A_{n\Delta}\cap A_{k\Delta}$, cf.
Figure 5 below. Taking the above together, we may, after a little algebra,
conclude that, as $n\rightarrow\infty$,

$\displaystyle\frac{F_{n}^{(1)}}{Leb(A_{k\Delta}\cap A_{n\Delta})}\rightarrow$
$\displaystyle\ \tilde{\nu}f(x_{1},x_{2};\theta)f(x_{3},x_{4};\theta)$ (A.5)
$\displaystyle-\tilde{\nu}f(x_{1},x_{2})\sum_{c\neq
0}\tilde{\eta}(c)\left((a_{1}+a_{3})f(x_{3}-c,x_{4}-c;\theta)+(a_{2}+a_{4})f(x_{3}-c,x_{4};\theta)\right).$

Turning now to $F_{n}^{(2)}$, similar calculations yield that, as
$n\rightarrow\infty$,

$\displaystyle\frac{F_{n}^{(2)}}{Leb(A_{k\Delta}\cap A_{n\Delta})}\rightarrow$
$\displaystyle-\tilde{\nu}f(x_{3},x_{4};\theta)\sum_{c\neq
0}\tilde{\eta}(c)\left((a_{1}+a_{2})f(x_{1},x_{2}-c;\theta)+(a_{3}+a_{4})f(x_{1}-c,x_{2}-c;\theta)\right)$
$\displaystyle+\tilde{\nu}a_{1}\sum_{c\neq
0}\tilde{\eta}(c)f(x_{1},x_{2}-c)f(x_{3}-c,x_{4}-c)$
$\displaystyle+\tilde{\nu}a_{2}\sum_{c\neq
0}\tilde{\eta}(c)f(x_{1},x_{2}-c)f(x_{3}-c,x_{4})$
$\displaystyle+\tilde{\nu}a_{3}\sum_{c\neq
0}\tilde{\eta}(c)f(x_{1}-c,x_{2}-c)f(x_{3}-c,x_{4}-c)$
$\displaystyle+\tilde{\nu}a_{4}\sum_{c\neq
0}\tilde{\eta}(c)f(x_{1}-c,x_{2}-c)f(x_{3}-c,x_{4}).$ (A.6)

Finally, recalling Equation (A.4), we can conclude that

$\displaystyle\lim_{n\rightarrow\infty}\frac{G_{n}(x_{1},x_{2},x_{3},x_{4};k,k^{\prime},\theta)}{Leb(A_{k\Delta}\cap
A_{n\Delta})}$
$\displaystyle=\lim_{n\rightarrow\infty}\frac{F_{n}^{(1)}+F_{n}^{(2)}}{Leb(A_{k\Delta}\cap
A_{n\Delta})}f_{k}(x_{1},x_{2};\theta)^{-1}f_{k^{\prime}}(x_{3},x_{4};\theta)^{-1}$
$\displaystyle=:G(x_{1},x_{2},x_{3},x_{4};k,k^{\prime},\theta),$ (A.7)

where $\lim_{n\rightarrow\infty}\frac{1}{Leb(A_{k\Delta}\cap
A_{n\Delta})}F_{n}^{(i)}$ for $i=1,2$ are given above in Equations
(A.5)–(A.6). (See the following Remark A.1 for how the expression for $G$
simplifies slightly in the case of an IVT process with Poisson Lévy basis and
Gamma trawl function.)

Figure 5: Illustration of the decomposition of the four trawl sets
$A_{0},A_{k\Delta},A_{n\Delta},A_{(n+k^{\prime})\Delta}$.

∎

######  Remark A.1.

Note that in the case of a Poisson Lévy basis (Example 2.1 in the main
article), we have $\tilde{\nu}=\nu$, $\tilde{\eta}(1)=1$, and
$\tilde{\eta}(c)=0$ for $c\neq 1$. Further, in the case of $d$ being a Gamma
trawl function (Example 2.5), it is straightforward to show that
$a_{1}=a_{2}=a_{4}=0$ and $a_{3}=1$. For this specification, the limit in the
proof of Lemma A.1 simplifies somewhat. Indeed, in this case, Equation (A.7)
yields

$\displaystyle
G(x_{1},x_{2},x_{3},x_{4};k,k^{\prime},\theta)=\nu\left(\frac{f_{k}(x_{1}-1,x_{2}-1;\theta)}{f_{k}(x_{1},x_{2};\theta)}-1\right)\left(\frac{f_{k^{\prime}}(x_{3}-1,x_{4}-1;\theta)}{f_{k^{\prime}}(x_{3},x_{4};\theta)}-1\right).$

######  Lemma A.2.

In the setting of the proof of Lemma A.1, we have the following two-part
result.

