, and

# Nonparametric estimation of trawl processes: Theory and Applications

Orimar<EMAIL_ADDRESS>[ Almut E. D.
<EMAIL_ADDRESS>[ Department of Mathematical
Sciences, Aalborg University, Skjernvej 4, A, 5-202, 9220 Aalborg Ø,
Denmarkpresep=, ]e1  Department of Mathematics, Imperial College London, 180
Queen’s Gate, London, SW7 2AZ, UKpresep=, ]e2

###### Abstract

Trawl processes belong to the class of continuous-time, strictly stationary,
infinitely divisible processes; they are defined as Lévy bases evaluated over
deterministic trawl sets. This article presents the first nonparametric
estimator of the trawl function characterising the trawl set and the serial
correlation of the process. Moreover, it establishes a detailed asymptotic
theory for the proposed estimator, including a law of large numbers and a
central limit theorem for various asymptotic relations between an in-fill and
a long-span asymptotic regime. In addition, it develops consistent estimators
for both the asymptotic bias and variance, which are subsequently used for
establishing feasible central limit theorems which can be applied to data. A
simulation study shows the good finite sample performance of the proposed
estimators and, in an empirical illustration, the new methodology is applied
to modelling and forecasting high-frequency financial spread data from a limit
order book.

62G05, 62G20,

60F05,60G10, 60G57, 60E07,

trawl process,

nonparametric estimation,

central limit theorem,

asymptotic variance estimation,

forecasting,

high-frequency financial data,

###### keywords:

[class=MSC]

###### keywords:

##

###  1 Introduction

####  1.1 Overview

This article develops the nonparametric estimation theory for _trawl
processes_ , investigates the finite sample properties of the corresponding
asymptotic results in a simulation study, and illustrates the new methodology
in an empirical application to high-frequency financial data.

Trawl processes are a class of continuous-time, strictly stationary,
infinitely divisible processes. A trawl process
$X=(X_{\\_}t)_{\\_}{t\in\mathbb{R}}$ is defined as a Lévy basis $L$ evaluated
over a trawl set $A_{\\_}t$, which is typically chosen as a subset of
$\mathbb{R}^{2}$. I.e. we set $X_{\\_}t=L(A_{\\_}t)$, for $t\in\mathbb{R}$;
the mathematical details will be provided in the next section. What makes
trawl processes great candidates for applications is the fact that their
marginal (infinitely divisible) distribution and their serial dependence can
be modelled independently of each other. What is more, they can be viewed as a
universal class for modelling stationary stochastic processes with infinitely
divisible marginal law, since for any infinitely divisible law, there exists a
trawl process which has that particular law as its marginal distribution, see
[2, p. 1535]. Combining this fact with the discussion in [34, p. 1527, 1535],
we can deduce that any stationary infinitely divisible process with a
differentiable covariance function that is non-increasing and convex on
$(0,\infty)$ can be written as a trawl process. Moreover, the type of serial
dependence is fully determined through the choice of the trawl set and allows
for both short- and long-memory settings. Given the breadth of phenomena which
can be modelled by trawl processes, there has been a lot of interest in trawl
processes recently; hence we will briefly survey the related literature.

####  1.2 Related work

Trawl processes were first mentioned in [2], motivated by the need for a
flexible but analytically tractable class of stochastic processes that can be
used in various applications, including the areas of turbulence and finance.
[34] considered stochastic processes in this realm under the name of
_”upstairs representation”_ in their study of the fractional Ornstein-
Uhlenbeck process. What we call a trawl (set) was simply referred to as a
”geometric figure in a higher-dimensional space”. Furthermore, restricted to
integer values, trawl-type processes were mentioned by [33] and referred to as
”random measure processes”.

[4] provide a systematic study of integer-valued trawl processes, describing
their probabilistic properties as well as simulation and inference methods.
This work was further extended to a multivariate framework by [30]. Recently,
[6] developed a composite likelihood estimation theory for integer-valued
trawl processes, as well as tools for model selection and forecasting. All of
these articles were motivated by applications to high-frequency financial
data. Related work also includes the article by [27], which proposes a new
modelling framework for high-frequency financial data using trawl processes,
and the work by [26], which develops likelihood inference for the special case
of exponential-trawl processes.

Trawl processes also feature prominently in recent work on hierarchical models
for extreme values using a peaks-over-threshold methodology. For instance,
[20] propose using a latent trawl process to model clusters in extremes of
environmental time series data, and [9] derive the corresponding asymptotic
theory for inference in such a model. [1] propose a spatio-temporal model for
extreme values based on extensions of trawls and successfully model extreme
precipitation data.

In a discrete-time setting, trawl processes were first introduced by [11], who
also derived functional limit theorems for partial sums of discrete-time trawl
processes. Motivated by these findings, there have been various articles
focussing on limit theorems for trawl processes (both in continuous and
discrete time). For example, [13] study limit theorems to characterise the
intermittency of the trawl process, and [22] derives limit theorems for
partial sums of linear processes with tapered innovations, including trawl
processes. Furthermore, [28] study the long-term behaviour of integrated trawl
processes with symmetric Lévy bases. Moreover, [21] derive limit theorems for
continuous-time trawl processes, including the asymptotic behaviour of partial
sums of discretised trawl processes and the functional convergence in
distribution of trawl processes.

Statistical inference for trawl processes has so far focused on fully
parametric settings and includes the use of the (generalised) method of
moments, see [4, 27, 30], likelihood methods, see [20, 1, 9, 26, 6], and
spectral methods, see [12].

####  1.3 Main contributions of this article

This article develops the first nonparametric estimation theory for trawl
processes. It presents an estimator for the trawl function which is based on
the idea that the derivative of the autocovariance function and the trawl
function are proportional to each other. It then establishes the consistency
of the estimator under mild regularity conditions, using a double-asymptotic
scheme based on both in-fill and long-span asymptotics. Next, we derived
suitable central limit theorems for the proposed estimator, taking various
asymptotic relations between the in-fill and the long-span asymptotic regimes
into account. The resulting central limit theorems are infeasible, in the
sense that they typically feature an asymptotic bias and/or an asymptotic
variance which is not directly observable. Hence we propose new estimators for
both the asymptotic bias and variance and prove their consistency. This
enables us to present feasible central limit theorems which can be implemented
in practice.

As an application of our theoretical results, we develop suitable estimators
for slices of trawl sets, which are disjoint subsets obtained via intersection
and set-difference operations applied to trawl sets. Such slice estimators can
be applied in the construction of nonparametric predictors of trawl processes.

Since our asymptotic theory depends on a delicate balance between an in-fill
and a long-span asymptotic regime, we assess the finite sample performance of
our proposed methods in detailed simulation experiments.

Finally, we apply the new methodology to modelling and forecasting high-
frequency financial data, consisting of spreads between the best bid and ask
prices of a limit order book. We show that our methodology works at least as
well as parametric methods and sometimes even better while being easy to
implement and avoiding the need for parametric model selection.

This article is complemented by two software releases: The R package ambit,
see [31], has been published on CRAN, collecting all the relevant
implementations of the new methodology as well as detailed documentation.
Moreover, the code used in the empirical study is made publicly available on
GitHub111https://github.com/almutveraart/Nonparametric-trawl-estimation and
archived in Zenodo, see [32].

####  1.4 Organisation of the article

The remainder of this article is structured as follows. Section 2 contains
background material on the definition of trawl processes and presents the
construction of the estimator for the trawl function. Section 3 lays out the
asymptotic theory, which is proven in the Supplementary Material [24, Section
B]. Section 4 discusses some of the results of our simulation study designed
to assess the finite sample performance of the proposed estimation theory,
more details are available in the Supplementary Material [24, Sections C, D,
E, F]. Finally, Section 5 applies the new estimation methodology to high-
frequency financial data from a limit order book and compares the
nonparametric estimation with parametric alternatives.

###  2 Background and construction of the estimators

####  2.1 Lévy bases and infinite divisibility

Let $\mu$ be a measure on $\mathcal{B}(\mathbb{R}^{d})$, the Borel sets on
$\mathbb{R}^{d}$, and let
$\mathcal{B}_{\\_}{b}^{\mu}(\mathbb{R}^{d}):=\\{A\in\mathcal{B}(\mathbb{R}^{d}):\mu(A)<\infty\\}.$
The family
$L=\\{L\left(A\right):A\in\mathcal{B}_{\\_}{b}^{\mu}(\mathbb{R}^{d})\\}$ of
real-valued random variables (r.v.s) will be called a Lévy basis if it is an
infinitely divisible (ID for short) independently scattered random measure,
that is, $L$ is $\sigma$-additive almost surely and such that for any
$A,B\in\mathcal{B}_{\\_}{b}^{\mu}(\mathbb{R}^{d})$, $L(A)$ and $L(B)$ are ID
r.v.s that are independent whenever $A\cap B=\emptyset$. The cumulant of an
r.v. $\xi$, in case it exists, will be denoted by
$\mathcal{C}(z\ddagger\xi):=\log\mathbb{E}(e^{iu\xi})$. We will say that $L$
is separable with control measure $\mu$, if $\mathcal{C}(z\ddagger
L(A))=\mu(A)\psi(z)$, $A\in\mathcal{B}_{\\_}{b}^{\mu}(\mathbb{R}^{d})$,
$z\in\mathbb{R}$, where $\psi(z):=i\gamma
z-\frac{1}{2}b^{2}z^{2}+\int_{\\_}{\mathbb{R}\backslash\\{0\\}}(e^{izx}-1-izx\mathbf{1}_{\\_}{\left|x\right|\leq
1})\nu(\mathrm{d}x)$, $z\in\mathbb{R}$ with $\gamma\in\mathbb{R},$ $b\geq 0$
and $\nu$ is a Lévy measure, i.e. $\nu(\\{0\\})=0$ and
$\int_{\\_}{\mathbb{R}\backslash\\{0\\}}(1\land\left|x\right|^{2})\nu(\mathrm{d}x)<\infty$.
When $\mu=\mathrm{Leb}$, in which $\mathrm{Leb}$ represents the Lebesgue
measure on $\mathbb{R}^{d}$, $L$ is called homogeneous. The ID r.v. associated
to the characteristic triplet $\left(\gamma,b,\nu\right)$ is called the Lévy
seed of $L$ and will be denoted by $L^{\prime}$. As usual,
$\left(\gamma,b,\nu\right)$ will be called the characteristic triplet of $L$
and $\psi$ its characteristic exponent.

####  2.2 Trawl processes and some of their properties

######  Definition 1.

Let $L$ be a homogeneous Lévy basis on $\mathbb{R}^{2}$ with characteristic
triplet $(\gamma,b,\nu)$. In addition, let
$a:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be a continuous non-increasing
integrable function and put $A=\left\\{(r,y):r\leq 0,0\leq y\leq
a(-r)\right\\}$. The process defined by $X_{\\_}{t}:=L(A_{\\_}{t})$, for
$t\in\mathbb{R}$, where $A_{\\_}{t}:=A+(t,0)$, is called a trawl process. From
now on, we will refer to $A$ and $a$, as the trawl set and the trawl function,
respectively.

It is well known that $X$ is strictly stationary and, in the case when $L$ is
square integrable, its autocovariance function is given by

$\varGamma_{\\_}{X}(h):=\mathrm{Var}(L^{\prime})\int_{\\_}{\left|h\right|}^{\infty}a(u)\mathrm{d}u,\,\,\,h\in\mathbb{R}.$
(1)

This relation implies that $\varGamma_{\\_}{X}$ uniquely characterizes $a$
within the class of trawl processes represented by $L$. More precisely, if $X$
and $\tilde{X}$ are two trawls processes associated with $L$ with trawls
functions $a$ and $\tilde{a}$, respectively, then $a=\tilde{a}$ a.e. if and
only if $\varGamma_{\\_}{X}=\varGamma_{\\_}{\tilde{X}}$.

For a detailed exposition on the basic properties of trawl processes we refer
to [4] and [3, Chapter 8]. In what follows we focus on the estimation of $a$
under the assumption that
$\mathrm{Var}(L^{\prime})=b^{2}+\int_{\\_}{\mathbb{R}}x^{2}\nu(dx)=1$.

####  2.3 Estimator construction

We aim to estimate the trawl function $a$ based on equidistant observations of
the trawl process $X$, say $(X_{\\_}{i\Delta_{\\_}{n}})_{\\_}{i=0}^{n-1}$.
From equation (1), we deduce the relation (valid when
$\mathrm{Var}(L^{\prime})=1$)

$\varGamma_{\\_}{X}^{\prime}=-a,$

which motivates the construction of our nonparametric estimator of $a$: We can
use an estimator for $\varGamma_{\\_}{X}$ and then approximate its derivative.
We do this via the sample autocovariance function. Specifically, by letting

$\hat{\varGamma}_{\\_}{l}:=\frac{1}{n}\sum_{\\_}{k=0}^{n-1-l}(X_{\\_}{(l+k)\Delta_{\\_}{n}}-\bar{X}_{\\_}{n})(X_{\\_}{k\Delta_{\\_}{n}}-\bar{X}_{\\_}{n}),\,\,\,l=0,\ldots,n-1,$

in which
$\bar{X}_{\\_}{n}=\frac{1}{n}\sum_{\\_}{k=1}^{n}X_{\\_}{(k-1)\Delta_{\\_}{n}},$
we propose to estimate $a(t)$, for $t>0$ by

$\displaystyle\hat{a}(t)$
$\displaystyle=-\Delta_{\\_}{n}^{-1}\left[\hat{\varGamma}_{\\_}{l+1}-\hat{\varGamma}_{\\_}{l}\right],\,\,\,\text{if
}\Delta_{\\_}{n}l\leq t<(l+1)\Delta_{\\_}{n},$ (2)

while for $t=0$, we suggest estimation via the averaged realised variance:

$\hat{a}(0)=\frac{1}{2\Delta_{\\_}{n}n}\sum_{\\_}{k=0}^{n-2}(\delta_{\\_}{k}X)^{2},\,\,\,\delta_{\\_}{k}X:=X_{\\_}{(k+1)\Delta_{\\_}{n}}-X_{\\_}{k\Delta_{\\_}{n}}.$
(3)

######  Remark 2.1.

Note that one can consistently (and probably in a more natural way) estimate
$a(0)$ via the formula (2) with $l=0$. However, when the process is Gaussian,
this alternative approach creates an asymptotic error (which does not fulfil
any type of second-order limit theorem) that in general cannot be de-biased
from available data. The estimator proposed in (3) avoids this situation in
practice and it makes our estimation procedure more robust.

###  3 Asymptotic theory

####  3.1 Consistency

It is clear that the bigger $n$, the better our approximation for
$\varGamma_{\\_}{X}$ becomes. A similar conclusion holds when
$\Delta_{\\_}{n}$ is small. In consequence, it is evident that for our
estimator to be consistent, the following basic assumption on the in-fill and
long-span sampling scheme must be satisfied:

######  Assumption 1.

We assume that, as $n\uparrow\infty$, $\Delta_{\\_}{n}\downarrow 0$ and
$n\Delta_{\\_}{n}\rightarrow+\infty.$

Under this set-up our estimator $\hat{a}(t)$ is consistent:

######  Theorem 1 (Consistency).

Let Assumption 1 hold and suppose that $\mathrm{Var}(L^{\prime})=1$ as well as
$\mathbb{E}(\mid L^{\prime}\mid^{4})<\infty$. Then, for all $t\geq 0$,
$\hat{a}(t)\stackrel{{\scriptstyle\mathbb{P}}}{{\to}}a(t)$.

####  3.2 Asymptotic Gaussianity – An infeasible result

In this part, we present a CLT for $\hat{a}(t)$. Not surprisingly, stronger
assumptions are required, which we will introduce next.

First, we refine the sampling scheme introduced in Assumption 1.

######  Assumption 2.

We assume that, as $n\uparrow\infty$, $\Delta_{\\_}{n}\downarrow 0$ and
$n\Delta_{\\_}{n}\rightarrow+\infty.$ Furthermore,
$n\Delta_{\\_}{n}^{3}\rightarrow\mu\in[0,\infty]$.

Second, we impose the following assumption on the trawl function:

######  Assumption 3.

The trawl function admits the representation
$a(s)=\int_{\\_}{s}^{\infty}\phi(y)dy$, for $s\geq 0$, where $\phi$ is a
strictly positive continuous function such that
$\phi(s)=\mathrm{O}(s^{-\alpha-1})$ as $s\rightarrow+\infty$ for some
$\alpha>1$.

######  Example 1.

Various choices for the trawl function are possible, see e.g. [4] for many
parametric examples. In this article, we will be focussing on two particular
examples for our simulation study and empirical work:

  1. 1.

Exponential trawl function: Here we choose $a(s)=\exp(-\lambda s)$, for
$\lambda>0$, $s\geq 0$. In this case, we have $\phi(s)=\lambda\exp(-\lambda
s)$, for $s\geq 0$.

  2. 2.

SupGamma trawl function: Here we choose
$a(s)=\left(1+\frac{s}{\overline{\alpha}}\right)^{-H}$, for
$\overline{\alpha}>0,H>1$, $s\geq 0$. In this case, we have
$\phi(s)=\frac{H}{\overline{\alpha}}\left(1+\frac{s}{\overline{\alpha}}\right)^{-H-1}$,
for $s\geq 0$. Note that, for $H\in(1,2]$, the resulting trawl process has
long memory and for $H>2$ short memory.

Finally, for the Lévy seed associated with the Lévy basis we make the
following moment assumption:

######  Assumption 4.

$\mathrm{Var}(L^{\prime})=1$ and for some $p_{\\_}{0}\geq 16/3$ we have that
$\mathbb{E}(\mid L^{\prime}\mid^{p_{\\_}{0}})<\infty$.

The following result fully describes the second-order asymptotic theory of our
proposed estimator under the preceding set-up.

######  Theorem 2.

Let Assumptions 2, 3 and 4 hold. Then, for all $t\geq 0$ as
$n\rightarrow\infty$, we have the following convergence results.

  1. 1.

If $\mu=0$, then

$\sqrt{n\Delta_{\\_}{n}}\left(\hat{a}(t)-a(t)\right)\overset{d}{\rightarrow}N(0,\sigma_{\\_}{a}^{2}(t)),$

where, for $c_{\\_}{4}(L^{\prime}):=\int x^{4}\nu(dx)$, we have set

$\displaystyle\sigma_{\\_}{a}^{2}(t)=$ $\displaystyle
c_{\\_}{4}(L^{\prime})a(t)+2\left\\{\int_{\\_}{0}^{\infty}a(s)^{2}ds+\int_{\\_}{0}^{\infty}a(\left|t-s\right|)a(t+s)\mathrm{sign}(t-s)ds\right\\}.$

  2. 2.

If $0<\mu<\infty$, then

$\sqrt{n\Delta_{\\_}{n}}\left(\hat{a}(t)-a([t/\Delta_{\\_}{n}]\Delta_{\\_}{n})\right)\overset{d}{\rightarrow}\frac{1}{2}\sqrt{\mu}a^{\prime}(t)+N(0,\sigma_{\\_}{a}^{2}(t)).$

  3. 3.

If $\mu=\infty$, then

$\frac{1}{\Delta_{\\_}{n}}\left(\hat{a}(t)-a([t/\Delta_{\\_}{n}]\Delta_{\\_}{n})\right)\overset{\mathbb{P}}{\rightarrow}\frac{1}{2}a^{\prime}(t).$

######  Remark 3.1.

The following remarks are in order:

  1. 1.

It is worth noting that the previous theorem does not rely on the memory of
$X$. This is an unexpected result since the second-order asymptotics of the
sample autocovariance function typically depend on whether the underlying
process has short or long-range dependence.

  2. 2.

The asymptotic bias appearing in the previous theorem comes from the
discretisation error created when one approximates $a(t)$ by

$\frac{1}{\Delta_{\\_}{n}}\int_{\\_}{[t/\Delta_{\\_}{n}]\Delta_{\\_}{n}}^{[t/\Delta_{\\_}{n}]\Delta_{\\_}{n}+\Delta_{\\_}{n}}a(s)ds=a([t/\Delta_{\\_}{n}]\Delta_{\\_}{n})+\Delta_{\\_}{n}\frac{1}{2}a^{\prime}([t/\Delta_{\\_}{n}]\Delta_{\\_}{n})+\mathrm{o}(\Delta_{\\_}{n}).$
(4)

Thus, under Assumption 2, this discretisation error is negligible as soon as
$n\Delta_{\\_}{n}^{3}\rightarrow 0$. When this is not the case, the quantity
in (4) no longer approximates $a(t)$ but
$a([t/\Delta_{\\_}{n}]\Delta_{\\_}{n})$, and when
$n\Delta_{\\_}{n}^{3}\rightarrow+\infty$, it will dominate the asymptotic
behaviour of $\hat{a}(t)$.

  3. 3.

One can unify the results of the previous theorem by centring around the
discretisation error discussed above. More precisely, within the framework of
Theorem 2, it holds that

$\sqrt{n\Delta_{\\_}{n}}\left(\hat{a}(t)-\frac{1}{\Delta_{\\_}{n}}\int_{\\_}{[t/\Delta_{\\_}{n}]\Delta_{\\_}{n}}^{[t/\Delta_{\\_}{n}]\Delta_{\\_}{n}+\Delta_{\\_}{n}}a(s)ds\right)\overset{d}{\rightarrow}N(0,\sigma_{\\_}{a}^{2}(t)).$

This result is robust to whether $n\Delta_{\\_}{n}^{3}$ is convergent or not.

#####  3.2.1 The case when $t=0$

The reader might have noticed that $\sigma_{\\_}{a}^{2}(0)$ vanishes when $L$
is purely Gaussian (i.e. $c_{\\_}{4}(L^{\prime})=0$), so a degenerate limit
appears in the case $\mu=0$. In order to get an intuition of why this is the
case, first, note that

$\hat{a}(0)-a(0)=\underbrace{\frac{1}{2\Delta_{\\_}{n}n}\sum_{\\_}{k=0}^{n-1}\left\\{(\delta_{\\_}{k}X)^{2}-\mathbb{E}((\delta_{\\_}{k}X)^{2})\right\\}}_{\\_}{\text{Statistical
Error}}+\underbrace{\mathrm{O}_{\\_}{\mathbb{P}}(\Delta_{\\_}{n})}_{\\_}{\text{Discretization
Error}}.$

Now, according to [5], when $L$ is Gaussian, $X$ can be decomposed as the sum
of a Brownian motion with parameter $a(0)$ plus a continuous process of finite
variation. This implies that for $\Delta_{\\_}{n}$ small, the increments of
$X$ are comparable with those of a Brownian motion. Thus, thanks to the self-
similar property of the latter, we heuristically deduce that

$\hat{a}(0)-a(0)\approx\mathrm{O}_{\\_}{\mathbb{P}}(1/\sqrt{n})+\mathrm{O}_{\\_}{\mathbb{P}}(\Delta_{\\_}{n}).$

This formal approximation reveals that the asymptotic properties of
$\hat{a}(0)$ in the Gaussian case will depend on the limiting behaviour of
$n\Delta_{\\_}{n}^{2}$. Therefore, our study for this particular case is
conducted under the following assumption:

######  Assumption 5.

We assume that, as $n\uparrow\infty$, $\Delta_{\\_}{n}\downarrow 0$ and
$n\Delta_{\\_}{n}\rightarrow+\infty.$ Furthermore,
$n\Delta_{\\_}{n}^{2}\rightarrow\mu_{\\_}{0}\in[0,\infty]$.

In the next result, we will use the so-called quarticity which is denoted as

$Q_{\\_}{n}:=\frac{1}{2\Delta_{\\_}{n}n}\sum_{\\_}{k=0}^{n-2}(\delta_{\\_}{k}X)^{4}.$
(5)

######  Theorem 3.

Let Assumptions 2 (with $\mu<\infty$), 3 and 4 hold. As long as
$c_{\\_}{4}(L^{\prime})>0$, it holds that

$\displaystyle\sqrt{\frac{n\Delta_{\\_}{n}}{Q_{\\_}{n}}}\left(\hat{a}(0)-a(0)\right)\overset{d}{\rightarrow}N(0,1)+\frac{a^{\prime}(0)}{2}\sqrt{\frac{\mu}{c_{\\_}{4}(L^{\prime})a(0)}},\,\,\,\text{as
}n\rightarrow\infty.$ (6)

If $L$ is purely Gaussian assume in addition that Assumption 5 holds. Then, we
have the following limits as $n\rightarrow\infty$.

  1. 1.

When $0\leq\mu_{\\_}{0}<\infty$, it holds that

$\displaystyle\sqrt{\frac{n\Delta_{\\_}{n}}{Q_{\\_}{n}}}\left(\hat{a}(0)-a(0)\right)\overset{d}{\rightarrow}N(0,1/3)+\frac{a^{\prime}(0)}{2}\sqrt{\frac{\mu_{\\_}{0}}{6a(0)^{2}}}.$
(7)

  2. 2.

In the case when $\mu_{\\_}{0}=\infty$, we have that

$\frac{1}{\Delta_{\\_}{n}}\left(\hat{a}(0)-a(0)\right)\overset{\mathbb{P}}{\rightarrow}\frac{1}{2}a^{\prime}(0).$

We conclude this part with a remark.

######  Remark 3.2.

It is very natural to ask about second-order asymptotics in Theorem 2 (3) when
$n\Delta_{\\_}{n}^{3}\rightarrow+\infty$
($n\Delta_{\\_}{n}^{2}\rightarrow+\infty$). It turns out that this will be
dependent on the relationship between how fast $n\Delta_{\\_}{n}^{3}$
($n\Delta_{\\_}{n}^{2}$) explodes and how smooth the trawl function is. Since
the analysis in this situation is based on estimators for high order
derivatives of $a$, we have decided to leave this problem for future research.
Therefore, for the rest of this paper we concentrate on the case when
$n\Delta_{\\_}{n}^{3}\rightarrow\mu\in[0,\infty)$
($n\Delta_{\\_}{n}^{2}\rightarrow\mu_{\\_}0\in[0,\infty)$).

The central limit theorems presented above are, in general, _infeasible_ in
the sense that the asymptotic variance and asymptotic bias (if present) are
not known a priori. Hence they need to be estimated as well to obtain a
_feasible_ asymptotic theory which can be used in practice.

Only in the cases of equations (6) and (7), when $\mu=0$ and $\mu_{\\_}0=0$,
the resulting limit theorems are feasible and can be directly implemented in
practice.

####  3.3 Asymptotic Gaussianity – Towards a feasible theory

In order to be able to use the asymptotic results presented in Theorems 2 and
3, we need to replace the asymptotic bias and variance (from now on ABI and
AVAR, respectively) by a suitable estimator. Our methodology (described below)
requires two additional tuning parameters
$K_{\\_}{n},N_{\\_}{n}\in\mathbb{N}$. The tuning parameter $K_{\\_}n$ features
in the estimator of ABI and $N_{\\_}n$ in the AVAR estimator.

######  Assumption 6.

The sequence $(K_{\\_}n)$ taking values in $\mathbb{N}$ satisfies the
following three conditions as $n\rightarrow\infty$:
$i)\,K_{\\_}{n}\uparrow+\infty$, $ii)\,K_{\\_}{n}\Delta_{\\_}{n}\rightarrow
0$, and $iii)\,n/K_{\\_}{n}\rightarrow+\infty$.

######  Assumption 7.

The sequence $(N_{\\_}n)$ taking values in $\mathbb{N}$ satisfies the
following three conditions as $n\rightarrow\infty$:
$i)\,N_{\\_}{n}\uparrow+\infty$,
$ii)\,N_{\\_}{n}\Delta_{\\_}{n}\rightarrow+\infty$, and
$iii)\,N_{\\_}{n}/n\rightarrow 0$.

#####  3.3.1 Estimating the ABI

Since the ABI involves the derivative of $a$, in this part we propose a
procedure for estimating such a function. A natural estimator within our
framework is

$\hat{a}^{\prime}_{\\_}d(t)=\frac{1}{\Delta_{\\_}{n}}[\hat{a}(t+\Delta_{\\_}{n})-\hat{a}(t)].$

However, we eliminate the sample mean error by modifying the previous
estimator as

$\hat{a}^{\prime}(t)=\frac{1}{n\Delta_{\\_}{n}^{2}}\sum_{\\_}{k=l+1}^{n-2}\delta_{\\_}{k}X\cdot\delta_{\\_}{k-l-1}X,\,\,\,\text{if
}\Delta_{\\_}{n}l\leq t<(l+1)\Delta_{\\_}{n}.$ (8)

In the Supplementary Material [24, Section B], we show the following result.

######  Proposition 1.

Suppose that Assumptions 2, 3 and 4 hold. Let $\hat{a}^{\prime}(t)$ be as in
(8). Then, for all $t\geq 0$,

$\hat{a}^{\prime}(t)=a^{\prime}(t)+\mathrm{O}_{\\_}{\mathbb{P}}\left(1/\sqrt{n\Delta_{\\_}{n}^{2}}\right)+\mathrm{o}(1),\,\,\,\text{as
}n\rightarrow\infty.$

Note that the ABI is only present when either
$n\Delta_{\\_}{n}^{3}\rightarrow\mu\in(0,\infty)$ or when the process is
Gaussian and $n\Delta_{\\_}{n}^{2}\rightarrow\mu_{\\_}{0}\in(0,\infty)$. In
the former situation $\hat{a}^{\prime}(t)$ is consistent while in the latter
is not. Nevertheless, if we have that
$n\Delta_{\\_}{n}^{2}\rightarrow\mu_{\\_}{0}\in(0,\infty)$, we can construct a
consistent estimator for $a^{\prime}$ by sampling “less” frequently as we
describe next: Put

$\tilde{\Delta}_{\\_}{n}:=K_{\\_}{n}\Delta_{\\_}{n};\,\,\,M_{\\_}{n}:=[(n-1)/K_{\\_}{n}],$

where $K_{\\_}{n}$ as in Assumption 6. Note that by construction,
$\tilde{\Delta}_{\\_}{n}\rightarrow 0$, $M_{\\_}{n}\uparrow+\infty$,
$M_{\\_}{n}\tilde{\Delta}_{\\_}{n}\rightarrow+\infty$, and
$M_{\\_}{n}\tilde{\Delta}_{\\_}{n}^{2}\rightarrow+\infty$. In consequence,
when $n\Delta_{\\_}{n}^{2}\rightarrow\mu_{\\_}{0}\in(0,\infty)$, the following
sequence (constructed using the sub-sample
$(X_{\\_}{i\tilde{\Delta}_{\\_}{n}})_{\\_}{i=0,\ldots,M_{\\_}n}$)

$\tilde{a}^{\prime}(t)=\frac{1}{M_{\\_}{n}\tilde{\Delta}_{\\_}{n}^{2}}\sum_{\\_}{k=l+1}^{M_{\\_}{n}-1}\tilde{\delta}_{\\_}{k}X\cdot\tilde{\delta}_{\\_}{k-l-1}X,\,\,\,\text{if
}\tilde{\Delta}_{\\_}{n}l\leq t<(l+1)\tilde{\Delta}_{\\_}{n},$ (9)

where

$\tilde{\delta}_{\\_}{k}X:=X_{\\_}{(k+1)\tilde{\Delta}_{\\_}{n}}-X_{\\_}{k\tilde{\Delta}_{\\_}{n}},\,\,k=0,\ldots,M_{\\_}{n}-1,$

satisfies, due to Proposition 1, that

$\tilde{a}^{\prime}(t)=a^{\prime}(t)+\mathrm{O}_{\\_}{\mathbb{P}}(1/\sqrt{K_{\\_}n})+\mathrm{o}(1),\,\,\,\text{as
}n\rightarrow\infty.$

#####  3.3.2 Feasible theory for $t=0$

######  Theorem 4.

Let the Assumptions of Theorem 2 hold with $\mu<\infty$. If
$c_{\\_}{4}(L^{\prime})>0$, then

$\sqrt{\frac{n\Delta_{\\_}{n}}{Q_{\\_}{n}}}\left(\hat{a}(0)-a(0)-\frac{1}{2}\Delta_{\\_}{n}\hat{a}^{\prime}(0)\right)\overset{d}{\rightarrow}N(0,1).$

If $c_{\\_}{4}(L^{\prime})=0$, assume in addition that Assumption 5 holds with
$\mu_{\\_}{0}<\infty$. Then, as $n\rightarrow\infty$

$\sqrt{\frac{n\Delta_{\\_}{n}}{Q_{\\_}{n}}}\left(\hat{a}(0)-a(0)-\frac{1}{2}\Delta_{\\_}{n}\tilde{a}^{\prime}(0)\right)\overset{d}{\rightarrow}N(0,1/3),$

in which $\tilde{a}^{\prime}(0)$ is defined as in (9).

#####  3.3.3 Estimating the AVAR

Next, we focus on estimating the AVAR given by

$\sigma_{\\_}{a}^{2}(t)=c_{\\_}{4}(L^{\prime})a(t)+2\left\\{\int_{\\_}{0}^{\infty}a(s)^{2}ds+\int_{\\_}{0}^{t}a(t-s)a(t+s)ds-\int_{\\_}{t}^{\infty}a(s-t)a(t+s)ds\right\\},$

for $t>0$ (the case $t=0$ is already covered in Theorem 4).

For each of the four components of $\sigma_{\\_}{a}^{2}(t)$, we construct an
estimator based on the observations
$(X_{\\_}{i\Delta_{\\_}{n}})_{\\_}{i=0}^{n-1}$. The tuning parameter
$N_{\\_}n$ satisfying Assumption 7 features in the estimates of the integrals
of the trawl functions.

Step 1:

Estimate $c_{\\_}{4}(L^{\prime})a(t)$: In the proof of Theorem 3 we verify
that $Q_{\\_}{n}$ is consistent for $c_{\\_}{4}(L^{\prime})a(0)$. Thus, we can
estimate $c_{\\_}{4}(L^{\prime})$ as
$\widehat{c_{\\_}{4}(L^{\prime})}:=Q_{\\_}{n}/\hat{a}(0)>0$. We set

$\hat{v}_{\\_}{1}(t):=\widehat{c_{\\_}{4}(L^{\prime})}\hat{a}(t).$

Step 2:

We use a step-function approximation, i.e.:
$2\int_{\\_}{0}^{\infty}a(s)^{2}ds\approx
2\int_{\\_}{0}^{N_{\\_}{n}\Delta_{\\_}{n}}a(s)^{2}ds$ is estimated by

$\hat{v}_{\\_}{2}(t):=2\sum_{\\_}{l=0}^{N_{\\_}{n}}\hat{a}^{2}(l\Delta_{\\_}{n})\Delta_{\\_}{n}.$

Step 3:

For $i=1,\ldots,n-1$, $\Delta_{\\_}{n}i\leq t<(i+1)\Delta_{\\_}{n}$, we
approximate
$2\int_{\\_}{0}^{t}a(i\Delta_{\\_}{n}-s)a(i\Delta_{\\_}{n}+s)ds\approx
2\int_{\\_}{0}^{i\Delta_{\\_}{n}}a(i\Delta_{\\_}{n}-s)a(i\Delta_{\\_}{n}+s)ds$
by

$\hat{v}_{\\_}{3}(t):=2\sum_{\\_}{l=0}^{\min\\{i,n-1-i\\}}\hat{a}((i-l)\Delta_{\\_}{n})\hat{a}((i+l)\Delta_{\\_}{n})\Delta_{\\_}{n}.$

Step 4:

For $i=0,\ldots,n-1$, $\Delta_{\\_}{n}i\leq t<(i+1)\Delta_{\\_}{n}$, we
approximate
$-2\int_{\\_}{t}^{\infty}a(s-i\Delta_{\\_}{n})a(i\Delta_{\\_}{n}+s)ds\approx-2\int_{\\_}{i\Delta_{\\_}{n}}^{\infty}a(s-i\Delta_{\\_}{n})a(i\Delta_{\\_}{n}+s)ds\approx-2\int_{\\_}{i\Delta_{\\_}{n}}^{N_{\\_}{n}\Delta_{\\_}{n}}a(s-i\Delta_{\\_}{n})a(i\Delta_{\\_}{n}+s)ds$
by

$\hat{v}_{\\_}{4}(t):=-2\sum_{\\_}{l=i}^{N_{\\_}{n}-i}\hat{a}((l-i)\Delta_{\\_}{n})\hat{a}((i+l)\Delta_{\\_}{n})\Delta_{\\_}{n}.$

Note that as soon as $N_{\\_}{n}-i<i$, we use the convention that the sum is
set to 0.

So, overall we have that, for $t>0$

$\displaystyle\widehat{\sigma_{\\_}{a}^{2}}(t):=\widehat{\sigma_{\\_}{a}^{2}}(i\Delta_{\\_}{n})=\hat{v}_{\\_}{1}(t)+\hat{v}_{\\_}{2}(t)+\hat{v}_{\\_}{3}(t)+\hat{v}_{\\_}{4}(t).$
(10)

Next, we verify the consistency of $\widehat{\sigma_{\\_}{a}^{2}}(t)$.

######  Proposition 2.

Suppose that Assumptions 2 (with $\mu<\infty$), 3 and 4 hold. Then for all
$t>0$, as $n\rightarrow\infty$,
$\widehat{\sigma_{\\_}{a}^{2}}(t)\overset{\mathbb{P}}{\rightarrow}\sigma_{\\_}{a}^{2}(t).$

######  Remark 3.3.

Note that we used a simple step function approximation of the integrals.
Alternative numerical schemes, e.g. a trapezoidal rule, could be used, but our
simulation study revealed little difference in such choices and hence we will
not further modify the proposed simple estimator.

#####  3.3.4 Feasible theory for $t>0$

Based on the results of Propositions 1 and 2, we can now formulate feasible
versions of Theorem 2 as follows:

######  Theorem 5.

Let the assumptions of Theorem 2 hold with $\mu<\infty$. Then, for all $t>0$
as $n\rightarrow\infty$, we have the following convergence results.

  1. 1.

If $\mu=0$, then

$\sqrt{\frac{n\Delta_{\\_}{n}}{\widehat{\sigma_{\\_}{a}^{2}}(t)}}\left(\hat{a}(t)-a(t)-\frac{1}{2}\Delta_{\\_}{n}\hat{a}^{\prime}(t)\right)\overset{d}{\rightarrow}N(0,1),$

in which $\hat{a}^{\prime}(t)$ is given by (8) and
$\widehat{\sigma_{\\_}{a}^{2}}(t)$ as in (10).

  2. 2.

If $0<\mu<\infty$, then

$\sqrt{\frac{n\Delta_{\\_}{n}}{\widehat{\sigma_{\\_}{a}^{2}}(t)}}\left(\hat{a}(t)-a([t/\Delta_{\\_}{n}]\Delta_{\\_}{n})-\frac{1}{2}\Delta_{\\_}{n}\hat{a}^{\prime}(t)\right)\overset{d}{\rightarrow}N(0,1).$

####  3.4 Application to slice estimation and nonparametric forecasting of
trawl processes

Let $\mathcal{F}_{\\_}t=\sigma((X_{\\_}s)_{\\_}{s\leq t})$, $t\geq 0$, and
suppose we wish to forecast the trawl process, where $h>0$ denotes the
forecast horizon. [4] discussed that, due to the non-Markovianity of trawl
processes, the predictive distribution $X_{\\_}{t+h}|\mathcal{F}_{\\_}t$ is
intractable and suggest approximating it by the conditional distribution
$X_{\\_}{t+h}|X_{\\_}t$ instead. Recently, [17] showed the following result:

$\displaystyle\mathbb{E}(X_{\\_}{t+h}|X_{\\_}t)=\frac{\mathrm{Leb}(A\cap
A_{\\_}h)}{Leb(A)}X_{\\_}t+\frac{\mathrm{Leb}(A\setminus
A_{\\_}h)}{\mathrm{Leb}(A)}\mathbb{E}(X_{\\_}t),$ (11)

see [6] for examples for integer-valued trawl processes.

Again, let us consider $n$ observations
$X_{\\_}0,\ldots,X_{\\_}{(n-1)\Delta_{\\_}n}$. Suppose we would liked to make
a prediction at time $t=(n-1)\Delta_{\\_}n$ for time $t+h$. Using our
estimator for the trawl function, we can now define a predictor
$\hat{X}_{\\_}{t+h|t}$ as follows.

$\displaystyle\hat{X}_{\\_}{t+h|t}:=\frac{\widehat{\mathrm{Leb}(A\cap
A_{\\_}h)}}{\widehat{\mathrm{Leb}(A)}}X_{\\_}t+\frac{\widehat{\mathrm{Leb}(A\setminus
A_{\\_}h)}}{\widehat{\mathrm{Leb}(A)}}\overline{X},$ (12)

where $\overline{X}=\frac{1}{n}\sum_{\\_}{l=0}^{n-1}X_{\\_}{l\Delta_{\\_}n}$,

$\displaystyle\begin{split}\widehat{\mathrm{Leb}(A)}&=\sum_{\\_}{l=0}^{n-1}\hat{a}(l\Delta_{\\_}n)\Delta_{\\_}n,\\\
\widehat{\mathrm{Leb}(A\cap
A_{\\_}h)}&=\sum_{\\_}{\begin{subarray}{c}l\in\\{0,\ldots,n-1\\}:\\\
l\Delta_{\\_}n\geq h\end{subarray}}\hat{a}(l\Delta_{\\_}n)\Delta_{\\_}n,\\\
\widehat{\mathrm{Leb}(A\setminus
A_{\\_}h)}&=\sum_{\\_}{\begin{subarray}{c}l\in\\{0,\ldots,n-1\\}:\\\
l\Delta_{\\_}n<h\end{subarray}}\hat{a}(l\Delta_{\\_}n)\Delta_{\\_}n.\end{split}$
(13)

We can justify the choice of the estimators by the following consistency
result whose proof follows from the fact that $X$ is mixing (due to Corollary
1 in [23]) and that

$\mathbb{E}[(X_{\\_}t-X_{\\_}s)^{2}]\leq C\mid
t-s\mid,\,\,\,t,s\in\mathbb{R}.$

######  Proposition 3.

Let the assumptions of Theorem 1 hold. Then, for all $h>0$, the statistics in
(13) are consistent for $\mathrm{Leb}(A)$, $\mathrm{Leb}(A\cap A_{\\_}h)$ and
$\mathrm{Leb}(A\setminus A_{\\_}h)$, respectively.