(First part) There exists a constant $\tilde{\nu}>0$, such that

$\displaystyle
1-\prod_{j=1}^{4}\mathbb{P}_{\theta}\left(L(C_{j})=0\right)=\tilde{\nu}Leb(A_{k\Delta}\cap
A_{n\Delta})+o\left(Leb(A_{k\Delta}\cap A_{n\Delta})\right),$

as $n\rightarrow\infty$.

(Second part) There exists a non-negative function $\tilde{\eta}$,
concentrated on the integers, such that, for $c_{j}\neq 0$,

$\displaystyle\mathbb{P}_{\theta}\left(L(C_{j})=c_{j}\right)=\tilde{\eta}(c_{j})\tilde{\nu}Leb(C_{j})e^{-\tilde{\nu}Leb(C_{j})}+o\left(Leb(C_{j})\right),$

as $n\rightarrow\infty$, while

$\displaystyle\mathbb{P}_{\theta}\left(L(C_{j})=0\right)=e^{-\tilde{\nu}Leb(C_{j})}+o\left(Leb(C_{j})\right),$

as $n\rightarrow\infty$.

###### Proof of Lemma A.2.

The proof of the lemma relies on the alternative representation of the Lévy
basis $L$ given at the start of the Appendix, see Equation (A.1).

(Proof of second part) Note that since $\eta(0)=0$, we have
$\mathbb{P}_{\theta}\left(Y_{i}=0\right)=0$. Using this in the setup of Lemma
A.1, we get, from Equations (A.1) and (A.2),

$\displaystyle\mathbb{P}_{\theta}\left(L(C_{j})=0\right)$
$\displaystyle=e^{-\tilde{\nu}Leb(C_{j})}+\sum_{q=2}^{\infty}\mathbb{P}_{\theta}\left(\sum_{i=1}^{q}Y_{i}=x\right)\mathbb{P}_{\theta}(\tilde{N}(C_{j})=q)$
$\displaystyle=e^{-\tilde{\nu}Leb(C_{j})}+o\left(Leb(C_{j})\right),$

as $n\rightarrow\infty$, while for $c_{j}\neq 0$,

$\displaystyle\mathbb{P}_{\theta}\left(L(C_{j})=c_{j}\right)$
$\displaystyle=\mathbb{P}_{\theta}\left(Y_{1}=c_{j}\right)\mathbb{P}_{\theta}(\tilde{N}(C_{j})=1)+\sum_{q=2}^{\infty}\mathbb{P}_{\theta}\left(\sum_{i=1}^{q}Y_{i}=x\right)\mathbb{P}_{\theta}(\tilde{N}(C_{j})=q)$
$\displaystyle=\tilde{\eta}(c_{j})\tilde{\nu}Leb(C_{j})e^{-\tilde{\nu}Leb(C_{j})}+o\left(Leb(C_{j})\right),$

as $n\rightarrow\infty$. This proves the second part of the lemma.

(Proof of first part) As for the first part, use Equations (A.1) and (A.2) to
write

$\displaystyle\prod_{j=1}^{4}\mathbb{P}_{\theta}\left(L(C_{j})=0\right)$
$\displaystyle=e^{-\tilde{\nu}\sum_{j=1}^{4}Leb(C_{j})}+o\left(\sum_{j=1}^{4}Leb(C_{j})\right)$
$\displaystyle=e^{-\tilde{\nu}Leb(A_{n\Delta}\cap
A_{k\Delta})}+o\left(Leb(A_{n\Delta}\cap A_{k\Delta})\right)$
$\displaystyle=1-\tilde{\nu}Leb(A_{n\Delta}\cap
A_{k\Delta})+o\left(Leb(A_{n\Delta}\cap A_{k\Delta})\right),$

as $n\rightarrow\infty$, where we in the last line Taylor expanded the
exponential function. This proves the first part of the lemma.