######  Remark 3.4.

Note that when estimating $\mathrm{Leb}(A)$ and the slices $\mathrm{Leb}(A\cap
A_{\\_}h)$ and $\mathrm{Leb}(A\setminus A_{\\_}h)$, we are only considering
linear functionals of the trawl function as integrands. In particular, by the
definition of $\hat{a}$ and the ergodicity of the process $X$ we deduce that

$\widehat{\mathrm{Leb}(A\cap
A_{\\_}h)}=\hat{\varGamma}_{\\_}{[h/\Delta_{\\_}{n}]}+\mathrm{o}_{\\_}{\mathbb{P}}(1)=\varGamma_{\\_}X(h)+\mathrm{o}_{\\_}{\mathbb{P}}(1).$

A similar relation holds for $\widehat{\mathrm{Leb}(A)}$ and
$\widehat{\mathrm{Leb}(A\setminus A_{\\_}h)}$. Here the situation is simpler
and hence a tuning parameter is not required. In contrast, when estimating the
AVAR, we had to consider non-linear functionals of $\hat{a}$. In this case, to
the best of our knowledge, we cannot apply directly the Ergodic Theorem which
forced us to introduce the tuning parameter $(N_{\\_}n)$.

When studying the expression for the conditional expectation in (11), we note
that we have the equivalent representation given by

$\displaystyle\mathbb{E}(X_{\\_}{t+h}|X_{\\_}t)$
$\displaystyle=\mathrm{Cor}(X_{\\_}t,X_{\\_}{t+h})X_{\\_}t+(1-\mathrm{Cor}(X_{\\_}t,X_{\\_}{t+h}))\mathbb{E}(X_{\\_}t).$

This suggests an alternative estimator of the form

$\displaystyle\begin{split}\widehat{\mathrm{Leb}(A)}&=\widehat{\mathrm{Var}(X)}=\hat{\Gamma}_{\\_}0,\\\
\widehat{\mathrm{Leb}(A\cap
A_{\\_}h)}&=\widehat{\mathrm{Cov}(X_{\\_}0,X_{\\_}h)}=\hat{\Gamma}_{\\_}{[h/\Delta_{\\_}n]},\\\
\widehat{\mathrm{Leb}(A\setminus
A_{\\_}h)}&=\widehat{\mathrm{Var}(X)}-\widehat{\mathrm{Cov}(X_{\\_}0,X_{\\_}h)}=\hat{\Gamma}_{\\_}0-\hat{\Gamma}_{\\_}{[h/\Delta_{\\_}n]},\end{split}$
(14)

where $\widehat{\mathrm{Var}(X)}$ and
$\widehat{\mathrm{Cov}(X_{\\_}0,X_{\\_}h)}$ denote the empirical variance and
autocovariance, respectively.

Hence an alternative predictor is given by

$\displaystyle\hat{X}_{\\_}{t+h|t}^{\mathrm{ACF}}:=\frac{\hat{\Gamma}_{\\_}{[h/\Delta_{\\_}n]}}{\hat{\Gamma}_{\\_}0}X_{\\_}t+\left(1-\frac{\hat{\Gamma}_{\\_}{[h/\Delta_{\\_}n]}}{\hat{\Gamma}_{\\_}0}\right)\overline{X}.$
(15)

While the two predictors are similar, they are not identical. We will compare
the two alternative slice ratio estimation methods in the subsequent
simulation study and compare the respective performance of both predictors in
the empirical study.

###  4 Simulation study

In order to assess the finite sample performance of our new methodology, we
ran extensive simulation studies, which are described in detail in the
Supplementary material, see [24]. The relevant R functions have been published
in the R package ambit on CRAN, see [31].

We simulate trawl processes with three different marginal distributions
(negative binomial, Gamma and Gaussian) and two choices of trawl functions
(exponential and supGamma). The parameters are chosen such that all processes
have the same mean and unit variance. While the exponential trawl function
allows for a possibly relatively fast decay in the corresponding
autocorrelation function, the parameters of the supGamma trawl are chosen such
that the resulting process exhibits long memory, in the sense that the
integral of its autocorrelation function equals infinity.

Throughout our simulation experiments, we assume that $\mu=0$ (and
$\mu_{\\_}0=0$) since this is possibly the situation of most practical
relevance: In practice, we would typically observe data on a fixed frequency
without being able to assess the relation between $\Delta_{\\_}n$ and $n$.
Hence, we vary both $\Delta_{\\_}n$ and $n$ and evaluate the performance of
the proposed estimators for various settings of the sampling scheme.

####  4.1 Consistency results

#####  4.1.1 Trawl function estimation

First of all, we consider the trawl function estimator $\widehat{a}(\cdot)$
and its bias-corrected version
$\widehat{a}(\cdot)-0.5\Delta_{\\_}n\widehat{a}^{\prime}(\cdot)$, which we
evaluate for various points in time. We compute the sample mean, bias and
standard deviation of the two estimators over 200 Monte Carlo runs for various
choices of $\Delta_{\\_}n$ and various choices of the sample size $n$, see
[24, Section D]. The results for a negative binomial marginal distribution
with an exponential trawl and a supGamma trawl are presented in Table
LABEL:tab:Con-NB-Exp-fixedt-main and LABEL:tab:Con-NB-LM-fixedt-main,
respectively.

We observe that, for an exponential trawl, the estimation bias is generally
small for both estimators, with the one for the bias-corrected version, as
expected, being even smaller. Also, for a supGamma trawl function with a
parameter choice in the long-memory regime, the consistency results are
generally very good, apart from one particular setting: When
$\Delta_{\\_}n=0.1$ and $t=0$, we note that $\hat{a}(0)$ is strongly biased
for all three marginal distributions considered, whereas the bias-corrected
estimator performs noticeably better in this particular setting. As soon as a
finer grid width $\Delta_{\\_}n=0.01$ is considered, the estimation results
are generally good for all time points considered.

We also observe that the particular choice of $n$ has almost no impact on the
size of the bias, but an increase in $n$ decreases the standard deviation of
the estimators as expected.

Overall, we conclude that the proposed estimators work well in finite samples
across different marginal distributions, short- and long-memory settings and
different in-fill and long-span asymptotic settings. In the long-memory case,
we advise the use of the bias-corrected estimator, in particular for small
times $t$.

#####  4.1.2 Consistency results of the slice (ratio) estimators

Next, we assess how well we can estimate the trawl set, its slices and the
corresponding ratios given by $\mathrm{\mathrm{Leb}}(A)$,
$\mathrm{\mathrm{Leb}}(A\cap A_{\\_}h)$, $\mathrm{\mathrm{Leb}}(A\setminus
A_{\\_}h)$, $\mathrm{\mathrm{Leb}}(A\cap A_{\\_}h)/\mathrm{\mathrm{Leb}}(A)$
and $\mathrm{\mathrm{Leb}}(A\setminus A_{\\_}h)/\mathrm{\mathrm{Leb}}(A)$.
Detailed results are available in [24, Section F].

In this part of the simulation study, we considered the asymptotic setting
closely related to our empirical study, by setting $\Delta_{\\_}n=0.1$ and
$n=5000$.

We considered three different methods for estimating these integrals of the
trawl function: Using a Riemann approximation of the integrals based on the
original trawl function estimator as in (13), using the Riemann approximation
based on the bias-corrected trawl function estimator and using the empirical
variance and autocovariance estimator presented in (14).

The results for the case of a supGamma trawl are presented in Table 1.

$h=$ |  |
---|---|---
$i\Delta_{\\_}n$ | $\widehat{\mathrm{Leb}(A)}$ | $\widehat{\mathrm{Leb}(A\cap A_{\\_}h)}$ | $\widehat{\mathrm{Leb}(A\setminus A_{\\_}h)}$ | $\widehat{\mathrm{Leb}(A\cap A_{\\_}h)}/\widehat{\mathrm{Leb}(A)}$ | $\widehat{\mathrm{Leb}(A\setminus A_{\\_}h)}/\widehat{\mathrm{Leb}(A)}$
for $i$ | Mean | Bias | SD | Mean | Bias | SD | Mean | Bias | SD | Mean | Bias | SD | Mean | Bias | SD
| Numerical integration of the estimated trawl without bias correction
| Negative binomial marginal distribution
1 | 0.197 | -0.003 | 0.067 | 0.197 | 0.056 | 0.067 | 0 | -0.059 | 0 | 1 | 0.293 | 0 | 0 | -0.293 | 0
2 | 0.197 | -0.003 | 0.067 | 0.139 | 0.024 | 0.067 | 0.058 | -0.026 | 0.006 | 0.681 | 0.104 | 0.078 | 0.319 | -0.104 | 0.078
5 | 0.197 | -0.003 | 0.067 | 0.087 | 0.005 | 0.064 | 0.11 | -0.008 | 0.013 | 0.401 | -0.007 | 0.131 | 0.599 | 0.007 | 0.131
10 | 0.197 | -0.003 | 0.067 | 0.061 | 0 | 0.061 | 0.137 | -0.003 | 0.019 | 0.262 | -0.039 | 0.147 | 0.738 | 0.039 | 0.147
| Gamma marginal distribution
1 | 0.194 | -0.006 | 0.125 | 0.194 | 0.053 | 0.125 | 0 | -0.059 | 0 | 1 | 0.293 | 0 | 0 | -0.293 | 0
2 | 0.194 | -0.006 | 0.125 | 0.134 | 0.019 | 0.122 | 0.06 | -0.025 | 0.011 | 0.649 | 0.071 | 0.099 | 0.351 | -0.071 | 0.099
5 | 0.194 | -0.006 | 0.125 | 0.081 | -0.001 | 0.116 | 0.113 | -0.005 | 0.024 | 0.348 | -0.06 | 0.151 | 0.652 | 0.06 | 0.151
10 | 0.194 | -0.006 | 0.125 | 0.053 | -0.007 | 0.111 | 0.141 | 0.001 | 0.037 | 0.205 | -0.096 | 0.154 | 0.795 | 0.096 | 0.154
| Gaussian marginal distribution
1 | 0.193 | -0.007 | 0.011 | 0.193 | 0.052 | 0.011 | 0 | -0.059 | 0 | 1 | 0.293 | 0 | 0 | -0.293 | 0
2 | 0.193 | -0.007 | 0.011 | 0.135 | 0.019 | 0.011 | 0.059 | -0.026 | 0.001 | 0.696 | 0.118 | 0.018 | 0.304 | -0.118 | 0.018
5 | 0.193 | -0.007 | 0.011 | 0.083 | 0.001 | 0.011 | 0.11 | -0.008 | 0.003 | 0.427 | 0.018 | 0.033 | 0.573 | -0.018 | 0.033
10 | 0.193 | -0.007 | 0.011 | 0.057 | -0.003 | 0.011 | 0.136 | -0.004 | 0.004 | 0.293 | -0.008 | 0.039 | 0.707 | 0.008 | 0.039
| Numerical integration of the estimated trawl with bias correction
| Negative binomial marginal distribution
1 | 0.227 | 0.027 | 0.068 | 0.227 | 0.085 | 0.068 | 0 | -0.059 | 0 | 1 | 0.293 | 0 | 0 | -0.293 | 0
2 | 0.227 | 0.027 | 0.068 | 0.152 | 0.037 | 0.067 | 0.074 | -0.01 | 0.008 | 0.651 | 0.073 | 0.078 | 0.349 | -0.073 | 0.078
5 | 0.227 | 0.027 | 0.068 | 0.091 | 0.009 | 0.064 | 0.136 | 0.017 | 0.014 | 0.368 | -0.04 | 0.125 | 0.632 | 0.04 | 0.125
10 | 0.227 | 0.027 | 0.068 | 0.062 | 0.002 | 0.061 | 0.165 | 0.025 | 0.02 | 0.237 | -0.064 | 0.138 | 0.763 | 0.064 | 0.138
| Gamma marginal distribution
1 | 0.224 | 0.024 | 0.126 | 0.224 | 0.083 | 0.126 | 0 | -0.059 | 0 | 1 | 0.293 | 0 | 0 | -0.293 | 0
2 | 0.224 | 0.024 | 0.126 | 0.147 | 0.032 | 0.123 | 0.077 | -0.008 | 0.015 | 0.62 | 0.042 | 0.1 | 0.38 | -0.042 | 0.1
5 | 0.224 | 0.024 | 0.126 | 0.085 | 0.003 | 0.117 | 0.139 | 0.021 | 0.027 | 0.321 | -0.088 | 0.144 | 0.679 | 0.088 | 0.144
10 | 0.224 | 0.024 | 0.126 | 0.055 | -0.005 | 0.111 | 0.169 | 0.029 | 0.04 | 0.186 | -0.115 | 0.145 | 0.814 | 0.115 | 0.145
| Gaussian marginal distribution
1 | 0.222 | 0.022 | 0.011 | 0.222 | 0.081 | 0.011 | 0 | -0.059 | 0 | 1 | 0.293 | 0 | 0 | -0.293 | 0
2 | 0.222 | 0.022 | 0.011 | 0.147 | 0.032 | 0.011 | 0.075 | -0.01 | 0.002 | 0.662 | 0.085 | 0.018 | 0.338 | -0.085 | 0.018
5 | 0.222 | 0.022 | 0.011 | 0.087 | 0.005 | 0.011 | 0.136 | 0.017 | 0.003 | 0.388 | -0.02 | 0.031 | 0.612 | 0.02 | 0.031
10 | 0.222 | 0.022 | 0.011 | 0.058 | -0.002 | 0.011 | 0.164 | 0.024 | 0.005 | 0.262 | -0.04 | 0.036 | 0.738 | 0.04 | 0.036
| Empirical ACF-based estimator
| Negative binomial marginal distribution
1 | 0.197 | -0.003 | 0.067 | 0.139 | -0.002 | 0.067 | 0.058 | 0 | 0.006 | 0.681 | -0.026 | 0.078 | 0.319 | 0.026 | 0.078
2 | 0.197 | -0.003 | 0.067 | 0.113 | -0.002 | 0.066 | 0.084 | 0 | 0.008 | 0.54 | -0.037 | 0.107 | 0.46 | 0.037 | 0.107
5 | 0.197 | -0.003 | 0.067 | 0.079 | -0.002 | 0.063 | 0.118 | 0 | 0.014 | 0.361 | -0.048 | 0.137 | 0.639 | 0.048 | 0.137
10 | 0.197 | -0.003 | 0.067 | 0.057 | -0.003 | 0.061 | 0.14 | 0 | 0.02 | 0.246 | -0.055 | 0.148 | 0.754 | 0.055 | 0.148
| Gamma marginal distribution
1 | 0.194 | -0.006 | 0.125 | 0.134 | -0.007 | 0.122 | 0.06 | 0.001 | 0.011 | 0.649 | -0.059 | 0.098 | 0.351 | 0.059 | 0.098
2 | 0.194 | -0.006 | 0.125 | 0.108 | -0.008 | 0.119 | 0.086 | 0.002 | 0.017 | 0.499 | -0.079 | 0.129 | 0.501 | 0.079 | 0.129
5 | 0.194 | -0.006 | 0.125 | 0.073 | -0.009 | 0.115 | 0.121 | 0.003 | 0.027 | 0.304 | -0.104 | 0.155 | 0.696 | 0.104 | 0.155
10 | 0.194 | -0.006 | 0.125 | 0.05 | -0.01 | 0.11 | 0.144 | 0.004 | 0.039 | 0.189 | -0.112 | 0.152 | 0.811 | 0.112 | 0.152
| Gaussian marginal distribution
1 | 0.193 | -0.007 | 0.011 | 0.135 | -0.007 | 0.011 | 0.059 | 0 | 0.001 | 0.696 | -0.012 | 0.018 | 0.304 | 0.012 | 0.018
2 | 0.193 | -0.007 | 0.011 | 0.109 | -0.007 | 0.011 | 0.085 | 0 | 0.002 | 0.561 | -0.016 | 0.025 | 0.439 | 0.016 | 0.025
5 | 0.193 | -0.007 | 0.011 | 0.075 | -0.007 | 0.011 | 0.118 | 0 | 0.003 | 0.386 | -0.022 | 0.035 | 0.614 | 0.022 | 0.035
10 | 0.193 | -0.007 | 0.011 | 0.054 | -0.006 | 0.011 | 0.139 | 0 | 0.005 | 0.278 | -0.024 | 0.04 | 0.722 | 0.024 | 0.04
Table 1:  Slice estimation for the supGamma trawl with $\alpha=0.1,H=1.5$ for
the case when $\Delta_{\\_}n=0.1$ and $n=5000$. We consider three different
methods for estimating these integrals of the trawl function: 1) Using a
Riemann approximation of the integrals based on the original trawl function
estimator as in (13), referred to as ”Numerical integration of the estimated
trawl without bias correction”; 2) using the Riemann approximation based on
the bias-corrected trawl function estimator, referred to as ”Numerical
integration of the estimated trawl with bias correction”; and 3) using the
empirical variance and autocovariance estimator presented in (14), referred to
as ”Empirical ACF-based estimator”. The results are based on 200 Monte Carlo
runs.

There are three main lessons we learned from this part of the simulation
study: First, the trawl set, the slices and the corresponding ratios can be
estimated with high precision. Second, the original trawl function estimator
performs better than its bias-corrected counterpart, which is possibly
surprising given that for the pointwise estimation the bias-corrected version
was often preferred. Third, the empirical variance and autocovariance-based
estimator perform best for estimating integrals of the trawl function for all
scenarios considered.

####  4.2 Asymptotic Gaussianity results

We also investigated how closely the infeasible statistic

$T^{IF}(t):=\sqrt{n\Delta_{\\_}{n}}\frac{\left(\hat{a}(t)-a(t)\right)}{\sqrt{\sigma_{\\_}a^{2}(t)}},$

the feasible statistic without bias correction

$T^{F}:=\sqrt{n\Delta_{\\_}{n}}\frac{\left(\hat{a}(t)-a(t)\right)}{\sqrt{\hat{\sigma}_{\\_}a^{2}(t)}},$

and the feasible statistic with bias correction

$T^{F}_{\\_}{\mathrm{bias\,corrected}}:=\sqrt{n\Delta_{\\_}{n}}\frac{\left(\hat{a}(t)-a(t)-\frac{1}{2}\Delta_{\\_}n\hat{a}^{\prime}(t)\right)}{\sqrt{\hat{\sigma}_{\\_}a^{2}(t)}},$

follow a standard normal distribution in finite samples. Detailed results are
available in [24, Section E].

While the consistency results reported above were generally very good
throughout our simulation experiments, the picture for the asymptotic
Gaussianity is slightly more mixed. Recall that for the consistency result, we
only needed simultaneously that $\Delta_{\\_}n$ is ”small”, $n$ is ”large” and
$n\Delta_{\\_}n$ is ”large”. In order to get asymptotic Gaussianity, we also
need that $n\Delta_{\\_}n^{3}$ is ”small” (or that $n\Delta_{\\_}n^{2}$ is
”small”) –when considering the case of $\mu=0$, $\mu_{\\_}0=0$.

For each of the three statistics, we compute the sample mean, standard
deviation and the 75%, 80%, 85%, 90%, 95% and 99% coverage probabilities over
200 Monte Carlo runs. We describe the specific findings for each of the
marginal distributions and the trawl function in detail in the Supplementary
material.

Here we present the results for a negative binomial marginal distribution with
an exponential trawl and a supGamma trawl in Table LABEL:tab:NB-Exp-fixedt-
main and LABEL:tab:NB-LM-fixedt-main, respectively.

Let us summarise our main findings:

First, for $T^{F}$ we often observe a finite sample bias, which is noticeably
reduced when $T^{F}_{\\_}{\mathrm{bias\,corrected}}$ is considered instead.
Bias correction is particularly advisable for smaller values of time $t$.
Second, it appears that the asymptotic variance estimator
$\hat{\sigma}_{\\_}a(\cdot)$ becomes less reliable when time increases; in
such cases, we occasionally observe a deviation from the (asymptotic) unit
standard deviation. This is not a surprise since the larger $t$, the less data
is used to compute $\hat{v}_{\\_}{4}(t)$. Third, decreasing $\Delta_{\\_}n$
does not necessarily lead to an improved finite sample performance, unless, as
expected, $n$ is also increased to get the ”right” balance between the in-fill
and long-span asymptotics. Fourth, in the purely Gaussian case, we observe
that the feasible test statistics are not asymptotically standard Gaussian
unless $\Delta_{\\_}n$ is ”comparably small” (in our case
$\Delta_{\\_}n=0.01$), $n$ is ”large” and a bias correction is used. These
results are in line with our theoretical findings, which require stronger
assumptions on the relation between $\Delta_{\\_}n$ and $n$. Fifth, when
comparing the performance of the test statistics in the short-and long-memory
case, we generally get better results in the short-memory settings. For long-
memory settings, we need rather small values for $\Delta_{\\_}n$ combined with
very large values for $n$ and still end up with poor finite sample performance
when $t$ is large, whereas the results are better for smaller values of $t$.
Finally, we recall that when estimating the asymptotic variance
$\sigma_{\\_}a^{2}(t)$ in the feasible statistics, we need to choose the
tuning parameter $N_{\\_}n$. In our simulation experiments, we found a low
sensitivity to varying choices of the tuning parameters. We typically set the
parameter such that a maximum number of observations can be used in the
estimation. However, since the monotonic trawl function decreases to 0 for
large time points, we found that the impact of including or omitting estimates
of the trawl function which are close to 0 is negligible in practice.

In summary, we conclude that the simulation study has revealed that the trawl
function and also integrals of the trawl function, such as the trawl set, its
slices and corresponding ratios can be estimated with a very small bias. If
one wants to use the asymptotic Gaussianity for constructing asymptotic
confidence intervals, then it is important to check the relation between the
interval width $\Delta_{\\_}n$ of the observations and the number of
observations $n$. While for short-memory trawl functions, the coverage
probabilities appear well-aligned with the theoretical ones for small enough
$\Delta_{\\_}n$ and large enough $n$, there seemed to be greater deviations in
the long-memory case.

###  5 Empirical illustration

We will now present an empirical illustration222The code which reproduces the
stated results is available on GitHub at
https://github.com/almutveraart/Nonparametric-trawl-estimation and archived in
Zenodo [32]. of our new methodology in the context of modelling the bid-ask
spread of equity prices. There is an extensive literature on this topic, see,
e.g., [15] and [8]. [4] proposed using integer-valued trawl processes for
modelling spread data and this work has been further extended in [6, 7]. Here
we use the same dataset as in the latter article, which considers 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 $4$, $2020$. The A
stock is traded on the New York Stock Exchange, which is open from 9.30 AM to
4 PM. We discard the first 60 minutes of data and consider the time period
from 10:30 AM to 4 PM only to obtain data which are not distorted by market
opening effects and are “more stationary”. The raw data is gathered from the
Trade and Quote database and cleaned using standard methods, see [7] for
details. The data is available at a very high frequency but to obtain
equidistant data, we sample the observations with $\Delta_{\\_}n=1/12\approx
0.083$ minutes (i.e. 5 seconds) time steps, using the previous tick approach,
starting at 10:30AM, resulting in $n=3961$ observations of the spread denoted
by $s_{\\_}t$ for $t\in\\{0,\Delta_{\\_}n,\ldots,3961\Delta_{\\_}n\\}$. 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 $x_{\\_}t=s_{\\_}t-1$. The times series
and its autocorrelation function are depicted in Figure 1.

(a)  (b)

Figure 1: The bid-ask spread data $s_{\\_}t$ from Agilent Technologies Inc.
stock on May $4$, $2020$, is sampled every 5s from 10:30 to 16:00. We set
$x_{\\_}t=s_{\\_}t-1$. (a) Time series $s_{\\_}t$, (b) Autocorrelation
function of $s_{\\_}t$.

####  5.1 Estimating the trawl function and confidence bounds

We compute the estimator of the trawl function $\widehat{a}$ and of its
asymptotic variance $\hat{\sigma}_{\\_}a^{2}$, which we present in Figure 2.

(a)  (b)

Figure 2: (a) Estimated trawl function $\hat{a}(l\Delta_{\\_}n)$ for
$l=0,\ldots,30$ (black, filled circles) with upper and lower 95% confidence
intervals (blue, unfilled circles). (b) Estimated asymptotic variance
$\hat{\sigma}_{\\_}a^{2}(l\Delta_{\\_}n)$ for $l=0,\ldots,30$.

Moreover, we compute asymptotic confidence bounds around the estimates of the
trawl function as follows:

$\mathbb{P}\left(a(t)\in\left(\hat{a}(t)-z_{\\_}{\beta/2}\sqrt{\frac{\hat{\sigma}^{2}_{\\_}a(t)}{n\Delta_{\\_}n}},\hat{a}(t)+z_{\\_}{\beta/2}\sqrt{\frac{\hat{\sigma}^{2}_{\\_}a(t)}{n\Delta_{\\_}n}}\right)\right)\approx
1-\beta,$

where $\beta\in(0,1)$ and $z_{\\_}{\beta/2}$ denotes the $1-\beta/2$-quantile
of the standard normal distribution. Figure 2 (a) depicts the estimated trawl
function (black filled circles) with the corresponding 95% confidence
intervals (blue empty circles) and (b) shows the estimated asymptotic
variance.

####  5.2 Forecasting

We can now demonstrate that our estimation method can be applied to
forecasting trawl processes using the results established in Subsection 3.4.
We now apply the predictor defined in (12) to our dataset. We first split our
dataset into a training sample of length $n:=n_{\\_}{training}:=3000$
observations on the $\Delta_{\\_}n=1/12$ grid, and then compute the 1-step,
2-step, $\ldots$, 20-step ahead forecasts. We then continue with a rolling-
window forecast, where we always use a window length of $n=3000$ for the
training sample length. Overall, we compute $3961-3000-20=941$ forecasts which
are 1 to 20 steps ahead. For each of these 20 forecast horizons, we compute
the mean squared error (MSE) and the mean absolute error (MAE) over the 941
forecasts.

(a)  (b)  (c)

(d)  (e)  (f)

Figure 3: (a) MSE ratio - 1: nonparametric trawl versus naive, (b) MSE ratio -
1: nonparametric trawl versus parametric trawl fit, (c) MSE ratio -1:
nonparametric trawl versus the ACF-based predictor
$\hat{X}_{\\_}{t+h|t}^{\mathrm{ACF}}$, (d) MAE ratio - 1: nonparametric trawl
versus naive, (e) MAE ratio - 1: nonparametric trawl versus parametric trawl
fit, (f) MAE ratio - 1: nonparametric trawl versus the ACF-based predictor
$\hat{X}_{\\_}{t+h|t}^{\mathrm{ACF}}$. We also computed the p-values ($p$) of
the Diebold-Mariano (DM) test of the null of equal forecasting performance
between the two respective procedures, against the alternative that the
nonparametric trawl method provides superior forecasts. We use the following
symbols: ”***” for $p\leq 0.001$, ”**” for $0.001<p\leq 0.01$, ”*” for
$0.01<p\leq 0.05$, ”+” for $0.05<p\leq 0.1$ and no symbol for $0.1<p\leq 1$.

We then compare our forecast with three alternative forecasting procedures.

First, we compare our forecast based on the trawl model to a naive forecast,
for which we also compute the corresponding MSE and MAE. We then consider the
ratio of the MSE (MAE) of the forecast based on the trawl model with the one
based on the naive forecast and subtract 1 from this ratio. If the resulting
”ratio-1” is smaller than 0, then the forecast based on the nonparametric
trawl estimation performs better than the naive one. The results are plotted
in Figures 3a and 3d, for the MSEs and the MAEs, respectively. Note that the
1-step ahead forecast resulted in the naive and the trawl-based forecast being
identical for this particular sample path, hence the ratio-1 is equal to 0 in
that case and hence not visible.

We also computed the p-values ($p$) of the Diebold-Mariano (DM) test, see
[10], of the null of equal forecasting performance between the two procedures,
against the alternative that the nonparametric trawl method provides superior
forecasts. We use the following symbols: ”***” for $p\leq 0.001$, ”**” for
$0.001<p\leq 0.01$, ”*” for $0.01<p\leq 0.05$, ”+” for $0.05<p\leq 0.1$ and no
symbol for $0.1<p\leq 1$.

When adding the symbols for the rejection levels of the null hypothesis, we
work with a loss function in the Diebold-Mariano test with power 1 when adding
the information to the MAE plot and with power 2 for the MSE plot.

We observe that according to the MSE and the DM test, the nonparametric trawl
procedure produces better forecasts than the naive model for all
$h\in\\{2,\ldots,20\\}$ forecast horizons, see Figure 3a, whereas the MAE
error shows that the short-term predictive performance of the naive forecast
is slightly better for lag 2, but the trawl model is performing better for
3-20 lags, see Figure 3d, where for 7 of these lags there is a statistically
significant difference in performance accuracy.

Second, we consider a parametric benchmark procedure. Note that the empirical
work carried out by [6, 7] on the same dataset suggests that a trawl process
with negative binomial distribution and supGamma trawl function provides a
good fit to the dataset. Hence we fit such a parametric model to the data and
compare the resulting forecasts with the ones using a nonparametric estimation
procedure above.

Consider the case that the Lévy seed
$L^{\prime}\sim\mathrm{NegBin}(m,\theta)$, for $\theta\in(0,1)$ and
$m=(1-\theta)^{2}/\theta$ to obtain $\mathrm{Var}(L^{\prime})=1$ (this
normalisation is different from the one used in [6, 7]). Then
$L(A_{\\_}t)\sim\mathrm{NegBin}(\mathrm{Leb}(A)(1-\theta)^{2}/\theta,\theta)$
and $\mathbb{E}(L_{\\_}t)=\mathrm{Leb}(A)(1-\theta)$,
$\mathrm{Var}(L_{\\_}t)=\mathrm{Leb}(A)$. Hence we have
$\mathbb{E}(X_{\\_}{t+h}|X_{\\_}t)=\frac{\mathrm{Leb}(A\cap
A_{\\_}h)}{\mathrm{Leb}(A)}X_{\\_}t+\mathrm{Leb}(A\setminus
A_{\\_}h)(1-\theta).$ For the supGamma-Gamma trawl function, we have
$a(x)=c\left(1+\frac{x}{\overline{\alpha}}\right)^{-H}$, for
$\overline{\alpha}>0,H>1,c>0,x\geq 0$,
$\mathrm{Leb}(A)=\int_{\\_}0^{\infty}a(x)dx=c\frac{\overline{\alpha}}{H-1}$,
$\mathrm{Leb}(A\cap
A_{\\_}h)=c\frac{\overline{\alpha}}{H-1}\left(1+\frac{h}{\overline{\alpha}}\right)^{1-H}$,
and $\mathrm{Leb}(A\setminus
A_{\\_}h)=c\frac{\overline{\alpha}}{H-1}\left[1-\left(1+\frac{h}{\overline{\alpha}}\right)^{1-H}\right]$.

######  Remark 5.1.

Note that the parametrisation of the negative-binomial-supGamma process
includes the additional multiplicative constant $c$ (compared to the Example
1), which was set to 1 in earlier works on trawl processes [6, 7]. Earlier
papers have considered parametrisations satisfying $a(0)=1$, whereas we do not
make this assumption in the current work, but instead assume that
$\mathrm{Var}(L^{\prime})=1$.

As before, we compute the ratio between the MSEs and the MAEs, respectively,
when estimating the trawl function and the mean nonparametrically (numerator)
or parametrically (nominator) and then subtract 1 from this ratio. If the
resulting ”ratio-1” is smaller than 0, then the nonparametric procedure
performs better than the parametric one. The results are plotted in Figures 3b
and 3e, for the MSEs and the MAEs, respectively, where the information from
the DM test is also included. As before, we consider the null hypothesis of
equal forecasting performance between the two procedures, against the
alternative that the nonparametric trawl method provides superior forecasts.

We observe that, according to the MAE and a power 1 in the loss function of
the DM-test, the nonparametric procedure outperforms the parametric one,
whereas, for the MSE and a power 2, we fail to reject the null hypothesis of
equal forecast accuracy for small lags, but reject in favour of the
nonparametric procedure for bigger forecast horizons.

Lastly, we compare the performance of the proposed predictor
$\hat{X}_{\\_}{t+h|t}$ with the nonparametric alternative based on the
empirical autocorrelation function $\hat{X}_{\\_}{t+h|t}^{\mathrm{ACF}}$. We
plot the ratios of the corresponding MSEs and MAEs with 1 subtracted in
Figures 3c and 3c, respectively, where the DM-test results are included.
Again, we consider the null hypothesis of equal forecasting performance
between the two procedures, against the alternative that the nonparametric
trawl method provides superior forecasts.

While according to the MSE the ACF-based estimator seems slightly better, the
picture is reversed when the MAE is considered. When running the DM test, we
fail to reject the null hypothesis for small lags $h\leq 6$ but reject it in
favour of higher forecast accuracy of the nonparametric trawl procedure for
higher lags.

###  6 Conclusion and discussion

Trawl processes constitute a large class of stationary, infinitely divisible
stochastic processes which can account for short and long memory and flexible
tail behaviour. This paper has established the first nonparametric estimation
theory for their trawl function, proved the consistency of the estimator and
derived both an infeasible and a feasible version of its asymptotic
Gaussianity. We stress that the selection of any tuning parameters is not
needed for the estimator itself, but is required when constructing asymptotic
confidence bounds. In our simulation experiments, however, we found a low
sensitivity to varying choices of the tuning parameters.

We recall that for guaranteeing the identifiability of the trawl function we
need to either fix the variance of the Lévy seed, which we did by setting
$\mathrm{Var}(L^{\prime})=1$, or the $\mathrm{Leb}(A)$, which could be set to
1. While the former assumption aligns well with equivalent assumptions in
Brownian- or general Lévy-driven models, the latter assumption has also
theoretical appeal: If trawl functions were chosen amongst the class of
(monotonic) density functions, which guarantees that $\mathrm{Leb}(A)=1$, then
the marginal law of the resulting trawl process $L(A_{\\_}t)$ would be exactly
the same as the marginal law of $L^{\prime}$, whereas in the case considered
in this article, the cumulant function of the trawl process and of the
corresponding Lévy seed differ by the proportionality factor
$\mathrm{Leb}(A)$.

Our simulation study has revealed an excellent finite sample performance for
the estimation of the trawl function, the trawl set and its slices over
various choices of the observation scheme. Moreover, when using the asymptotic
Gaussianity for approximating confidence bounds, we advise using the bias-
corrected version of the trawl function estimator and carefully checking
whether the observation scheme, i.e. the choice of $n$ and $\Delta_{\\_}n$,
approximates the theoretical assumptions well.

Finally, we note that nonparametric methods for estimating and forecasting
trawl processes are powerful and simple alternatives to parametric procedures.
They do not require the choice of any tuning parameters, as is often the case
for other nonparametric methods. They avoid the need for parametric model
selection, are easy to implement and have a comparable and sometimes even
superior forecast performance as illustrated in our empirical work.

[Acknowledgments] The authors would like to thank the Center for Research in
Econometric Analysis of Time Series (CREATES) at Aarhus University, Denmark,
for providing the data and Mikkel Bennedsen for preprocessing, cleaning and
subsampling the high-frequency data.

Proofs of the technical results and details from the simulation study The
supplementary material contains all the proofs of the technical results
presented in the main article. Moreover, it presents a detailed description of
the simulation study, tables of numerical results and a summary of the
assessment of the finite sample performance of the proposed estimation theory.

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The supplementary material contains all the proofs of the technical results
presented in the main article [25]. Moreover, it presents a detailed
description of the simulation study, tables of numerical results and a summary
of the assessment of the finite sample performance of the proposed estimation
theory.

## Appendix

###  Appendix A Introduction

This supplementary material for the article [25] is structured as follows.
Section B contains all the proofs of the asymptotic theory presented in the
main article [25]. Sections C, D, E and F contain details on the simulation
study: Section C describes the simulation set-up, Section D presents the
consistency results of the trawl function estimation, Section E assesses the
finite sample performance of the asymptotic Gaussianity and Section F focuses
on the trawl function, slice and slice ratio estimation.

###  Appendix B Proofs of the main results

Throughout all our proofs, the non-random positive constants will be denoted
by the generic symbol $C>0$, and they may change from line to line.
Furthermore, for every $t\geq 0$ and $n\in\mathbb{N}$ we will let
$l_{\\_}{n}:=[t/\Delta_{\\_}{n}]$. Let us now introduce some notation. The
centred version of $X$ will be denoted as
$\tilde{X}_{\\_}{t}:=X_{\\_}{t}-\mathbb{E}(X_{\\_}{t})$, $t\in\mathbb{R}$.
Similarly, we will let $\tilde{L}(\cdot):=L(\cdot)-\mathbb{E}(L(\cdot))$,
which is clearly a centered Lévy basis satisfying that
$\tilde{X}_{\\_}{t}=\tilde{L}(A_{\\_}{t})$. The partial sums of $\tilde{X}$
will be denoted as
$S_{\\_}{n}:=\sum_{\\_}{k=0}^{n-1}\tilde{X}_{\\_}{k\Delta_{\\_}{n}}$.

We proceed to state two key lemmas that will be constantly used during our
proofs. The first one, whose proof is straightforward, concerns the moments of
homogeneous Lévy bases.

######  Lemma 1.

If $L$ is a centered homogeneous Lévy basis, then

$\displaystyle\mathbb{E}(L(A)^{2})$
$\displaystyle=\mathrm{Leb}(A)(\sigma^{2}+\int x^{2}\nu(dx)).$
$\displaystyle\mathbb{E}(L(A)^{3})$ $\displaystyle=\mathrm{Leb}(A)\int
x^{3}\nu(dx).$ $\displaystyle\mathbb{E}(L(A)^{4})$
$\displaystyle=\mathrm{Leb}(A)\left\\{c_{\\_}{4}(L^{\prime})+3\mathrm{Leb}(A)(\sigma^{2}+\int
x^{2}\nu(dx))^{2}\right\\}.$

Moreover, if $A\subseteq B\subseteq C$, then

$\mathbb{E}\left[L(A)L(B)L(C)\right]=\mathbb{E}\left[L(A)^{3}\right],$

and

$\mathbb{E}\left[L(A)^{2}L(B)L(C)\right]=\mathbb{E}\left[L(A)^{4}\right]+\mathbb{E}\left[L(A)^{2}\right]\mathbb{E}\left[L(B\backslash
A)^{2}\right].$

The second key result gives conditions for the negligibility of sums of
martingale differences. For a proof see Lemma 2.2.11 in [16].

######  Lemma 2.

Let $(\Omega,\mathscr{F},\mathbb{P})$ be a complete probability space endowed
with a sequence of discrete-time filtrations
$(\mathscr{H}_{\\_}{i}^{n})_{\\_}{i\geq 1}$. Suppose that
$(D_{\\_}{i}^{n}:i,n\geq 1)$ is a square-integrable
$(\mathscr{H}_{\\_}{i}^{n})_{\\_}{i\geq 1}$-martingale difference and that
$N_{\\_}{n}$ is an $\mathbb{N}$-valued $(\mathscr{H}_{\\_}{i}^{n})_{\\_}{i\geq
1}$-stopping time increasing to $+\infty$. Then

$\sum_{\\_}{i=1}^{N_{\\_}{n}}D_{\\_}{i}^{n}\overset{\mathbb{P}}{\rightarrow}0,$

whenever

$\sum_{\\_}{i=1}^{N_{\\_}{n}}\mathbb{E}\left[(D_{\\_}{i}^{n})^{2}\mid\mathscr{H}_{\\_}{i-1}^{n}\right]\overset{\mathbb{P}}{\rightarrow}0.$

Now, for $i,j=0,1,\ldots$ and $j\geq i$ define

$\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,j):=\\{(r,s):a(t_{\\_}{j+1}-s)<r\leq
a(t_{\\_}{j}-s),t_{\\_}{i-1}<s\leq t_{\\_}{i}\\},$ (16)

where $t_{\\_}{i}=i\Delta_{\\_}{n}$ with the convention that
$t_{\\_}{-1}=-\infty$. Using these sets we introduce the filtrations:

$\displaystyle\mathscr{F}_{\\_}{k}^{n}$
$\displaystyle:=\sigma\left(\\{L(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(m,j))\\}_{\\_}{j\geq
m}:m=0,\ldots,k\right),\,\,\,\,k\geq 0,$ (17)

and

$\mathscr{G}_{\\_}{j}^{n}:=\sigma\left(\\{L(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,m))\\}_{\\_}{0\leq
i\leq m}:m=0,\ldots,j\right),\,\,\,j\geq 0.$ (18)

The following local approximations for $a$ and $\phi$ will be often used
through our computations:

$\psi_{\\_}{n}(x):=\int_{\\_}{0}^{1}a(s\Delta_{\\_}{n}+x)ds,\,\,\psi_{\\_}{n}^{\prime}(x):=\int_{\\_}{0}^{1}\int_{\\_}{0}^{1}\phi((s+r)\Delta_{\\_}{n}+x)dsdr\,\,x\geq
0.$ (19)

####  B.1 Proof of the consistency of $\hat{a}(t)$

Now we present the proof of Theorem 1 in [25]. First, we derive a key
decomposition for $\hat{a}(t)$ that will be used throughout all our proofs:
For all $t\geq 0$,

$\hat{a}(t)=\tilde{a}(t)+\mathfrak{b}_{\\_}{n}(t),$ (20)

where $\mathfrak{b}_{\\_}{n}(0)=0$ and for $t>0$

$\mathfrak{b}_{\\_}{n}(t):=\frac{1}{n\Delta_{\\_}{n}}(X_{\\_}{(n-1)\Delta_{\\_}{n}}-X_{\\_}{l_{\\_}{n}\Delta_{\\_}{n}})\left(\frac{1}{n}S_{\\_}{n}\right)+\frac{1}{n\Delta_{\\_}{n}}(X_{\\_}{(n-1)\Delta_{\\_}{n}}-\bar{X}_{\\_}{n})(X_{\\_}{(n-1-l_{\\_}{n})\Delta_{\\_}{n}}-\bar{X}_{\\_}{n}),$

and

$\tilde{a}(t):=\begin{cases}\frac{1}{2n\Delta_{\\_}{n}}\sum_{\\_}{k=0}^{n-2}(\delta_{\\_}{k}\tilde{X})^{2}&\text{if
}t=0;\\\
-\frac{1}{n\Delta_{\\_}{n}}\sum_{\\_}{k=l_{\\_}{n}}^{n-2}\tilde{X}_{\\_}{(k-l_{\\_}{n})\Delta_{\\_}{n}}\delta_{\\_}{k}\tilde{X}&\text{if
}t>0,\end{cases}$

in which
$\delta_{\\_}{k}\tilde{X}:=\left(\tilde{X}_{\\_}{(k+1)\Delta_{\\_}{n}}-\tilde{X}_{\\_}{k\Delta_{\\_}{n}}\right)$.