∎

###### Proof of Theorem 3.4.

With similar calculations to those used in the proof of Theorem 3.3, we can
write

$\displaystyle
n^{H/2}(\hat{\theta}-\theta_{0})=-\left(\frac{1}{n}\frac{\partial}{\partial\theta^{\prime}}s_{n}(\bar{\theta})\right)^{-1}n^{H/2-1}s_{n}(\theta_{0}).$

We again have that
$-\frac{1}{n}\frac{\partial}{\partial\theta}s_{n}(\bar{\theta})^{\prime}\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}H(\theta_{0})$
as $n\rightarrow\infty$. As in the proof of Theorem 3.3, we can write

$\displaystyle Var\left(s_{n}(\theta_{0})\right)=A_{1,n}+A_{2,n},$

where $A_{1,n}$ is $O(n)$. Further, Lemma A.1 implies that

$\displaystyle A_{2,n}=O(n^{2}Leb(A_{n\Delta}\cap A_{0})),$

as $n\rightarrow\infty$. Equation (2.5) along with the condition on $\rho$
imposed in the theorem thus yields

$\displaystyle A_{2,n}=O(n^{2}L_{\infty}(n\Delta)n^{-H}),$

as $n\rightarrow\infty$. Using this, we get that, for all $\epsilon>0$,

$\displaystyle n^{H-2\pm
2\epsilon}Var\left(s_{n}(\theta_{0})\right)=O(n^{H-1\pm
2\epsilon})+O(L_{\infty}(n\Delta)n^{\pm 2\epsilon}),$ (A.8)

as $n\rightarrow\infty$. Finally, we recall the so-called _Potter bounds_ for
slowly varying functions: Since $L_{\infty}$ is a slowly varying function, for
all $\epsilon>0$ it holds that (Bingham et al. 1989, Theorem 1.5.6(ii))

$\displaystyle
L_{\infty}(n\Delta)n^{\epsilon}\rightarrow\infty\quad\textnormal{ and }\quad
L_{\infty}(n\Delta)n^{-\epsilon}\rightarrow 0,$

as $n\rightarrow\infty$. Combining the Potter bounds with (A.8) yields the
required results. ∎

###### Proof of Theorem 3.5.

Set, for simplicity, $K=1$. From the proof of Theorem 3.3, it is clear that
the asymptotic behaviour of $\hat{\nu}^{CL}$ is governed by the asymptotic
behaviour of the score function

$\displaystyle s_{n}(\nu_{0})$
$\displaystyle=\sum_{i=1}^{n}\left.\frac{\partial}{\partial\nu}\log
f(X_{i\Delta},X_{(i+1)\Delta})\right|_{\nu=\nu_{0}}=\sum_{i=1}^{n}\left.\frac{1}{f(X_{i\Delta},X_{(i+1)\Delta})}\frac{\partial}{\partial\nu}f(X_{i\Delta},X_{(i+1)\Delta})\right|_{\nu=\nu_{0}}.$
(A.9)

Let $B$ be a Borel set. Since $L^{\prime}\sim Poi(\nu)$, we have

$\displaystyle\mathbb{P}\left(L(B)=x\right)=\frac{[\nu Leb(B)]^{x}\exp(-\nu
Leb(B))}{x!},\quad x=0,1,\ldots.$

Hence,

$\displaystyle\frac{\partial}{\partial\nu}\mathbb{P}\left(L(B)=x\right)=\left(x\nu^{-1}-Leb(B)\right)\mathbb{P}\left(L(B)=x\right),\quad
x=0,1,\ldots.$

Using this, along with Proposition 3.1, it is straightforward to show that

$\displaystyle\frac{1}{f(X_{(i+1)\Delta},X_{i\Delta})}\frac{\partial}{\partial\nu}f(X_{i\Delta},X_{(i+1)\Delta})$
$\displaystyle=\nu^{-1}(X_{(i+1)\Delta}+X_{i\Delta})-C_{i}-D_{i}$ (A.10)

where

$\displaystyle C_{i}$
$\displaystyle=\frac{\nu^{-1}}{f(X_{(i+1)\Delta},X_{i\Delta})}\sum_{c=0}^{\min(X_{i\Delta},X_{(i+1)\Delta})}c\cdot
P(L(A_{(i+1)\Delta})=X_{(i+1)\Delta},L(A_{i\Delta})=X_{i\Delta},(L(A_{i\Delta}\cap
A_{(i+1)\Delta})=c)$
$\displaystyle=\nu^{-1}\sum_{c=0}^{\min(X_{i\Delta},X_{(i+1)\Delta})}c\cdot
P((L(A_{i\Delta}\cap
A_{(i+1)\Delta})=c|L(A_{(i+1)\Delta})=X_{(i+1)\Delta},L(A_{i\Delta})=X_{i\Delta})$
$\displaystyle=\nu^{-1}\mathbb{E}[L(A_{i\Delta}\cap
A_{(i+1)\Delta})|X_{i\Delta},X_{(i+1)\Delta}]$
$\displaystyle=\nu^{-1}\mathbb{E}[L(A_{i+k})-L(A_{(i+1)\Delta}\setminus
A_{i\Delta})|X_{i\Delta},X_{(i+1)\Delta}]$
$\displaystyle=\nu^{-1}\mathbb{E}[X_{(i+1)\Delta}-L(A_{(i+1)\Delta}\setminus
A_{i\Delta})|X_{i\Delta},X_{(i+1)\Delta}]$
$\displaystyle=\nu^{-1}\left(X_{(i+1)\Delta}-\mathbb{E}[L(A_{(i+1)\Delta}\setminus
A_{i\Delta})|X_{i\Delta},X_{(i+1)\Delta}]\right)$ (A.11)

and

$\displaystyle D_{i}$ $\displaystyle=Leb(A_{(i+1)\Delta}\setminus
A_{i\Delta})+Leb(A_{i\Delta}\setminus A_{(i+1)\Delta})+Leb(A_{i\Delta}\cap
A_{(i+1)\Delta})$ $\displaystyle=Leb(A)+Leb(A_{(i+1)\Delta}\setminus
A_{i\Delta}).$ (A.12)

Define $U_{i}:=\mathbb{E}[L(A_{(i+1)\Delta}\setminus
A_{i\Delta})|X_{i\Delta},X_{(i+1)\Delta}]$, $i=1,2,\ldots,n$. Recall that
$\mathbb{E}[L(B)]=\nu Leb(B)$ from which we deduce that

$\displaystyle\mathbb{E}[X_{i\Delta}]=\nu Leb(A),\quad\textnormal{ and
}\quad\mathbb{E}[U_{i}]=\nu Leb(A_{(i+1)\Delta}\setminus A_{i\Delta}).$

Using the above and Equations (A.10)–(A.11) in Equation (A.9), we get

$\displaystyle s_{n}(\nu_{0})=\nu_{0}^{-1}S_{n}(X)+\nu_{0}^{-1}S_{n}(U),$

Using arguments similar to those in the proof of Theorem 3.3, the result now
follows from Theorem 3.6.

∎

Before we provide the proof of Theorem 3.6, we will need the following two
lemmas. For two sequences $a_{n},b_{n}$, we will write “$a_{n}\sim b_{n}$” if
$a_{n}/b_{n}$ tends to a non-zero constant as $n\rightarrow\infty$.

######  Lemma A.3.

Let Assumption 3.4 hold and define the sets

$\displaystyle B_{i,k}=\\{(x,s):(i-1)\Delta\leq s\leq
i\Delta,d(s-(k+1)\Delta)\leq x<d(s-k\Delta)\\},$ (A.13)

for $i=1,2,\ldots,n$ and $k\in\mathbb{N}$. Then, for all $i=1,2,\ldots,n$,

$\displaystyle Leb(B_{i,k})=\Delta^{2}d^{\prime}(-(k-i+1+r_{2})\Delta),$

where $r_{2}=r_{2}(i,k)\in(-1,1)$. In particular,

$\displaystyle Leb(B_{1,N})\sim g_{2}(N)N^{-H-2},\quad N\rightarrow\infty,$

and

$\displaystyle\sum_{k=1}^{\infty}kLeb(B_{1,k})<\infty.$

###### Proof of Lemma A.3.

Fix $i\in\\{1,2,\ldots,n\\}$ and $k\geq 1$. Using the mean value theorem
twice, we may write,

$\displaystyle Leb(B_{i,k})$
$\displaystyle=\int_{(i-1)\Delta}^{i\Delta}\left(d(x-k\Delta)-d(x-(k+1)\Delta)\right)dx$
$\displaystyle=\Delta\int_{(i-1)\Delta}^{i\Delta}d^{\prime}(x-(k+r_{1})\Delta)dx$
$\displaystyle=\Delta[d(-(k+r_{1}-i)\Delta)-d(-(k+r_{1}-i+1)\Delta)]$
$\displaystyle=\Delta^{2}d^{\prime}(-(k-i+1+r_{2})\Delta),$

where $r_{1}\in(0,1)$ and $r_{2}\in(-1,1)$, which proves the first part of the
lemma. Note that $r_{1}$ and $r_{2}$ will in general depend on $k$ and $i$,
but we will suppress this for ease of notation. The latter parts of the lemma
now follow from Assumption 3.4. ∎

######  Lemma A.4.

Let Assumption 3.4 hold and define the sets

$\displaystyle\bar{B}_{1,k}=\cup_{j=k+1}^{\infty}B_{1,j},\qquad
k\in\mathbb{N}.$ (A.14)