###### Proof.

Clearly $\mathfrak{b}_{\\_}{n}(t)=\mathrm{o}_{\\_}\mathbb{P}(1)$. Moreover, as
$n\rightarrow\infty$

$\mathbb{E}(\tilde{a}(t))=(1-l_{\\_}n/n)\frac{1}{\Delta_{\\_}{n}}\int_{\\_}{l_{\\_}{n}\Delta_{\\_}{n}}^{(l_{\\_}{n}+1)\Delta_{\\_}{n}}a(s)ds\rightarrow
a(t),$ (21)

due to the stationarity of $X$, Assumption 1, and the continuity of $a$. It is
then left to show that $\tilde{a}(t)-\mathbb{E}(\tilde{a}(t))$ is
asymptotically negligible. For $t>0$, we use the following identities, valid
for $l_{\\_}{n}\leq k\leq j\leq n-1$,

$-\delta_{\\_}{k}\tilde{X}=\chi_{\\_}{k}-\varrho_{\\_}{k};\,\,\,X_{\\_}{(k-l_{\\_}{n})\Delta_{\\_}{n}}=\beta_{\\_}{k,j}^{(1)}+\beta_{\\_}{k,j}^{(2)}+\sum_{\\_}{m=j+1}^{n}\beta_{\\_}{k,m}^{(1)}+\beta_{\\_}{k}^{(3)},$
(22)

where

$\chi_{\\_}{k}:=\tilde{L}(\cup_{\\_}{i=0}^{k}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,k)),\,\,\,\varrho_{\\_}{k}:=\tilde{L}((\cup_{\\_}{j\geq
k+1}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,j)),$ (23)

and

$\displaystyle\beta_{\\_}{k,j}^{(1)}$
$\displaystyle=\tilde{L}((\cup_{\\_}{i=0}^{k-l_{\\_}{n}}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,j));$
(24) $\displaystyle\beta_{\\_}{k,j}^{(2)}$
$\displaystyle=\tilde{L}((\cup_{\\_}{i=0}^{k-l_{\\_}{n}}\cup_{\\_}{m=k-l_{\\_}{n}}^{j-1}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,m));$
$\displaystyle\beta_{\\_}{k}^{(3)}$
$\displaystyle=\tilde{L}(L(\cup_{\\_}{i=0}^{k-l_{\\_}{n}}\cup_{\\_}{m\geq
n+1}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,m)).$

Note that the array
$(\tilde{X}_{\\_}{(k-l_{\\_}{n})\Delta_{\\_}{n}}\varrho_{\\_}{k})_{\\_}{k\geq
l_{\\_}n}$ is an $\mathscr{F}_{\\_}{k+1}^{n}$ square integrable increment
martingale with
$\mathbb{E}[(\tilde{X}_{\\_}{(k-l_{\\_}{n})\Delta_{\\_}{n}}\varrho_{\\_}{k})^{2}]\leq
C\Delta_{\\_}{n}$, due to Lemma 1. From Lemma 2

$\tilde{a}(t)=\frac{1}{n\Delta_{\\_}{n}}\sum_{\\_}{k=l_{\\_}{n}}^{n-1}\tilde{X}_{\\_}{(k-l_{\\_}{n})\Delta_{\\_}{n}}\chi_{\\_}k+\mathrm{o}_{\\_}\mathbb{P}(1).\
$

The consistency now follows from the relations (33) and (39) below as well as
the boundedness and integrability of $a$. The case $t=0$ can be shown
analogously. ∎

####  B.2 Proof of the infeasible central limit theorem

Next, we turn to the proof of Theorem 2 in [25]. Before presenting our proof,
let us state some simple facts. Identities (20) and (21) together with
Assumption 3, reveal that for $t\geq 0$

$\hat{a}(t)-a(l_{\\_}{n}\Delta_{\\_}{n})=\tilde{a}(t)-\mathbb{E}(\tilde{a}(t))+\frac{1}{\Delta_{\\_}{n}}\int_{\\_}{l_{\\_}{n}\Delta_{\\_}{n}}^{(l_{\\_}{n}+1)\Delta_{\\_}{n}}\left[a(s)-a(l_{\\_}{n}\Delta_{\\_}{n})\right]ds+\mathfrak{b}_{\\_}{n}(t)+\mathrm{O}(1/n\Delta_{\\_}{n}).$
(25)

In view that $\tilde{X}_{\\_}{t}=\tilde{L}(A_{\\_}{t})$, where
$\tilde{L}(\cdot):=L(\cdot)-\mathbb{E}(L(\cdot))$ is a centered Lévy basis,
(25) indicates that without loss of generality we may and do assume that $L$
is centred which in particular implies that $\tilde{X}=X$.

###### Proof.

First note that by [21, Lemma 5.1], Assumption 2 and the stationarity of $X$
we get

$\mathfrak{b}_{\\_}{n}(t)=\mathrm{o}_{\\_}{\mathbb{P}}(1/\sqrt{n\Delta_{\\_}{n}}).$

Furthermore, Assumption 3 easily implies that

$\frac{1}{\Delta_{\\_}{n}}\int_{\\_}{l_{\\_}{n}\Delta_{\\_}{n}}^{(l_{\\_}{n}+1)\Delta_{\\_}{n}}\left[a(s)-a(l_{\\_}{n}\Delta_{\\_}{n})\right]ds=-\Delta_{\\_}{n}\int_{\\_}{0}^{1}\int_{\\_}{0}^{y}\phi(\Delta_{\\_}{n}x+l_{\\_}{n}\Delta_{\\_}{n})dxdy.$
(26)

Therefore, thanks to (25) and the Dominated Convergence Theorem, we deduce
that

$\hat{a}(t)-a(l_{\\_}{n}\Delta_{\\_}{n})=[\tilde{a}(t)-\mathbb{E}(\tilde{a}(t))]-\frac{\Delta_{\\_}{n}}{2}\phi(t)+\mathrm{o}_{\\_}{\mathbb{P}}(1/\sqrt{n\Delta_{\\_}{n}})+\mathrm{o}(\Delta_{\\_}{n}).$
(27)

The key part of the proof is showing that

$\mathcal{E}_{\\_}{t}:=\sqrt{n\Delta_{\\_}{n}}[\tilde{a}(t)-\mathbb{E}(\tilde{a}(t))]\overset{d}{\rightarrow}N(0,\sigma_{\\_}{a}^{2}(t)).$
(28)

If this were true, then by (27) and Assumption 2, we would have that for
$0\leq\mu<\infty$

$\sqrt{n\Delta_{\\_}{n}}(\hat{a}(t)-a(l_{\\_}{n}\Delta_{\\_}{n}))=-\frac{1}{2}\sqrt{\mu}\phi(t)+N(0,\sigma_{\\_}{a}^{2}(t))+\mathrm{o}_{\\_}{\mathbb{P}}(1),$

while for $\mu=+\infty$

$\frac{\hat{a}(t)-a(l_{\\_}{n}\Delta_{\\_}{n})}{\Delta_{\\_}{n}}=-\frac{1}{2}\phi(t)+\mathrm{o}_{\\_}{\mathbb{P}}(1),$

which is enough. The verification of (28) in divided in three parts: In the
first step we write $\mathcal{E}_{\\_}{t}$ as a sum of an increment martingale
plus a negligible term. In the last two steps we derive the asymptotic
variance and check that the Lyapunov condition is satisfied under our
assumptions.

##### Step 1: An increment martingale decomposition for $\mathcal{E}_{\\_}{t}$

Let us start by assuming that $t>0$ and observe that in this situation (recall
that $L$ is assumed to be centred)

$\mathcal{E}_{\\_}{t}=-\frac{1}{\sqrt{n\Delta_{\\_}{n}}}\sum_{\\_}{k=l_{\\_}{n}}^{n-1}\left\\{X_{\\_}{(k-l_{\\_}{n})\Delta_{\\_}{n}}\delta_{\\_}{k}X-\mathbb{E}\left(X_{\\_}{(k-l_{\\_}{n})\Delta_{\\_}{n}}\delta_{\\_}{k}X\right)\right\\}+\mathrm{o}_{\\_}{\mathbb{P}}(1),$

Furthermore, using (22) and that

$-\delta_{\\_}{k}X=\chi_{\\_}{k}-\sum_{\\_}{j=k+1}^{n}\alpha_{\\_}{k+1,j}-\gamma_{\\_}{k},$
(29)

with

$\alpha_{\\_}{k,j}=L(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k,j)),\,\,\,\gamma_{\\_}{k}:=L(\cup_{\\_}{j\geq
n+1}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,j)),$ (30)

we further deduce that

$\mathcal{E}_{\\_}{t}=\frac{1}{\sqrt{n\Delta_{\\_}{n}}}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\zeta_{\\_}{j,n}+\sum_{\\_}{q=1}^{3}r_{\\_}{n,q}+\mathrm{o}_{\\_}{\mathbb{P}}(1),$
(31)

where $\zeta_{\\_}{j,n}=\sum_{\\_}{p=1}^{5}\xi_{\\_}{j,n}^{(p)}$ in which

$\displaystyle\xi_{\\_}{j,n}^{(1)}$
$\displaystyle=\left[\chi_{\\_}{j}\beta_{\\_}{j,j}^{(1)}-\mathbb{E}(\chi_{\\_}{j}\beta_{\\_}{j,j}^{(1)})\right];\,\,\,\xi_{\\_}{j,n}^{(2)}=\chi_{\\_}{j}\beta_{\\_}{j,j}^{(2)},\,\,\,\xi_{\\_}{j,n}^{(3)}=\sum_{\\_}{k=l_{\\_}{n}}^{j-1}\chi_{\\_}{k}\beta_{\\_}{k,j}^{(1)};$
(32) $\displaystyle\xi_{\\_}{j,n}^{(4)}$
$\displaystyle=-\sum_{\\_}{k=l_{\\_}{n}}^{j-1}\alpha_{\\_}{k+1,j}\beta_{\\_}{k,j}^{(2)};\,\,\,\xi_{\\_}{j,n}^{(5)}=-\sum_{\\_}{m=l_{\\_}{n}+1}^{j}\sum_{\\_}{k=l_{\\_}{n}}^{m-1}\alpha_{\\_}{k+1,m}\beta_{\\_}{k,j}^{(1)}=-\sum_{\\_}{k=l_{\\_}{n}}^{j-1}\sum_{\\_}{m=k+1}^{j}\alpha_{\\_}{k+1,m}\beta_{\\_}{k,j}^{(1)}$

Furthermore,

$\displaystyle r_{\\_}{n,1}=$
$\displaystyle\frac{1}{\sqrt{n\Delta_{\\_}{n}}}\sum_{\\_}{k=l_{\\_}{n}}^{n-1}\chi_{\\_}{k}\beta_{\\_}{k}^{(3)},\,\,r_{\\_}{n,2}=-\frac{1}{\sqrt{n\Delta_{\\_}{n}}}\sum_{\\_}{k=l_{\\_}{n}}^{n-1}X_{\\_}{(k-l_{\\_}{n})\Delta_{\\_}{n}}\gamma_{\\_}{k}$
$\displaystyle r_{\\_}{n,3}=$
$\displaystyle-\frac{1}{\sqrt{n\Delta_{\\_}{n}}}\sum_{\\_}{k=l_{\\_}{n}}^{n-1}\beta_{\\_}{k}^{(3)}L(\cup_{\\_}{j=k+1}^{n}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,j)).$

We proceed now to show that $r_{\\_}{n,q}$ are asymptotically negligible for
$q=1,2,3$. Observe first that since
$(\chi_{\\_}{k})_{\\_}{k=l_{\\_}{n},\ldots,n-1}$ are i.i.d., centered, and
independent of $(\beta_{\\_}{k}^{(3)})_{\\_}{k=l_{\\_}{n},\ldots,n-1}$, then
the array $(\chi_{\\_}{k}\beta_{\\_}{k}^{(3)})_{\\_}{k=l_{\\_}{n},\ldots,n-1}$
is a martingale difference w.r.t. the filtration

$\mathscr{G}_{\\_}{j}^{n}\lor\sigma(\\{\beta_{\\_}{k}^{(3)}:k=l_{\\_}{n},\ldots,n-1\\}),\,\,j=l_{\\_}{n},\ldots,n-1.$

It follows from this and the fact that $l_{\\_}{n}/n=\mathrm{o}(1)$ that

$\displaystyle\mathbb{E}(r_{\\_}{n,1}^{2})$
$\displaystyle=\frac{1}{n\Delta_{\\_}{n}}\int_{\\_}{0}^{\Delta_{\\_}{n}}a(s)ds\sum_{\\_}{k=l_{\\_}{n}}^{n-1}\int_{\\_}{-\infty}^{t_{\\_}{k}-l_{\\_}{n}\Delta_{\\_}{n}}a((n+1)\Delta_{\\_}{n}-s)ds$
(33)
$\displaystyle=\mathrm{O}(1)\times\frac{1}{n\Delta_{\\_}{n}}\int_{\\_}{0}^{n\Delta_{\\_}{n}}\int_{\\_}{(n+1)\Delta_{\\_}{n}-[r/\Delta_{\\_}{n}]\Delta_{\\_}{n}}^{\infty}a(s)dsdr+\mathrm{o}(1)$
$\displaystyle=\mathrm{O}(1)\times\int_{\\_}{0}^{1}\int_{\\_}{(\frac{n+1-[nx]}{n})\Delta_{\\_}{n}n}^{\infty}a(s)dsdx+\mathrm{o}(1)\rightarrow
0,$

where in the last step we made the change of variables $x=r/n\Delta_{\\_}{n}$
as well as applied the Dominated Convergence Theorem. Similar arguments can be
used for $q=2,3$. Hence, we have that

$\mathcal{E}_{\\_}{t}=\frac{1}{\sqrt{n\Delta_{\\_}{n}}}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\zeta_{\\_}{j,n}+\mathrm{o}_{\\_}{\mathbb{P}}(1),\,\,\,t>0.$
(34)

where $\zeta_{\\_}{j,n}$ is defined as a sum of the r.v.’s introduced in (32).
On the other hand, by arguing as above, it is easy to see that (34) is valid
also for $t=0$ but in this situation
$\zeta_{\\_}{j,n}=\frac{1}{2}\sum_{\\_}{q=1}^{2}\xi_{\\_}{j,n}^{(q)}$ in which

$\displaystyle\xi_{\\_}{j,n}^{(1)}$
$\displaystyle=\chi_{\\_}{j}^{2}-\mathbb{E}(\chi_{\\_}{j}^{2});\,\,\,\xi_{\\_}{j,n}^{(2)}=\sum_{\\_}{k=0}^{j-1}[\alpha_{\\_}{k+1,j}^{2}-\mathbb{E}(\alpha_{\\_}{k+1,j}^{2})].$
(35)

Note that by construction $\zeta_{\\_}{j,n}$ is
$\mathscr{G}_{\\_}{j}^{n}$-measurable (see (18)) and such that
$\mathbb{E}(\zeta_{\\_}{j,n}\mid\mathscr{G}_{\\_}{j-1}^{n})=0$, i.e. the array
$(\zeta_{\\_}{j,n})_{\\_}{j=l_{\\_}{n}+1,\ldots,n-1}$ is an increment
martingale. Hence, according to Theorem 2.2.13 in [16], in order to finish the
proof, we must verify that for some $p>2$

$\displaystyle
1.\,\frac{1}{\Delta_{\\_}{n}n}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\mathbb{E}(\zeta_{\\_}{j,n}^{2}\mid\mathscr{G}_{\\_}{j-1}^{n})$
$\displaystyle\overset{\mathbb{P}}{\rightarrow}\sigma_{\\_}{a}^{2}(t),$ (36)
$\displaystyle
2.\,\frac{1}{(\Delta_{\\_}{n}n)^{p}}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\mathbb{E}(\mid\zeta_{\\_}{j,n}\mid^{p}\mid\mathscr{G}_{\\_}{j-1}^{n})$
$\displaystyle\overset{\mathbb{P}}{\rightarrow}0.$ (37)

For the rest of the proof, we will verify that this is indeed the case.