Then,

$\displaystyle Leb(\bar{B}_{1,N})$
$\displaystyle=\int_{0}^{\Delta}d(s-(N+1)\Delta)ds\sim g_{1}(N)N^{-H-1},$

as $N\rightarrow\infty$. In particular, using also Lemma A.3,

$\displaystyle\sum_{k=1}^{\infty}kLeb(B_{1,k})^{1/2}Leb(\bar{B}_{1,k})<\infty.$

###### Proof of Lemma A.4.

Using a change-of-variables, Assumption 3.4, Karamata’s theorem, the mean
value theorem, and standard properties of slowly varying functions, we get

$\displaystyle Leb(\bar{B}_{1,N})$
$\displaystyle=\int_{0}^{\Delta}d(s-(N+1)\Delta)ds$
$\displaystyle=\int_{N\Delta}^{(N+1)\Delta}d(-y)dy$
$\displaystyle=\int_{N\Delta}^{\infty}g_{1}(y)y^{-H-1}dy-\int_{(N+1)\Delta}^{\infty}g_{1}(y)y^{-H-1}dy$
$\displaystyle\sim\frac{1}{H}g_{1}(N\Delta)(N\Delta)^{-H}-\frac{1}{H}g_{1}((N+1)\Delta)((N+1)\Delta)^{-H}$
$\displaystyle=\frac{\Delta^{-H}}{H}g_{1}(N\Delta)\left(N^{-H}-\frac{g_{1}((N+1)\Delta)}{g_{1}(N\Delta)}(N+1)^{-H}\right)$
$\displaystyle=\frac{\Delta^{-H}}{H}g_{1}(N\Delta)\left(N^{-H}-(N+1)^{-H}-\left(\frac{g_{1}((N+1)\Delta)}{g_{1}(N\Delta)}-1\right)(N+1)^{-H}\right)$
$\displaystyle\sim\frac{\Delta^{-H}}{H}g_{1}(N\Delta)\left(N^{-H}-(N+1)^{-H}\right)$
$\displaystyle=\Delta^{-H}g_{1}(N\Delta)(N+r_{1})^{-H-1}$ $\displaystyle\sim
g_{1}(N\Delta)N^{-H-1},$ $\displaystyle\sim g_{1}(N)N^{-H-1},$

as $N\rightarrow\infty$, where $r_{1}\in(0,1)$. The latter parts of the lemma
now follow from Assumption 3.4. ∎

###### Proof of Theorem 3.6.

Suppose for simplicity that $L^{\prime}\sim Poi(\nu)$ for some $\nu>0$. The
general case follows along similar lines to the proof in the Poisson case,
using the compound Poisson representation of the IVT process as presented
above.

We follow the strategy of the proof outlined in Theorem 2 of Doukhan et al.
(2019), adapted to our purposes. First, define the sequence of random
variables

$\displaystyle Z_{i}$ $\displaystyle=Z_{i\Delta}=L(A_{i\Delta}\setminus
A_{(i-1)\Delta})+\sum_{j=1}^{\infty}L((A_{i\Delta}\cap
A_{(i-j)\Delta})\setminus A_{(i-j-1)\Delta}+(0,j\Delta)),\quad
i=1,2,\ldots,n,$

where the sum converges almost surely by Kolmogorov’s three-series theorem.
Note that

$\displaystyle(A_{i\Delta}\cap A_{(i-j)\Delta})\setminus
A_{(i-j-1)\Delta}=\\{(x,s):(i-j-1)\Delta<s\leq(i-j)\Delta,0\leq
d(s-i\Delta)\\},$

and, hence, applying a time shift of $j\Delta$ leads to

$(A_{i\Delta}\cap A_{(i-j)\Delta})\setminus A_{(i-j-1)\Delta}+(0,j\Delta)\\\
=\\{(x,s):(i-1)\Delta<s\leq i\Delta,0\leq d(s-(i+j)\Delta)\\}.$

We observe that we can write $Z_{i}$ as a sum of independent variables,

$\displaystyle Z_{i}$ $\displaystyle=\sum_{k=1}^{\infty}kL(B_{i,k}),$

where $B_{i,k}$ are disjoint sets, given as in Equation (A.14). The
construction of $Z_{1}$, and the decomposition into independent variables, are
illustrated in Figure 6. Note furthermore that, by construction, the sequence
$\\{Z_{i}\\}_{i=1}^{\infty}$ is iid. Let $S_{n}(\cdot)$ denote the de-meaned
partial sums of a process, e.g.
$S_{n}(X)=\sum_{i=1}^{n}(X_{i}-\mathbb{E}[X_{i}])$. Here we use the short-hand
notation that $X_{i}=X_{i\Delta}$. Our aim is to show that

  1. (i)

$\mathbb{P}(Z_{1}>y)\sim g(y)y^{-H-1}$, as $y\rightarrow\infty$, where $g$ is
a slowly varying function.

  2. (ii)

$\mathbb{E}[|S_{n}(X)-S_{n}(Z)|]=o(n^{1/(H+1)})$, as $n\rightarrow\infty$.

Indeed, the result of the present theorem follows from (i) and (ii). To see
this, note that (i) implies that (Ibragimov & Linnik 1971, Theorem 2.6.7)

$\displaystyle
n^{-1/(H+1)}S_{n}(Z)\stackrel{{\scriptstyle(d)}}{{\rightarrow}}L_{1+H},\quad
n\rightarrow\infty,$

where $L_{1+H}$ is an $(1+H)$-stable random variable with characteristic
function as given in Equation (3.6). Secondly, (ii) now implies that

$\displaystyle
n^{-1/(H+1)}S_{n}(X)=n^{-1/(H+1)}S_{n}(Z)+o_{\mathbb{P}}(1)\stackrel{{\scriptstyle(d)}}{{\rightarrow}}L_{1+H},\quad
n\rightarrow\infty,$

which is what we wanted to show.

We proceed to prove (i). First, let $Z^{*}$ be given by

$\displaystyle Z^{*}=\sum_{k=1}^{\infty}kI(L(B_{1,k})>0),$

where $I(A)$ is the indicator of the set $A$. Note that $Z_{1}\geq Z^{*}\geq
0$. Second, define $Z^{**}=Z_{1}-Z^{*}$. Let $g$ be a slowly varying function.
As in Doukhan et al. (2019), we prove that (a)
$\mathbb{P}(Z^{*}>y)=g(y)y^{-H-1}+o(y^{-H-1})$ and (b)
$\mathbb{P}(Z^{**}>y)=o(y^{-H-1})$ as $y\rightarrow\infty$, which allows us to
deduce (i), i.e. $\mathbb{P}(Z_{1}>y)\sim g(y)y^{-H-1}$ as
$y\rightarrow\infty$.

To prove (a), define the random variables

$\displaystyle Z_{1}^{*}$
$\displaystyle:=\sum_{k=1}^{\infty}kI(L(B_{1,k})>0)I(L(\bar{B}_{1,k})=0),$
$\displaystyle Z_{2}^{*}$
$\displaystyle:=\sum_{k=1}^{\infty}kI(L(B_{1,k})>0)I(L(\bar{B}_{1,k})=1),$

where, as before,

$\displaystyle\bar{B}_{1,k}=\\{(x,s):0\leq s\leq\Delta,0\leq
x<d(s-(k+1)\Delta)\\}=\cup_{j=k+1}^{\infty}B_{1,j}.$ (A.15)

The intuition is that $Z_{1}^{*}$ takes the value $k$ if there is an
observation in $B_{1,k}$ but not in $B_{1,k+1},B_{1,k+2},\ldots$, while
$Z_{2}^{*}$ takes the value $k$ if there is an observation in $B_{1,k}$ and
precisely one observation in $B_{1,k+1},B_{1,k+2},\ldots$. Note that
$Z^{*}\geq Z_{1}^{*}+Z_{2}^{*}\geq 0$. Define, analogously to above
$Z_{3}^{*}=Z^{*}-Z_{1}^{*}-Z_{2}^{*}$, which is $Z^{*}$ with the two largest
“points” removed.666It can be shown that if $H\in(1/2,1)$, then we can set
$Z_{2}^{*}=0$, i.e., in this case, we only need to remove the largest point
from $Z^{*}$ for the proof to go through. We prove that (a’)
$\mathbb{P}(Z_{1}^{*}>y)\sim g(y)y^{-H-1}$ and (b’)
$\mathbb{P}(Z_{2}^{*}>y)=o(y^{-H-1})$ and
$\mathbb{P}(Z_{3}^{*}>y)=o(y^{-H-1})$ as $y\rightarrow\infty$, which allows us
to deduce (a), i.e. $\mathbb{P}(Z^{*}>y)\sim g(y)y^{-H-1}$ as
$y\rightarrow\infty$. As noted in Doukhan et al. (2019), to prove (a’), we
only need to prove $\mathbb{P}(Z_{1}^{*}>N)\sim g(N)N^{-H-1}+o(N^{-H-1})$ for
$N\rightarrow\infty$, where $N\in\mathbb{N}$. Let therefore $N\in\mathbb{N}$.
We get, using Lemma A.4 and the fact that $L^{\prime}\sim Poi(\nu)$,