##### Step 2: The asymptotic variance of $\mathcal{E}_{\\_}{t}$

In Lemma 3 below, we show that

$\frac{1}{\Delta_{\\_}{n}n}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\mathbb{E}(\zeta_{\\_}{j,n}^{2}\mid\mathscr{G}_{\\_}{j-1}^{n})=\frac{1}{\Delta_{\\_}{n}n}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\mathbb{E}(\zeta_{\\_}{j,n}^{2})+\mathrm{o}_{\\_}{\mathbb{P}}(1).$
(38)

Once again, we start by considering $t>0$. By the independent scattered
property of $L$ we have that

$\mathbb{E}(\zeta_{\\_}{j,n}^{2})=\sum_{\\_}{p=1}^{5}\mathbb{E}[(\xi_{\\_}{j,n}^{(p)})^{2}]+2\sum_{\\_}{p=3}^{5}\sum_{\\_}{q=2}^{p-1}\mathbb{E}[\xi_{\\_}{j,n}^{(p)}\xi_{\\_}{j,n}^{(q)}].$

From Lemma 1 (recall that $\mathrm{Var}(L^{\prime})^{2}=1$) and the proof of
Lemma 3 below, we deduce that

$\displaystyle\mathbb{E}[(\xi_{\\_}{j,n}^{(1)})^{2}]=$
$\displaystyle\Delta_{\\_}{n}c_{\\_}{4}(L^{\prime})\psi_{\\_}{n}(l_{\\_}{n}\Delta_{\\_}{n})+\mathrm{O}(\Delta_{\\_}{n}^{2});$
(39) $\displaystyle\mathbb{E}[(\xi_{\\_}{j,n}^{(2)})^{2}]=$
$\displaystyle\Delta_{\\_}{n}\psi_{\\_}{n}(0)\int_{\\_}{0}^{l_{\\_}{n}\Delta_{\\_}{n}}a(s)ds;$
$\displaystyle\mathbb{E}[(\xi_{\\_}{j,n}^{(3)})^{2}]=$
$\displaystyle\Delta_{\\_}{n}\psi_{\\_}{n}(0)\int_{\\_}{l_{\\_}{n}\Delta_{\\_}{n}}^{t_{\\_}{j}}\psi_{\\_}{n}(\left\lceil
s/\Delta_{\\_}{n}\right\rceil\Delta_{\\_}{n})ds;$
$\displaystyle\mathbb{E}[(\xi_{\\_}{j,n}^{(4)})^{2}]=$
$\displaystyle\int_{\\_}{0}^{\Delta_{\\_}{n}}[\varGamma_{\\_}{X}(0)-\varGamma_{\\_}{X}(\left\lceil
s/\Delta_{\\_}{n}\right\rceil\Delta_{\\_}{n}+l_{\\_}{n}\Delta_{\\_}{n})]a(s)ds$
$\displaystyle+\Delta_{\\_}{n}\int_{\\_}{\Delta_{\\_}{n}}^{t_{\\_}{j}-l_{\\_}{n}\Delta_{\\_}{n}}\psi_{\\_}{n}(\left\lceil
s/\Delta_{\\_}{n}\right\rceil\Delta_{\\_}{n}+(l_{\\_}{n}-1)\Delta_{\\_}{n})a(s)ds$
$\displaystyle-\int_{\\_}{t_{\\_}{j}}^{t_{\\_}{j+1}}[\varGamma_{\\_}{X}(0)-\varGamma_{\\_}{X}(\left\lceil
s/\Delta_{\\_}{n}\right\rceil\Delta_{\\_}{n}-\Delta_{\\_}{n})]a(s-l_{\\_}{n}\Delta_{\\_}{n})ds;$
$\displaystyle\mathbb{E}[(\xi_{\\_}{j,n}^{(5)})^{2}]=$
$\displaystyle\Delta_{\\_}{n}\int_{\\_}{\Delta_{\\_}{n}}^{t_{\\_}{j}-l_{\\_}{n}\Delta_{\\_}{n}}\psi_{\\_}{n}(\left\lceil
s/\Delta_{\\_}{n}\right\rceil\Delta_{\\_}{n}+l_{\\_}{n}\Delta_{\\_}{n})\left(\psi_{\\_}{n}(0)-a(s)\right)ds.$

as well as

$\displaystyle-\mathbb{E}[\xi_{\\_}{j,n}^{(3)}\xi_{\\_}{j,n}^{(4)}]=$
$\displaystyle\Delta_{\\_}{n}\sum_{\\_}{k=2l_{\\_}{n}+1}^{j-1}\int_{\\_}{l_{\\_}{n}\Delta_{\\_}{n}}^{t_{\\_}{k}-l_{\\_}{n}\Delta_{\\_}{n}}\left[a(t_{\\_}{j}-s)-a(t_{\\_}{j+1}-s)\right]$
(40)
$\displaystyle\times\psi_{\\_}{n}(t_{\\_}{k}-[s/\Delta_{\\_}{n}]\Delta_{\\_}{n}+l_{\\_}{n}\Delta_{\\_}{n})ds$
$\displaystyle-\mathbb{E}[\xi_{\\_}{j,n}^{(3)}\xi_{\\_}{j,n}^{(5)}]=$
$\displaystyle\Delta_{\\_}{n}\int_{\\_}{0}^{t_{\\_}{j}-l_{\\_}{n}\Delta_{\\_}{n}}\psi_{\\_}{n}(\left\lceil
s/\Delta_{\\_}{n}\right\rceil\Delta_{\\_}{n}+l_{\\_}{n}\Delta_{\\_}{n})\left[\psi_{\\_}{n}(0)-a(s)\right]ds;$
$\displaystyle\mathbb{E}[\xi_{\\_}{j,n}^{(4)}\xi_{\\_}{j,n}^{(5)}]=$
$\displaystyle 0.$

Furthermore, by the definition of $\xi_{\\_}{j,n}^{(2)}$, if $t=0$, then
$\mathbb{E}[\xi_{\\_}{j,n}^{(2)}\xi_{\\_}{j,n}^{(p)}]=0$ for all
$p=1,\ldots,5$, while for $t>0$

$\displaystyle\mathbb{E}[\xi_{\\_}{j,n}^{(2)}\xi_{\\_}{j,n}^{(3)}]=$
$\displaystyle\Delta_{\\_}{n}\int_{\\_}{0}^{l_{\\_}{n}\Delta_{\\_}{n}}\psi_{\\_}{n}(\left\lceil
s/\Delta_{\\_}{n}\right\rceil\Delta_{\\_}{n}+l_{\\_}{n}\Delta_{\\_}{n})\psi_{\\_}{n}(l_{\\_}{n}\Delta_{\\_}{n}-\left\lceil
s/\Delta_{\\_}{n}\right\rceil\Delta_{\\_}{n})ds$ (41)
$\displaystyle-\mathbb{E}[\xi_{\\_}{j,n}^{(2)}\xi_{\\_}{j,n}^{(4)}]=$
$\displaystyle\Delta_{\\_}{n}\int_{\\_}{\Delta_{\\_}{n}}^{t_{\\_}{j}-l_{\\_}{n}\Delta_{\\_}{n}}\left[a(s)-a(s+l_{\\_}{n}\Delta_{\\_}{n})\right]\left[\psi_{\\_}{n}(0)-\psi_{\\_}{n}(\left\lceil
s/\Delta_{\\_}{n}\right\rceil\Delta_{\\_}{n}-\Delta_{\\_}{n})\right]ds$
$\displaystyle+\Delta_{\\_}{n}\int_{\\_}{t_{\\_}{j}-l_{\\_}{n}\Delta_{\\_}{n}}^{t_{\\_}{j}}a(s)ds(\psi_{\\_}{n}(0)-\psi_{\\_}{n}(t_{\\_}{j}-l_{\\_}{n}\Delta_{\\_}{n}))$
$\displaystyle-\mathbb{E}[\xi_{\\_}{j,n}^{(2)}\xi_{\\_}{j,n}^{(5)}]=$
$\displaystyle\Delta_{\\_}{n}\int_{\\_}{l_{\\_}{n}\Delta_{\\_}{n}}^{t_{\\_}{j}-l_{\\_}{n}\Delta_{\\_}{n}}\left[a(s-l_{\\_}{n}\Delta_{\\_}{n})-a(s)\right]\psi_{\\_}{n}(\left\lceil
s/\Delta_{\\_}{n}\right\rceil\Delta_{\\_}{n}+l_{\\_}{n}\Delta_{\\_}{n})ds,$

Note that by swapping the order of summation and making the change of
variables $x=r/n\Delta_{\\_}{n}$, we obtain that as $n\rightarrow\infty$

$\displaystyle\frac{1}{\Delta_{\\_}{n}n}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\mathbb{E}[\xi_{\\_}{j,n}^{(3)}\xi_{\\_}{j,n}^{(4)}]$
(42) $\displaystyle=$
$\displaystyle-\frac{1}{n\Delta_{\\_}{n}}\int_{\\_}{(2l_{\\_}{n}+1)\Delta_{\\_}{n}}^{(n-1)\Delta_{\\_}{n}}\int_{\\_}{l_{\\_}{n}\Delta_{\\_}{n}}^{([r/\Delta_{\\_}{n}]-l_{\\_}{n})\Delta_{\\_}{n}}\psi_{\\_}{n}(\left\lceil
s/\Delta_{\\_}{n}\right\rceil\Delta_{\\_}{n}+l_{\\_}{n}\Delta_{\\_}{n})a(s+\Delta_{\\_}{n})dsdr+o(1)$
$\displaystyle\rightarrow$ $\displaystyle-\int_{\\_}{t}^{\infty}a(s+t)a(s)ds,$

where in the last step we have used the Dominated Convergence Theorem (recall
that $a$ is continuous, integrable and totally bounded). Similar arguments can
be applied to (39), (40) and (41), to conclude that (36) holds.

Suppose now that $t=0$. Then

$\displaystyle\mathbb{E}[(\xi_{\\_}{j,n}^{(1)})^{2}\mid\mathscr{G}_{\\_}{j-1}^{n}]=\mathbb{E}[(\xi_{\\_}{j,n}^{(1)})^{2}]=$
$\displaystyle\Delta_{\\_}{n}c_{\\_}{4}(L^{\prime})\psi_{\\_}{n}(0)+\mathrm{O}(\Delta_{\\_}{n}^{2});$
$\displaystyle\mathbb{E}[(\xi_{\\_}{j,n}^{(2)})^{2}\mid\mathscr{G}_{\\_}{j-1}^{n}]=\mathbb{E}[(\xi_{\\_}{j,n}^{(2)})^{2}]=$
$\displaystyle
c_{\\_}{4}(L^{\prime})\sum_{\\_}{k=0}^{j-1}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,j))(1+\mathrm{O}(\Delta_{\\_}{n}));$
$\displaystyle\mathbb{E}[\xi_{\\_}{j,n}^{(1)}\xi_{\\_}{j,n}^{(2)}\mid\mathscr{G}_{\\_}{j-1}^{n}]=\mathbb{E}[\xi_{\\_}{j,n}^{(1)}\xi_{\\_}{j,n}^{(2)}]=$
$\displaystyle=c_{\\_}{4}(L^{\prime})\sum_{\\_}{k=0}^{j-1}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,j))(1+\mathrm{O}(\Delta_{\\_}{n})).$

Relation (36) follows easily from this.

##### Step 3: The Lyapunov condition

Here we only consider the case $t>0$ since the case $t=0$ can be analysed in
the same way. Thus, in order to achieve (37), we will check that for some
$p>2$ and every $M=1,2,\ldots,5$ as $n\rightarrow\infty$

$\frac{1}{(\Delta_{\\_}{n}n)^{p/2}}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\mathbb{E}(\mid\xi_{\\_}{j,n}^{(M)}\mid^{p})\rightarrow
0.$ (43)

From Assumption 4, there is $p_{\\_}{0}\geq 16/3$, such that
$\mathbb{E}(\left|L(A)\right|^{q})<\infty$ for all $q\leq p_{\\_}{0}$ and all
$A\in\mathcal{B}(\mathbb{R}^{2})$. Since $L(A)=Y_{\\_}{\mathrm{Leb}(A)}$,
where $Y$ is a Lévy process such that $Y_{\\_}{1}\sim L^{\prime}$, we deduce
from [19] (c.f. [29, Corollary 1.2.7.]) that for all
$A\in\mathcal{B}(\mathbb{R}^{2})$ with $\,\mathrm{Leb}(A)\leq 1$, it holds

$\mathbb{E}\left(\left|L(A)\right|^{q}\right)\leq
C\begin{cases}\mathrm{Leb}(A)&\text{if }p_{\\_}{0}\geq q\geq 2;\\\
\mathrm{Leb}(A)^{q/2}&\text{if }1\leq q<2,\end{cases}$ (44)

for a constant $C>0$ only depending on $\nu$ and $q$. Using this estimate, we
easily get that for all $p_{\\_}{0}/2\geq p>2$ uniformly on $j$

$\mathbb{E}(\mid\xi_{\\_}{j,n}^{(1)}\mid^{p})=\mathbb{E}(\mid\xi_{\\_}{j,n}^{(2)}\mid^{p})=\mathrm{O}(\Delta_{\\_}{n}),$
(45)

from which it is easily deduced that (43) holds for $M=1,2$. We now proceed to
analyse $\mathbb{E}(\mid\xi_{\\_}{j,n}^{(3)}\mid^{p})$. Rosenthal’s inequality
(see [14, Theorem 2.12] and [18]) applied to the filtration
$\mathscr{F}_{\\_}{k}^{n}\lor\sigma(\beta_{\\_}{k,j}^{(1)})_{\\_}{k=l_{\\_}{n},\ldots,j-1}$,
let us infer that for all $p_{\\_}{0}/2\geq p>2$

$\displaystyle\mathbb{E}(\mid\xi_{\\_}{j,n}^{(3)}\mid^{p})$ $\displaystyle\leq
C\left\\{\mathbb{E}\left[\left(\sum_{\\_}{k=l_{\\_}{n}}^{j-1}(\beta_{\\_}{k,j}^{(1)})^{2}\mathbb{E}(\chi_{\\_}{k}^{2})\right)^{p/2}\right]+\sum_{\\_}{k=l_{\\_}{n}}^{j-1}\mathbb{E}(\mid\chi_{\\_}{k}\beta_{\\_}{k,j}^{(1)}\mid^{p})\right\\}$
(46)
$\displaystyle=C\mathbb{E}\left[\left(\Delta_{\\_}{n}\sum_{\\_}{k=l_{\\_}{n}}^{j-1}(\beta_{\\_}{k,j}^{(1)})^{2}\right)^{p/2}\right]+\mathrm{O}(\Delta_{\\_}{n}),$

uniformly on $j$, wherein the second inequality we additionally used (44).
Observe that

$\sum_{\\_}{k=l_{\\_}{n}}^{j-1}(\beta_{\\_}{k,j}^{(1)})^{2}=\sum_{\\_}{i=0}^{j-1-l_{\\_}{n}}(\alpha_{\\_}{i,j}^{2}+\eta_{\\_}{i,j})(j-i-
l_{\\_}{n}),$

where $\alpha_{\\_}{i,j}$ are as in (30) and
$\eta_{\\_}{i,j}:=\alpha_{\\_}{i,j}\sum_{\\_}{i^{\prime}=0}^{i-1}\alpha_{\\_}{i^{\prime},j}\mathbf{1}_{\\_}{i\geq
1}$. If we in addition take $p_{\\_}{0}/2\land 4\geq p>2$, we deduce on the
one hand, by Rosenthal’s inequality and Lemma 1 that

$\displaystyle\mathbb{E}\left[\left(\Delta_{\\_}{n}\sum_{\\_}{i=0}^{j-1-l_{\\_}{n}}\alpha_{\\_}{i,j}^{2}(j-i-
l_{\\_}{n})\right)^{p/2}\right]\leq$ $\displaystyle
C\sum_{\\_}{i=0}^{j-1-l_{\\_}{n}}\mathbb{E}(\mid\alpha_{\\_}{i,j}\mid^{p})(t_{\\_}{j}-t_{\\_}{i}-l_{\\_}{n}\Delta_{\\_}{n})^{p/2}$
$\displaystyle+C\left(\sum_{\\_}{i=0}^{j-1-l_{\\_}{n}}\mathbb{E}(\alpha_{\\_}{i,j}^{2})(t_{\\_}{j}-t_{\\_}{i}-l_{\\_}{n}\Delta_{\\_}{n})\right)^{p/2}$
$\displaystyle=C\sum_{\\_}{i=0}^{j-1-l_{\\_}{n}}\mathbb{E}(\mid\alpha_{\\_}{i,j}\mid^{p})(t_{\\_}{j}-t_{\\_}{i}-l_{\\_}{n}\Delta_{\\_}{n})^{p/2}+\mathrm{O}(\Delta_{\\_}{n}),$

while on the other hand, by the von Bahr-Esseen inequality (see [18, 9.3.b]),
it holds

$\mathbb{E}\left[\left|\Delta_{\\_}{n}\sum_{\\_}{i=1}^{j-1-l_{\\_}{n}}\eta_{\\_}{i,j}(j-i-
l_{\\_}{n})\right|^{p/2}\right]\leq
C\sum_{\\_}{i=1}^{j-1-l_{\\_}{n}}\mathbb{E}(\mid\eta_{\\_}{i,j}\mid^{p/2})(t_{\\_}{j}-t_{\\_}{i}-l_{\\_}{n}\Delta_{\\_}{n})^{p/2}.$
(47)

Using (44) and the Mean-Value Theorem, we infer that

$\displaystyle\sum_{\\_}{i=0}^{j-1-l_{\\_}{n}}\mathbb{E}(\mid\alpha_{\\_}{i,j}\mid^{p})(t_{\\_}{j}-t_{\\_}{i}-l_{\\_}{n}\Delta_{\\_}{n})^{p/2}\leq$
$\displaystyle
C\sum_{\\_}{i=0}^{j-1-l_{\\_}{n}}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,j))(t_{\\_}{j}-t_{\\_}{i}-l_{\\_}{n}\Delta_{\\_}{n})^{p/2}$
(48) $\displaystyle\leq$
$\displaystyle\Delta_{\\_}{n}\int_{\\_}{l_{\\_}{n}\Delta_{\\_}{n}}^{t_{\\_}{j}}a(s)s^{p/2-1}ds$
$\displaystyle-\int_{\\_}{t_{\\_}{j}}^{t_{\\_}{j+1}}a(s)(\left\lceil
s/\Delta_{\\_}{n}\right\rceil\Delta_{\\_}{n}-(l_{\\_}{n}+2)\Delta_{\\_}{n})^{p/2}ds$
$\displaystyle+\mathrm{O}(\Delta_{\\_}{n})+\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(0,j))(t_{\\_}{j}-l_{\\_}{n}\Delta_{\\_}{n})^{p/2}$
$\displaystyle\leq$
$\displaystyle\Delta_{\\_}{n}\int_{\\_}{l_{\\_}{n}\Delta_{\\_}{n}}^{t_{\\_}{j}}a(s)s^{p/2-1}ds+\Delta_{\\_}{n}\int_{\\_}{t_{\\_}{j}}^{t_{\\_}{j+1}}a(s)s^{p/2-1}ds$
$\displaystyle+\mathrm{O}(\Delta_{\\_}{n})$ $\displaystyle=$
$\displaystyle\mathrm{O}(\Delta_{\\_}{n})+\mathrm{O}(\Delta_{\\_}{n}(\Delta_{\\_}{n}n)^{p/2-\alpha}),$

where in the last step we used Assumption 3. In order to find a final estimate
for $\mathbb{E}(\mid\xi_{\\_}{j,n}^{(3)}\mid^{p})$, we need to distinguish
whether $p_{\\_}{0}\geq 8$ or $p_{\\_}{0}<8$. Suppose first that
$p_{\\_}{0}\geq 8$. Taking $p=4$ and (44), we deduce that the right-hand side
of (47) is bounded up to a constant by

$\Delta_{\\_}{n}(\Delta_{\\_}{n}n)^{p/2}\sum_{\\_}{i=0}^{j}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,j)).$

Contrastingly, if $p_{\\_}{0}<8$, (44) gives that in this situation

$\displaystyle\sum_{\\_}{i=1}^{j-1-l_{\\_}{n}}\mathbb{E}(\mid\eta_{\\_}{i,j}\mid^{p/2})(t_{\\_}{j}-t_{\\_}{i}-l_{\\_}{n}\Delta_{\\_}{n})^{p/2}\leq$
$\displaystyle\Delta_{\\_}{n}^{\frac{3p}{4}-1}\int_{\\_}{0}^{n\Delta_{\\_}{n}}\left[\psi_{\\_}{n}^{\prime}(\left\lceil
s/\Delta_{\\_}{n}\right\rceil\Delta_{\\_}{n})a(s)\right]^{\frac{p}{4}}s^{p/2}ds$
$\displaystyle=$
$\displaystyle\Delta_{\\_}{n}^{\frac{3p}{4}-1}(\mathrm{O}(1)+\mathrm{O}((\Delta_{\\_}{n}n)^{1-\frac{p}{4}(2\alpha+1)+p/2})),$

in which we utilize once again Assumption 3. Applying the previous estimates
to (46), we conclude that (43) holds for $M=3$ if $p\geq 8/3$. By replacing,
$\mathbb{E}(\chi_{\\_}{k}^{2})$ by
$\mathbb{E}(L(\cup_{\\_}{m=k+1}^{j}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,m)^{2})$
in the preceding argument allow us to conclude that (43) holds for $M=5$.
Lastly, we examine $\mathbb{E}(\mid\xi_{\\_}{j,n}^{(4)}\mid^{p})$. We start by
writing

$\xi_{\\_}{j,n}^{(4)}=\sum_{\\_}{i=0}^{j-l_{\\_}{n}-1}\sum_{\\_}{m=i}^{j-1}\alpha_{\\_}{i,m}L(\cup_{\\_}{k=i+l_{\\_}{n}}^{(j-1)\land(m+l_{\\_}{n})}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,j)).$

Analogous arguments used above, allow us to conclude that for
$p_{\\_}{0}/2\land 4\geq p>2$

$\displaystyle\mathbb{E}(\mid\xi_{\\_}{j,n}^{(4)}\mid^{p})$ $\displaystyle\leq
C\sum_{\\_}{k=l_{\\_}{n}+1}^{j-1}\mathbb{E}(\mid\alpha_{\\_}{k+1,j}\mid^{p/2})\mathbb{E}(\mid\sum_{\\_}{k^{\prime}=l_{\\_}{n}}^{k-1}\alpha_{\\_}{k^{\prime}+1,j}\mathrm{Leb}(\cup_{\\_}{i=0}^{k^{\prime}-l_{\\_}{n}}\cup_{\\_}{m=k-l_{\\_}{n}}^{j-1}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,m))\mid^{p/2})$
$\displaystyle+C\sum_{\\_}{i=0}^{j-l_{\\_}{n}-1}\sum_{\\_}{k=i+l_{\\_}{n}+1}^{j-1}\mathbb{E}(\mid\alpha_{\\_}{k+1,j}\mid^{p/2})\mathbb{E}(\mid
L(\cup_{\\_}{k^{\prime}=i+l_{\\_}{n}}^{k-1}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k^{\prime}+1,j))\mid^{p/2})$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\times\mathrm{Leb}(\cup_{\\_}{m=k-l_{\\_}{n}}^{j-1}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,m))^{p/2}+\mathrm{O}(\Delta_{\\_}{n})$
$\displaystyle:=S_{\\_}{1}+S_{\\_}{2}+\mathrm{O}(\Delta_{\\_}{n})$

We once again distinguish between the cases in which $p_{\\_}{0}\geq 8$ and
$p_{\\_}{0}<8$. As before, in the former case by choosing $p=4$ we obtain that
$\mathbb{E}(\mid\xi_{\\_}{j,n}^{(4)}\mid^{p})=\mathrm{O}(\Delta_{\\_}{n}).$ If
instead $p_{\\_}{0}<8$, (44) implies that

$S_{\\_}{1}\leq C\Delta_{\\_}{n}^{\frac{3p}{4}-1}ds,\,\,S_{\\_}{2}\leq
C\Delta_{\\_}{n}^{p/2+\frac{3p}{4}-2}ds.$

Therefore, if $p\geq 8/3$, we conclude that (43) is also valid for $M=4$.

∎

####  B.3 Proof of the infeasible central limit theorem for the Gaussian case
when $t=0$

Now we present the proof of Theorem 3, which follows the same lines as the
proof of Theorem 2. We first observe that it is enough to show that
$Q_{\\_}{n}\overset{\mathbb{P}}{\rightarrow}c_{\\_}{4}(L^{\prime})a(0)$ and
that in the Gaussian case we further have that

$\sqrt{n}\left(\hat{a}(0)-a(0)\right)\overset{d}{\rightarrow}N(0,2a(0)^{2})-\frac{1}{2}\sqrt{\mu_{\\_}{0}}\phi(0);\,\,\,\text{and
}\frac{1}{\Delta_{\\_}{n}}Q_{\\_}{n}\overset{\mathbb{P}}{\rightarrow}6a(0)^{2}.$
(49)

###### Proof.

Let us first check that in general,
$Q_{\\_}{n}\overset{\mathbb{P}}{\rightarrow}c_{\\_}{4}(L^{\prime})a(0)$.
Indeed, using (29), Lemma 1 and putting $\varrho_{\\_}{k}:=L(\cup_{\\_}{j\geq
k+1}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,j))$, we get that

$\displaystyle
Q_{\\_}{n}-\frac{\int_{\\_}{0}^{\Delta_{\\_}{n}}a(s)ds}{\Delta_{\\_}{n}}\left(c_{\\_}{4}(L^{\prime})+3\int_{\\_}{0}^{\Delta_{\\_}{n}}a(s)ds\right)=$
$\displaystyle\frac{1}{2n\Delta_{\\_}{n}}\sum_{\\_}{k=0}^{n-2}[\chi_{\\_}{k}^{4}-\mathbb{E}(\chi_{\\_}{k}^{4})]$
(50) $\displaystyle+$
$\displaystyle\frac{1}{2n\Delta_{\\_}{n}}\sum_{\\_}{k=0}^{n-2}[\varrho_{\\_}{k}^{4}-\mathbb{E}(\varrho_{\\_}{k}^{4})]$
$\displaystyle+\frac{2}{n\Delta_{\\_}{n}}\sum_{\\_}{k=0}^{n-2}\chi_{\\_}{k}^{3}\varrho_{\\_}{k}+\frac{2}{n\Delta_{\\_}{n}}\sum_{\\_}{k=0}^{n-2}\chi_{\\_}{k}[\varrho_{\\_}{k}^{3}-\mathbb{E}(\varrho_{\\_}{k}^{3})]$
$\displaystyle+\frac{2\int
x^{3}\nu(dx)\int_{\\_}{0}^{\Delta_{\\_}{n}}a(s)}{n\Delta_{\\_}{n}}\sum_{\\_}{k=0}^{n-2}\chi_{\\_}{k}$
$\displaystyle+\frac{3}{n\Delta_{\\_}{n}}\sum_{\\_}{k=0}^{n-2}\chi_{\\_}{k}^{2}\varrho_{\\_}{k}^{2}$
$\displaystyle=:$ $\displaystyle
S_{\\_}{1}+S_{\\_}{2}+S_{\\_}{3}+S_{\\_}{4}+S_{\\_}{5}+S_{\\_}{6},$

plus an error of order $\mathrm{O}(1/n)$. It is clear, due to Lemma 1, that
$S_{\\_}{6}\overset{\mathbb{P}}{\rightarrow}0$. Furthermore, for every
$q=1,\ldots,5$, the summands in $S_{\\_}{q}$ are martingale differences. An
application of the von Bahr-Esseen inequality (see [18, 9.3.b]) and (44) yield
to the asymptotic negligibility of $S_{\\_}{q}$.

Now assume that $L$ is Gaussian. We start by showing that in this case
$\frac{1}{\Delta_{\\_}{n}}Q_{\\_}{n}\overset{\mathbb{P}}{\rightarrow}6a(0)^{2}$.
Relation (50) still holds with $c_{\\_}{4}(L^{\prime})=0$. Thus, in view that
$(\int_{\\_}{0}^{\Delta_{\\_}{n}}a(s)ds)^{-1/2}\chi_{\\_}{k}$ are i.i.d.
centered Gaussian r.v.s., it follows trivially from the law of large numbers
that $\frac{1}{\Delta_{\\_}{n}}S_{\\_}{1}\overset{\mathbb{P}}{\rightarrow}0.$
Using the exact argument as well as the fact that the summands in $S_{\\_}{q}$
are martingale differences allow us to conclude that
$\frac{1}{\Delta_{\\_}{n}}S_{\\_}{q}\overset{\mathbb{P}}{\rightarrow}0$ for
$q=1,\ldots,5,$ also holds in this situation. We finally show that
$\frac{1}{\Delta_{\\_}{n}}S_{\\_}{6}\overset{\mathbb{P}}{\rightarrow}3a(0)^{2}.$
To do this, we note that by the Gaussianity of $L$ we have that

$\frac{1}{\Delta_{\\_}{n}}S_{\\_}{6}=3a(0)^{2}+\mathrm{o}_{\\_}{\mathbb{P}}(1)+\frac{3}{n\Delta_{\\_}{n}^{2}}\sum_{\\_}{k=0}^{n-2}\chi_{\\_}{k}^{2}[\varrho_{\\_}{k}^{2}-\mathbb{E}(\varrho_{\\_}{k}^{2})].$
(51)

In view that the array
$\chi_{\\_}{k}^{2}[\varrho_{\\_}{k}^{2}-\mathbb{E}(\varrho_{\\_}{k}^{2})]$ is
martingale difference and
$\mathbb{E}\left[(\chi_{\\_}{k}^{2}[\varrho_{\\_}{k}^{2}-\mathbb{E}(\varrho_{\\_}{k}^{2})])^{2}\right]=\mathrm{O}(\Delta_{\\_}{n}^{4})$,
uniformly on $k$, the last summand in (51) is also
$\mathrm{o}_{\\_}{\mathbb{P}}(1)$, which gives the desired convergence.
Therefore, it is left to show that

$\sqrt{n}\left(\hat{a}(0)-a(0)\right)\overset{d}{\rightarrow}N(0,2a(0)^{2})-\frac{1}{2}\sqrt{\mu_{\\_}{0}}\phi(0).$

To verify this, as in the proof of Theorem 3, we decompose

$\sqrt{n}\left(\hat{a}(0)-a(0)\right)=\mathcal{\tilde{E}}_{\\_}{0}-\frac{1}{2}\sqrt{\mu_{\\_}{0}}\phi(0)+\mathrm{o}(1),$
(52)

where

$\mathcal{\tilde{E}}_{\\_}{0}=\frac{1}{2\Delta_{\\_}{n}\sqrt{n}}\sum_{\\_}{j=1}^{n-1}\tilde{\zeta}_{\\_}{j,n},$

where
$\tilde{\zeta}_{\\_}{j,n}=\sum_{\\_}{q=1}^{4}\acute{\xi}_{\\_}{j,n}^{(q)}$

$\displaystyle\acute{\xi}_{\\_}{j,n}^{(1)}$
$\displaystyle=\chi_{\\_}{j}^{2}-\mathbb{E}(\chi_{\\_}{j}^{2});\,\,\,\acute{\xi}_{\\_}{j,n}^{(2)}=\sum_{\\_}{k=0}^{j-1}[\alpha_{\\_}{k+1,j}^{2}-\mathbb{E}(\alpha_{\\_}{k+1,j}^{2})];$
$\displaystyle\acute{\xi}_{\\_}{j,n}^{(3)}$
$\displaystyle=2\sum_{\\_}{k=0}^{j-1}\sum_{\\_}{j^{\prime}=k+1}^{j-1}\alpha_{\\_}{k+1,j}\alpha_{\\_}{k+1,j^{\prime}}\mathbf{1}_{\\_}{j\geq
k+2};\,\,\,\acute{\xi}_{\\_}{j,n}^{(4)}=-2\sum_{\\_}{k=0}^{j-1}\chi_{\\_}{k}\alpha_{\\_}{k+1,j}.$

Furthermore, as $n\rightarrow\infty$,
$\frac{1}{\Delta_{\\_}{n}^{8}n^{4}}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\mathbb{E}(\mid\acute{\zeta}_{\\_}{j,n}\mid^{8})\rightarrow
0$ and

$\frac{1}{4\Delta_{\\_}{n}^{2}n}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\mathbb{E}(\acute{\zeta}_{\\_}{j,n}^{2}\mid\mathscr{G}_{\\_}{j-1}^{n})=\frac{1}{4\Delta_{\\_}{n}^{2}n}\sum_{\\_}{j=1}^{n-1}\mathbb{E}(\acute{\zeta}_{\\_}{j,n}^{2})+\mathrm{o}_{\\_}{\mathbb{P}}(1).$

Since $L$ is Gaussian and centred, we have by Lemma 1

$\displaystyle\mathbb{E}[(\xi_{\\_}{j,n}^{(1)})^{2}]$
$\displaystyle=2(\Delta_{\\_}{n}\psi_{\\_}{n}(0))^{2};\,\,\mathbb{E}[(\xi_{\\_}{j,n}^{(2)})^{2}]=2\sum_{\\_}{k=0}^{j-1}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,j))^{2};$
$\displaystyle\mathbb{E}[(\xi_{\\_}{j,n}^{(3)})^{2}]$
$\displaystyle=4\sum_{\\_}{k=0}^{j-1}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,j))\sum_{\\_}{j^{\prime}=k+1}^{j-1}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,j^{\prime}))\mathbf{1}_{\\_}{j\geq
k+2};$ $\displaystyle\mathbb{E}[(\xi_{\\_}{j,n}^{(4)})^{2}]$
$\displaystyle=4\Delta_{\\_}{n}\psi_{\\_}{n}(0)\sum_{\\_}{k=0}^{j-1}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,j));$
$\displaystyle\mathbb{E}[\xi_{\\_}{j,n}^{(2)}\xi_{\\_}{j,n}^{(3)}]$
$\displaystyle=\mathbb{E}[\xi_{\\_}{j,n}^{(1)}\xi_{\\_}{j,n}^{(3)}]=\mathbb{E}[\xi_{\\_}{j,n}^{(1)}\xi_{\\_}{j,n}^{(4)}]=\mathbb{E}[\xi_{\\_}{j,n}^{(2)}\xi_{\\_}{j,n}^{(4)}]=\mathbb{E}[\xi_{\\_}{j,n}^{(3)}\xi_{\\_}{j,n}^{(4)}]=0.$

as well as, uniformly on $j$,
$\mathbb{E}[\xi_{\\_}{j,n}^{(1)}\xi_{\\_}{j,n}^{(2)}]=2\sum_{\\_}{k=0}^{j-1}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,j))^{2}=\mathrm{O}(\Delta_{\\_}{n}^{3})$,
due to Assumption 3. The first part in (49) now follows by summing over $j$
and rearranging terms.∎

####  B.4 Proof of the consistency of the ABI, AVAR and the trawl set and
slice estimators

We now tackle the proofs of Propositions 1, 2 and 3. During this part, we keep
the notation introduced in the previous sections. We will also use that, under
Assumptions 2, 3, and 4, the estimate

$\mathbb{E}[(\tilde{a}(l\Delta_{\\_}{n})-\mathbb{E}(\tilde{a}(l\Delta_{\\_}{n})))^{2}]\leq
C/n\Delta_{\\_}{n},$ (53)

is valid uniformly on $l$, which is obtained by combining (31), the arguments
in (33), and (39).

###### Proof of Proposition 1.

By means of (29), we decompose

$\displaystyle\hat{a}^{\prime}(t)-\mathbb{E}(\hat{a}^{\prime}(t))=$
$\displaystyle-\frac{1}{n\Delta_{\\_}{n}^{2}}\sum_{\\_}{k=l_{\\_}{n}+1}^{n-2}\left[\chi_{\\_}{k}\lambda_{\\_}{k}-\mathbb{E}(\chi_{\\_}{k}\lambda_{\\_}{k})\right]$
(54)
$\displaystyle-\frac{1}{n\Delta_{\\_}{n}^{2}}\sum_{\\_}{j=l_{\\_}{n}+2}^{n-1}\sum_{\\_}{k=l_{\\_}{n}+1}^{j-1}\alpha_{\\_}{k-l_{\\_}{n},j}\chi_{\\_}{k}-\frac{1}{n\Delta_{\\_}{n}^{2}}\sum_{\\_}{k=l_{\\_}{n}+1}^{n-2}\gamma_{\\_}{k-l_{\\_}{n}-1}\chi_{\\_}{k}$
$\displaystyle+\frac{1}{n\Delta_{\\_}{n}^{2}}\sum_{\\_}{k=l_{\\_}{n}+1}^{n-2}\delta_{\\_}{k-l_{\\_}{n}-1}X\cdot\varrho_{\\_}{k}+\frac{1}{n\Delta_{\\_}{n}^{2}}\sum_{\\_}{k=l_{\\_}{n}+1}^{n-2}\chi_{\\_}{k-l_{\\_}{n}-1}\chi_{\\_}{k},$

where
$\lambda_{\\_}{k}:=L(\cup_{\\_}{j=k-l_{\\_}{n}}^{k}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k-l_{\\_}{n},j))$.
From Assumption 3, we further have that

$\displaystyle\mathbb{E}(\hat{a}^{\prime}(t))$
$\displaystyle=\frac{1}{\Delta_{\\_}{n}}\int_{\\_}{0}^{1}[a(s\Delta_{\\_}{n}+(l_{\\_}{n}+1)\Delta_{\\_}{n})-a(s\Delta_{\\_}{n}+l_{\\_}{n}\Delta_{\\_}{n})]ds+\mathrm{O}(1/n\Delta_{\\_}{n})$
$\displaystyle=a^{\prime}(t)+\mathrm{o}(1).$

Thus, in order to complete the proof, we only need to check that the sums in
(54) are $\mathrm{O}_{\\_}{\mathbb{P}}(1/\sqrt{n\Delta_{\\_}{n}^{2}})$. This
follows easily from the fact that each summand is a squared integrable
martingale difference whose second moment (due to Lemma 1 and Assumption 3) is
bounded uniformly up to a constant by $\Delta_{\\_}{n}^{2}$.∎

###### Proof of Proposition 2.

We have already verified in the previous subsection that
$Q_{\\_}{n}\overset{\mathbb{P}}{\rightarrow}c_{\\_}{4}(L^{\prime})a(0)$.
Furthermore, since the arguments used to analyse $\hat{v}_{\\_}{2}(t)$ can be
extended to $v_{\\_}{3}(t)$ and $v_{\\_}{4}(t)$ without any difficulty, for
the rest of the proof, we focus on checking that

$\hat{v}_{\\_}{2}(t)\overset{\mathbb{P}}{\rightarrow}2\int_{\\_}{0}^{\infty}a(s)^{2}ds.$
(55)

An application of (20), (21), and (53) result in

$\Delta_{\\_}{n}\sum_{\\_}{l=0}^{N_{\\_}{n}}[\hat{a}^{2}(l\Delta_{\\_}{n})-a^{2}(l\Delta_{\\_}{n})]=\Delta_{\\_}{n}\sum_{\\_}{l=0}^{N_{\\_}{n}}[\hat{a}(l\Delta_{\\_}{n})-a(l\Delta_{\\_}{n})]^{2}+\mathrm{o}_{\\_}{\mathbb{P}}(1),$

where we also used that (thanks to Assumptions 3, 7, and the Generalized
Dominated Convergence Theorem)

$\Delta_{\\_}{n}\sum_{\\_}{l=1}^{N_{\\_}{n}}\left((1-l/n)\frac{1}{\Delta_{\\_}{n}}\int_{\\_}{l\Delta_{\\_}{n}}^{(l+1)\Delta_{\\_}{n}}a(s)ds\right)^{p}=\int_{\\_}{0}^{\infty}a(s)^{p}ds+\mathrm{o}(1),\,\,\,p\geq
1.$

Since $n\Delta_{\\_}{n}^{3}$ is bounded, we may utilize (25) and (26) once
again to conclude that

$\Delta_{\\_}{n}\sum_{\\_}{l=0}^{N_{\\_}{n}}[\hat{a}(l\Delta_{\\_}{n})-a(l\Delta_{\\_}{n})]^{2}\leq
C\Delta_{\\_}{n}\sum_{\\_}{l=0}^{N_{\\_}{n}}[\tilde{a}(l\Delta_{\\_}{n})-\mathbb{E}(\tilde{a}(l\Delta_{\\_}{n}))]^{2}+\mathrm{o}_{\\_}{\mathbb{P}}(1)=\mathrm{o}_{\\_}{\mathbb{P}}(1),$

where the last equation is obtained by means of (53) and Assumption 7. This
concludes the proof of (55).∎

###### Proof of Proposition 3.

The consistency of the sample mean is obvious. The proof of the consistency of
the three estimators for the trawl (sub) sets follows similar arguments as in
the proof of Proposition 2. To see this, note that we work with the
approximations
$\mathrm{Leb}(A)=\int_{\\_}0^{\infty}a(s)ds\approx\int_{\\_}0^{n\Delta_{\\_}n}a(s)ds\approx\sum_{\\_}{l=0}^{n-1}\hat{a}(l\Delta_{\\_}n)\Delta_{\\_}n=:\widehat{\mathrm{Leb}(A)}$,$\mathrm{Leb}(A\cap
A_{\\_}h)=\int_{\\_}h^{\infty}a(s)ds\approx\int_{\\_}h^{n\Delta_{\\_}n}a(s)ds\approx\sum_{\\_}{\begin{subarray}{c}l\in\\{0,\ldots,n-1\\}:\\\
l\Delta_{\\_}n\geq
h\end{subarray}}\hat{a}(l\Delta_{\\_}n)\Delta_{\\_}n=:\widehat{\mathrm{Leb}(A\cap
A_{\\_}h)}$, $\mathrm{Leb}(A\setminus
A_{\\_}h)=\int_{\\_}0^{h}a(s)ds\approx\sum_{\\_}{\begin{subarray}{c}l\in\\{0,\ldots,n-1\\}:\\\
l\Delta_{\\_}n<h\end{subarray}}\hat{a}(l\Delta_{\\_}n)\Delta_{\\_}n=:\widehat{\mathrm{Leb}(A\setminus
A_{\\_}h)}$. Moreover, since we only consider linear functionals of the trawl
function in the integrands, we do not need to use the additional tuning
parameter $N_{\\_}n$.

∎

####  B.5 Approximating the Asymptotic Variance

In this part, we verify that equation (38) holds, i.e.

######  Lemma 3.

Let the assumptions of Theorem 2 in [25] holds. Then, we have that

$\frac{1}{\Delta_{\\_}{n}n}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\mathbb{E}(\zeta_{\\_}{j,n}^{2}\mid\mathscr{G}_{\\_}{j-1}^{n})=\frac{1}{\Delta_{\\_}{n}n}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\mathbb{E}(\zeta_{\\_}{j,n}^{2})+\mathrm{o}_{\\_}{\mathbb{P}}(1),$

where $\zeta_{\\_}{j,n}=\sum_{\\_}{p=1}^{5}\xi_{\\_}{j,n}^{(p)}$ and
$\xi_{\\_}{j,n}^{(p)}$ are defined as in (32).

The proof of the previous lemma consists of expressing the approximation error
as sums of martingale differences. Thus, Lemma 2 will be extensively used.