$\displaystyle\mathbb{P}(Z_{1}^{*}>N)$
$\displaystyle=\mathbb{P}\left(\sum_{k=1}^{\infty}kI(L(B_{1,k})>0)I(L(\bar{B}_{1,k})=0)>N\right)$
$\displaystyle=\mathbb{P}\left(L(\bar{B}_{1,N})>0\right)$
$\displaystyle=1-\mathbb{P}\left(L(\bar{B}_{1,N})=0\right)$
$\displaystyle=1-e^{-\nu Leb(\bar{B}_{1,N})}$
$\displaystyle=Leb(\bar{B}_{1,N})+o(Leb(\bar{B}_{1,N}))$ $\displaystyle\sim
g_{1}(N)N^{-H-1},$

as $N\rightarrow\infty$.

To prove (b’), note that, by Lemma A.4,

$\displaystyle\mathbb{P}(Z_{2}^{*}>N)$
$\displaystyle=\mathbb{P}\left(\sum_{k=1}^{\infty}kI(L(B_{1,k})>0)I(L(\bar{B}_{1,k})=1)>N\right)$
$\displaystyle=\mathbb{P}\left(L(\bar{B}_{1,N})>1)\right)$
$\displaystyle=1-\mathbb{P}\left(L(\bar{B}_{1,N})=0\right)-\mathbb{P}\left(L(\bar{B}_{1,N})=1)\right)$
$\displaystyle=1-e^{-\nu Leb(\bar{B}_{1,N})}-\nu Leb(\bar{B}_{1,N})e^{-\nu
Leb(\bar{B}_{1,N})}$ $\displaystyle\leq\nu Leb(\bar{B}_{1,N})-\nu
Leb(\bar{B}_{1,N})e^{-\nu Leb(\bar{B}_{1,N})}$ $\displaystyle=\nu
Leb(\bar{B}_{1,N})(1-e^{-\nu Leb(\bar{B}_{1,N})})$
$\displaystyle\leq\nu^{2}Leb(\bar{B}_{1,N})^{2}$ $\displaystyle\sim
g_{1}(N)^{2}N^{-2-2H}$ $\displaystyle=o(N^{-1-H}),$

as $N\rightarrow\infty$, by Lemma A.4. To show that
$\mathbb{P}(Z_{3}^{*}>y)=o(y^{-H-1})$ as $y\rightarrow\infty$, we show that
$Z_{3}^{*}$ is bounded in $L^{2}$. Let $\epsilon\in(0,H)$. Using Minkowski’s
inequality, the independence of $L(B_{1,k})$ and $L(\bar{B}_{1,k})$, and
Lemmas A.3 and A.4, we get

$\displaystyle\mathbb{E}[(Z_{3}^{*})^{2}]^{1/2}$
$\displaystyle=\mathbb{E}\left[\left(Z^{*}-Z_{1}^{*}-Z_{2}^{*}\right)^{2}\right]^{1/2}$
$\displaystyle=\mathbb{E}\left[\left(\sum_{k=1}^{\infty}kI(L(B_{1,k})>0)\left(1-I(L(\bar{B}_{1,k})=0)-I(L(\bar{B}_{1,k})=1)\right)\right)^{2}\right]^{1/2}$
$\displaystyle\leq\sum_{k=1}^{\infty}\mathbb{E}\left[k^{2}I(L(B_{1,k})>0)^{2}\left(1-I(L(\bar{B}_{1,k})=0)-I(L(\bar{B}_{1,k})=1)\right)^{2}\right]^{1/2}$
$\displaystyle=\sum_{k=1}^{\infty}k\mathbb{E}\left[I(L(B_{1,k})>0)I(L(\bar{B}_{1,k})>1)\right]^{1/2}$
$\displaystyle=\sum_{k=1}^{\infty}k\mathbb{P}(L(B_{1,k})>0)^{1/2}\mathbb{P}(L(\bar{B}_{1,k})>1)^{1/2}$
$\displaystyle=\sum_{k=1}^{\infty}k\left(1-\mathbb{P}(L(B_{1,k})=0)\right)^{1/2}\left(1-\mathbb{P}(L(\bar{B}_{1,k})=0)-\mathbb{P}(L(\bar{B}_{1,k})=1)\right)^{1/2}$
$\displaystyle=\sum_{k=1}^{\infty}k\left(1-\exp(-\nu
Leb(B_{1,k})\right)^{1/2}\left(1-\exp(-\nu Leb(\bar{B}_{1,k}))-\nu
Leb(\bar{B}_{1,k})\exp(-\nu Leb(\bar{B}_{1,k}))\right)^{1/2}$
$\displaystyle\leq\sum_{k=1}^{\infty}k\nu^{1/2}Leb(B_{1,k})^{1/2}\left(\nu
Leb(\bar{B}_{1,k})-\nu Leb(\bar{B}_{1,k})\exp(-\nu
Leb(\bar{B}_{1,k}))\right)^{1/2}$
$\displaystyle=\sum_{k=1}^{\infty}k\nu^{1/2}Leb(B_{1,k})^{1/2}\nu^{1/2}Leb(\bar{B}_{1,k})^{1/2}\left(1-\exp(-\nu
Leb(\bar{B}_{1,k}))\right)^{1/2}$
$\displaystyle\leq\nu^{3/2}\sum_{k=1}^{\infty}kLeb(B_{1,k})^{1/2}Leb(\bar{B}_{1,k})$
$\displaystyle<\infty,$

where we used, repeatedly, that $1-e^{-x}\leq x$.

To prove (b), we show that $Z^{**}$ is bounded in $L^{2}$. We utilize the
assumption that $L^{\prime}\sim Poi(\nu)$, which implies that $L(B)=L(B)$ for
all Borel sets $B$. This allows us to write

$\displaystyle Z_{1}$ $\displaystyle=\sum_{k=1}^{\infty}kL(B_{1,k})$
$\displaystyle=\sum_{k=1}^{\infty}kL(B_{1,k})I(L(B_{1,k})=1)+\sum_{k=1}^{\infty}kL(B_{1,k})I(L(B_{1,k})>1)$
$\displaystyle=\sum_{k=1}^{\infty}kI(L(B_{1,k})=1)+\sum_{k=1}^{\infty}kL(B_{1,k})I(L(B_{1,k})>1),$

and

$\displaystyle Z_{1}^{*}$ $\displaystyle=\sum_{k=1}^{\infty}kI(L(B_{1,k})>0)$
$\displaystyle=\sum_{k=1}^{\infty}kI(L(B_{1,k})=1)+\sum_{k=1}^{\infty}kI(L(B_{1,k})>1).$

We deduce that

$\displaystyle Z_{1}^{**}$ $\displaystyle=Z_{1}-Z_{1}^{*}$
$\displaystyle=\sum_{k=1}^{\infty}kL(B_{1,k})I(L(B_{1,k})>1)-\sum_{k=1}^{\infty}kI(L(B_{1,k})>1)$
$\displaystyle=\sum_{k=1}^{\infty}kI(L(B_{1,k})>1)(L(B_{1,k})-1)$
$\displaystyle\leq\sum_{k=1}^{\infty}kI(L(B_{1,k})>1)L(B_{1,k}).$

Using this and then Minkowski’s inequality, we may write

$\displaystyle\mathbb{E}[(Z_{1}^{**})^{2}]^{1/2}$
$\displaystyle\leq\mathbb{E}[(\sum_{k=1}^{\infty}kI(L(B_{1,k})>1)L(B_{1,k}))^{2}]^{1/2}$
$\displaystyle\leq\sum_{k=1}^{\infty}\mathbb{E}[k^{2}I(L(B_{1,k})>1)L(B_{1,k})^{2}]^{1/2}$
$\displaystyle=\sum_{k=1}^{\infty}k\mathbb{E}[I(L(B_{1,k})>1)L(B_{1,k})^{2}]^{1/2}$
$\displaystyle=\sum_{k=1}^{\infty}k\mathbb{E}[\sum_{j=2}^{\infty}(I(L(B_{1,k})=j))L(B_{1,k})^{2}]^{1/2}$
$\displaystyle=\sum_{k=1}^{\infty}k\left(\sum_{j=2}^{\infty}\mathbb{E}[I(L(B_{1,k})=j)j^{2}]\right)^{1/2}$
$\displaystyle=\sum_{k=1}^{\infty}k\left(\sum_{j=2}^{\infty}\mathbb{P}(I(L(B_{1,k})=j)j^{2}\right)^{1/2}$
$\displaystyle=\sum_{k=1}^{\infty}k\left(\sum_{j=2}^{\infty}\frac{(\nu
Leb(B_{1,k}))^{j}e^{-\nu Leb(B_{1,k})}}{j!}j^{2}\right)^{1/2}$