###### Proof of Lemma 3.

Let us start by choosing $n_{\\_}{0}$ such that
$l_{\\_}{n}=[t/\Delta_{\\_}{n}]>0$, for all $n\geq n_{\\_}{0}$. As we
mentioned above, most of the proof consists of writing the error terms as the
sum of martingale differences and then applying Lemma 2. Denote by
$\delta_{\\_}{k,j}=L(\cup_{\\_}{m=k+1}^{j-1}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,m))$
and note that in this situation

$\mathbb{E}(\zeta_{\\_}{j,n}^{2}\mid\mathscr{G}_{\\_}{j-1}^{n})=\sum_{\\_}{p,q=1}^{5}\mathbb{E}(\xi_{\\_}{j,n}^{(p)}\xi_{\\_}{j,n}^{(q)}\mid\mathscr{G}_{\\_}{j-1}^{n}),$

with a total of $15$ different summands, each of which equals (thanks to Lemma
1) to

$\displaystyle\begin{aligned}
\mathbb{E}((\xi_{\\_}{j,n}^{(1)})^{2}\mid\mathscr{G}_{\\_}{j-1}^{n})=&\mathbb{E}((\xi_{\\_}{j,n}^{(1)})^{2});\\\
\mathbb{E}((\xi_{\\_}{j,n}^{(2)})^{2}\mid\mathscr{G}_{\\_}{j-1}^{n})=&\mathbb{E}((\xi_{\\_}{j,n}^{(2)})^{2})+\mathrm{O}(\Delta_{\\_}{n})\left((\beta_{\\_}{j,j}^{(2)})^{2}-\mathbb{E}[(\beta_{\\_}{k,j}^{(2)})^{2}]\right);\\\
\mathbb{E}((\xi_{\\_}{j,n}^{(3)})^{2}\mid\mathscr{G}_{\\_}{j-1}^{n})=&\sum_{\\_}{k,k^{\prime}=l_{\\_}{n}}^{j-1}\chi_{\\_}{k}\chi_{\\_}{k^{\prime}}\mathrm{Leb}(\cup_{\\_}{i=0}^{k\land
k^{\prime}-l_{\\_}{n}}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,j));\\\
\mathbb{E}((\xi_{\\_}{j,n}^{(4)})^{2}\mid\mathscr{G}_{\\_}{j-1}^{n})=&\mathbb{E}[(\xi_{\\_}{j,n}^{(4)})^{2}]+\sum_{\\_}{k=l_{\\_}{n}}^{j-1}\left((\beta_{\\_}{k,j}^{(2)})^{2}-\mathbb{E}[(\beta_{\\_}{k,j}^{(2)})^{2}]\right)\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,j));\\\
\mathbb{E}((\xi_{\\_}{j,n}^{(5)})^{2}\mid\mathscr{G}_{\\_}{j-1}^{n})=&\sum_{\\_}{k=l_{\\_}{n}}^{j-1}\mathbb{E}[(\beta_{\\_}{k,j}^{(1)})^{2}]\mathbb{E}[\alpha_{\\_}{k+1,j}^{2}]+\sum_{\\_}{k,k^{\prime}=l_{\\_}{n}}^{j-2}\mathbb{E}[(\beta_{\\_}{k\land
k^{\prime},j}^{(1)})^{2}]\delta_{\\_}{k,j}\delta_{\\_}{k^{\prime},j},\end{aligned}$
(56)

and

$\displaystyle\mathbb{E}(\xi_{\\_}{j,n}^{(1)}\xi_{\\_}{j,n}^{(2)}\mid\mathscr{G}_{\\_}{j-1}^{n})=\beta_{\\_}{j,j}^{(2)}\mathbb{E}[(\beta_{\\_}{j,j}^{(1)})^{3}];$
$\displaystyle\,\,\,\mathbb{E}(\xi_{\\_}{j,n}^{(1)}\xi_{\\_}{j,n}^{(3)}\mid\mathscr{G}_{\\_}{j-1}^{n})=\sum_{\\_}{k=l_{\\_}{n}}^{j-1}\chi_{\\_}{k}\mathbb{E}[(\beta_{\\_}{k,j}^{(1)})^{3}];$
(57)
$\displaystyle\mathbb{E}(\xi_{\\_}{j,n}^{(1)}\xi_{\\_}{j,n}^{(4)}\mid\mathscr{G}_{\\_}{j-1}^{n})=$
$\displaystyle-\sum_{\\_}{k=l_{\\_}{n}}^{j-1}\beta_{\\_}{k,j}^{(2)}\mathbb{E}[\alpha_{\\_}{k+1,j}^{3}];$
$\displaystyle\mathbb{E}(\xi_{\\_}{j,n}^{(1)}\xi_{\\_}{j,n}^{(5)}\mid\mathscr{G}_{\\_}{j-1}^{n})=$
$\displaystyle-\sum_{\\_}{k=l_{\\_}{n}}^{j-1}\mathbb{E}[(\beta_{\\_}{k,j}^{(1)})^{2}]\mathbb{E}[\alpha_{\\_}{k+1,j}^{2}](1+\mathbf{1}_{\\_}{k\leq
j-l_{\\_}{n}-1})$
$\displaystyle-\sum_{\\_}{k=l_{\\_}{n}}^{j-2}\delta_{\\_}{k,j}\mathbb{E}[(\beta_{\\_}{k,j}^{(1)})^{3}];$
$\displaystyle\mathbb{E}(\xi_{\\_}{j,n}^{(2)}\xi_{\\_}{j,n}^{(3)}\mid\mathscr{G}_{\\_}{j-1}^{n})=$
$\displaystyle\beta_{\\_}{j,j}^{(2)}\sum_{\\_}{k=l_{\\_}{n}}^{j-1}\chi_{\\_}{k}\mathbb{E}[(\beta_{\\_}{k,j}^{(1)})^{2}];$
$\displaystyle\mathbb{E}(\xi_{\\_}{j,n}^{(2)}\xi_{\\_}{j,n}^{(4)}\mid\mathscr{G}_{\\_}{j-1}^{n})=$
$\displaystyle-\beta_{\\_}{j,j}^{(2)}\sum_{\\_}{k=l_{\\_}{n}}^{j-1}\beta_{\\_}{k,j}^{(2)}\mathbb{E}[\alpha_{\\_}{k+1,j}^{2}];$
$\displaystyle\mathbb{E}(\xi_{\\_}{j,n}^{(2)}\xi_{\\_}{j,n}^{(5)}\mid\mathscr{G}_{\\_}{j-1}^{n})=$
$\displaystyle-\beta_{\\_}{j,j}^{(2)}\sum_{\\_}{k=l_{\\_}{n}}^{j-2}\delta_{\\_}{k,j}\mathbb{E}[(\beta_{\\_}{k,j}^{(1)})^{2}];$
$\displaystyle\mathbb{E}(\xi_{\\_}{j,n}^{(3)}\xi_{\\_}{j,n}^{(4)}\mid\mathscr{G}_{\\_}{j-1}^{n})=$
$\displaystyle-\sum_{\\_}{k,k^{\prime}=l_{\\_}{n}}^{j-1}\chi_{\\_}{k}\beta_{\\_}{k^{\prime},j}^{(2)}\mathbb{E}[\alpha_{\\_}{k^{\prime}+1,j}^{2}]\mathbf{1}_{\\_}{k^{\prime}\leq
k-l_{\\_}{n}-1};$
$\displaystyle\mathbb{E}(\xi_{\\_}{j,n}^{(3)}\xi_{\\_}{j,n}^{(5)}\mid\mathscr{G}_{\\_}{j-1}^{n})=$
$\displaystyle-\sum_{\\_}{k,k^{\prime}=l_{\\_}{n}}^{j-2}\chi_{\\_}{k^{\prime}}\delta_{\\_}{k,j}\mathbb{E}[(\beta_{\\_}{k\land
k^{\prime},j}^{(1)})^{2}];$
$\displaystyle\mathbb{E}(\xi_{\\_}{j,n}^{(4)}\xi_{\\_}{j,n}^{(5)}\mid\mathscr{G}_{\\_}{j-1}^{n})=$
$\displaystyle\sum_{\\_}{k,k^{\prime}=l_{\\_}{n}}^{j-2}\delta_{\\_}{k,j}\beta_{\\_}{k^{\prime},j}^{(2)}\mathbb{E}[\alpha_{\\_}{k^{\prime}+1,j}^{2}]\mathbf{1}_{\\_}{k^{\prime}\leq
k-l_{\\_}{n}-1}.$

We will verify that for all $p=1,\ldots,5$ and for all $q=p,\ldots,5$ we have
that

$\frac{1}{\Delta_{\\_}{n}n}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\mathbb{E}(\xi_{\\_}{j,n}^{(p)}\xi_{\\_}{j,n}^{(q)}\mid\mathscr{G}_{\\_}{j-1}^{n})=\frac{1}{\Delta_{\\_}{n}n}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\mathbb{E}(\xi_{\\_}{j,n}^{(p)}\xi_{\\_}{j,n}^{(q)})+\mathrm{o}_{\\_}{\mathbb{P}}(1).$
(58)

We start analysing the case $p=q$, for $p=2,\ldots,4$ (the case $p=1$ follows
trivially from (56)).

##### Variances

$p=2$:

In view that $\beta_{\\_}{j,j}^{(2)}$ is $l_{\\_}{n}$ dependent and
$\mathbb{E}\left[\left((\beta_{\\_}{j,j}^{(2)})^{2}-\mathbb{E}[(\beta_{\\_}{k,j}^{(2)})^{2}]\right)^{2}\right]=\mathrm{O}(1)$
uniformly on $j$ (due to Lemma 1), it follows from the Cauchy–Schwarz
inequality that

$\frac{1}{n^{2}}\mathrm{Var}\left[\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\left((\beta_{\\_}{j,j}^{(2)})^{2}-\mathbb{E}[(\beta_{\\_}{k,j}^{(2)})^{2}]\right)\right]=\mathrm{O}(1/n)+\mathrm{O}(l_{\\_}{n}/n)\rightarrow
0,$ (59)

which is enough.

$p=3$:

Plainly

$\displaystyle\sum_{\\_}{k,k^{\prime}=l_{\\_}{n}}^{j-1}\chi_{\\_}{k}\chi_{\\_}{k^{\prime}}\mathrm{Leb}(\cup_{\\_}{i=0}^{k\land
k^{\prime}-l_{\\_}{n}}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,j))=$
$\displaystyle\mathbb{E}((\xi_{\\_}{j,n}^{(3)})^{2})+\sum_{\\_}{k=l_{\\_}{n}}^{j-1}[\chi_{\\_}{k}^{2}-\mathbb{E}(\chi_{\\_}{k}^{2})]\mathrm{Leb}(\cup_{\\_}{i=0}^{k-l_{\\_}{n}}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,j))$
$\displaystyle+2\sum_{\\_}{k=l_{\\_}{n}+1}^{j-1}\chi_{\\_}{k}\sum_{\\_}{k^{\prime}=l_{\\_}{n}}^{k-1}\chi_{\\_}{k^{\prime}}\mathrm{Leb}(\cup_{\\_}{i=0}^{k-l_{\\_}{n}}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,j)).$

Consequently,

$\displaystyle\frac{1}{n\Delta_{\\_}{n}}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\mathbb{E}((\xi_{\\_}{j,n}^{(3)})^{2}\mid\mathscr{G}_{\\_}{j-1}^{n})=$
$\displaystyle\frac{1}{n\Delta_{\\_}{n}}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\mathbb{E}((\xi_{\\_}{j,n}^{(3)})^{2})$
$\displaystyle+\frac{1}{n\Delta_{\\_}{n}}\sum_{\\_}{k=l_{\\_}{n}}^{n-2}[\chi_{\\_}{k}^{2}-\mathbb{E}(\chi_{\\_}{k}^{2})]\mathrm{Leb}(\cup_{\\_}{i=0}^{k-l_{\\_}{n}}\cup_{\\_}{j=k+1}^{n-1}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,j))$
$\displaystyle+2\mathrm{Var}(L^{\prime})\frac{1}{n\Delta_{\\_}{n}}\sum_{\\_}{k=l_{\\_}{n}+1}^{n-2}\chi_{\\_}{k}\nu_{\\_}{k}^{(1)},$

where
$\nu_{\\_}{k}^{(1)}:=\sum_{\\_}{k^{\prime}=l_{\\_}{n}}^{k-1}\chi_{\\_}{k^{\prime}}\mathrm{Leb}(\cup_{\\_}{i=0}^{k^{\prime}-l_{\\_}{n}}\cup_{\\_}{j=k+1}^{n-1}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,j))$.
Since $(\chi_{\\_}{k})_{\\_}{k\geq 0}$ are i.i.d.,
$\mathbb{E}[(\chi_{\\_}{k}^{2}-\mathbb{E}(\chi_{\\_}{k}^{2}))^{2}]=\mathrm{O}(\Delta_{\\_}{n})$
uniformly on $k$, and
$\mathrm{Leb}(\cup_{\\_}{i=0}^{k-l_{\\_}{n}}\cup_{\\_}{j=k+1}^{n+l_{\\_}{n}}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,j))\leq\mathrm{Leb}(A)$,
one trivially deduce that

$\frac{1}{n\Delta_{\\_}{n}}\sum_{\\_}{k=l_{\\_}{n}}^{n-2}[\chi_{\\_}{k}^{2}-\mathbb{E}(\chi_{\\_}{k}^{2})]\mathrm{Leb}(\cup_{\\_}{i=0}^{k-l_{\\_}{n}}\cup_{\\_}{j=k+1}^{n-1}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,j))=\mathrm{o}_{\\_}{\mathbb{P}}(1).$

Note now that $(\chi_{\\_}{k}\nu_{\\_}{k}^{(1)})_{\\_}{k\geq 1}$ is an
$(\mathscr{F}_{\\_}{k}^{n})$-martingale difference satisfying that

$\mathbb{E}[(\chi_{\\_}{k}\nu_{\\_}{k}^{(1)})^{2}]\leq
C\Delta_{\\_}{n}\int_{\\_}{0}^{t_{\\_}{k+1}}(\int_{\\_}{s}^{\infty}a(r)dr)^{2}ds.$

Therefore,

$\displaystyle\frac{1}{(n\Delta_{\\_}{n})^{2}}\sum_{\\_}{k=l_{\\_}{n}+1}^{n-2}\mathbb{E}\left[(\chi_{\\_}{k}\nu_{\\_}{k}^{(1)})^{2}\right]$
$\displaystyle\leq
C\frac{1}{(n\Delta_{\\_}{n})^{2}}\int_{\\_}{0}^{n\Delta_{\\_}{n}}\int_{\\_}{0}^{x}(\int_{\\_}{s}^{\infty}a(r)dr)^{2}dsdx$
$\displaystyle=C\int_{\\_}{0}^{1}\int_{\\_}{0}^{y}(\int_{\\_}{n\Delta_{\\_}{n}}^{\infty}a(r)dr)^{2}dudy\rightarrow
0,$

where we have done the change of variables $y=x/n\Delta_{\\_}{n}$ and
$s=u/n\Delta_{\\_}{n}$ as well as applied the Dominated Convergence Theorem. A
simple application of Lemma 2 gives that (58) is valid in this case.

$p=4$:

As for $p=3,$ by decomposing

$\displaystyle\sum_{\\_}{k=l_{\\_}{n}}^{j-1}\left((\beta_{\\_}{k,j}^{(2)})^{2}-\mathbb{E}[(\beta_{\\_}{k,j}^{(2)})^{2}]\right)\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,j))$
$\displaystyle=\sum_{\\_}{i=0}^{j-l_{\\_}{n}-1}\sum_{\\_}{m=i}^{j-1}[\alpha_{\\_}{i,m}^{2}-\mathbb{E}(\alpha_{\\_}{i,m}^{2})]\sum_{\\_}{k=i+l_{\\_}{n}}^{(j-1)\land(m+l_{\\_}{n})}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,j))$
$\displaystyle+2\sum_{\\_}{i=0}^{j-l_{\\_}{n}-1}\sum_{\\_}{m=i+1}^{j-1}\sum_{\\_}{m^{\prime}=i}^{m-1}\alpha_{\\_}{i,m}\alpha_{\\_}{i,m^{\prime}}\sum_{\\_}{k=i+l_{\\_}{n}}^{(j-1)\land(m^{\prime}+l_{\\_}{n})}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,j))$
$\displaystyle+2\sum_{\\_}{i=1}^{j-l_{\\_}{n}-1}\sum_{\\_}{i^{\prime}=0}^{i-1}\sum_{\\_}{m,m^{\prime}=i}^{j-1}\alpha_{\\_}{i,m}\alpha_{\\_}{i^{\prime},m^{\prime}}\sum_{\\_}{k=i+l_{\\_}{n}}^{(j-1)\land(m^{\prime}+l_{\\_}{n})\land(m+l_{\\_}{n})}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k+1,j))$

we obtain that

$\displaystyle\frac{1}{n\Delta_{\\_}{n}}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\mathbb{E}((\xi_{\\_}{j,n}^{(4)})^{2}\mid\mathscr{G}_{\\_}{j-1}^{n})=$
$\displaystyle\frac{1}{\Delta_{\\_}{n}n}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\mathbb{E}((\xi_{\\_}{j,n}^{(4)})^{2})$
(60)
$\displaystyle+\frac{1}{n\Delta_{\\_}{n}}\sum_{\\_}{i=0}^{n-1-l_{\\_}{n}}\sum_{\\_}{m=i}^{n-2}[\alpha_{\\_}{i,m}^{2}-\mathbb{E}(\alpha_{\\_}{i,m}^{2})]\theta_{\\_}{i,m,m^{\prime}}^{(1)}$
$\displaystyle+\frac{2}{n\Delta_{\\_}{n}}\sum_{\\_}{i=0}^{n-1-l_{\\_}{n}}\sum_{\\_}{m=i+1}^{n-2}\sum_{\\_}{m^{\prime}=i}^{m-1}\alpha_{\\_}{i,m}\alpha_{\\_}{i,m^{\prime}}\theta_{\\_}{i,m,m^{\prime}}^{(2)}$
$\displaystyle+\frac{2}{n\Delta_{\\_}{n}}\sum_{\\_}{i=0}^{n-1-l_{\\_}{n}}\sum_{\\_}{i^{\prime}=0}^{i-1}\sum_{\\_}{m,m^{\prime}=i}^{n-2}\alpha_{\\_}{i,m}\alpha_{\\_}{i^{\prime},m^{\prime}}\theta_{\\_}{i,m,m^{\prime}}^{(3)}$

where

$\displaystyle\theta_{\\_}{i,m,m^{\prime}}^{(1)}$
$\displaystyle=\sum_{\\_}{j=(i+l_{\\_}{n}+1)\lor(m+1)}^{n-1}\sum_{\\_}{k=i+l_{\\_}{n}+1}^{j\land(m+l_{\\_}{n}+1)}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k,j));$
(61) $\displaystyle\theta_{\\_}{i,m,m^{\prime}}^{(2)}$
$\displaystyle=\sum_{\\_}{j=(i+l_{\\_}{n}+1)\lor(m+1)}^{n-1}\sum_{\\_}{k=i+l_{\\_}{n}+1}^{j\land(m^{\prime}+l_{\\_}{n}+1)}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k,j));$
$\displaystyle\theta_{\\_}{i,m,m^{\prime}}^{(3)}$
$\displaystyle=\sum_{\\_}{j=(i+l_{\\_}{n}+1)\lor(m+1)\lor(m^{\prime}+1)}^{n-1}\sum_{\\_}{k=i+l_{\\_}{n}+1}^{j\land(m^{\prime}+l_{\\_}{n}+1)\land(m+l_{\\_}{n}+1)}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k,j)).$

Observe that all the error terms in (60) are sums of
$(\mathscr{F}_{\\_}{i}^{n})$-martingale differences and for $\ell=1,2,3$

$\displaystyle\left|\theta_{\\_}{i,m,m^{\prime}}^{(\ell)}\right|$
$\displaystyle\leq\sum_{\\_}{j=(i+l_{\\_}{n}+1)\lor(m+1)}^{m+l_{\\_}{n}}\sum_{\\_}{k=i+l_{\\_}{n}+1}^{j}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k,j))+\sum_{\\_}{j=m+l_{\\_}{n}+1}^{n}\sum_{\\_}{k=i+l_{\\_}{n}+1}^{m+l_{\\_}{n}+1}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(k,j))$
$\displaystyle\leq C\Delta_{\\_}{n}l_{\\_}{n}+\mathrm{Leb}(A)=\mathrm{O}(1).$

An application of the previous bound, Lemma 1 in [25], and the independent
scattered property of $L$ gives us that

$\displaystyle\sum_{\\_}{i=0}^{n-1-l_{\\_}{n}}\mathbb{E}\left[\left(\sum_{\\_}{m=i}^{n-2}[\alpha_{\\_}{i,m}^{2}-\mathbb{E}(\alpha_{\\_}{i,m}^{2})]\theta_{\\_}{i,m,m^{\prime}}^{(1)}\right)^{2}\right]$
$\displaystyle\leq
C\sum_{\\_}{i=0}^{n}\sum_{\\_}{m=i}^{n}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,m))\leq
C\Delta_{\\_}{n}n,$
$\displaystyle\sum_{\\_}{i=0}^{n-1-l_{\\_}{n}}\mathbb{E}\left[\left(\sum_{\\_}{m=i+1}^{n-2}\alpha_{\\_}{i,m}\sum_{\\_}{m^{\prime}=i}^{m-1}\alpha_{\\_}{i,m^{\prime}}\theta_{\\_}{i,m,m^{\prime}}^{(2)}\right)^{2}\right]$
$\displaystyle\leq
C\sum_{\\_}{i=0}^{n}\left(\sum_{\\_}{m=i}^{n}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,m))\right)^{2}\leq
C\Delta_{\\_}{n}n,$

and that

$\displaystyle\sum_{\\_}{i=0}^{n-1-l_{\\_}{n}}\mathbb{E}\left(\sum_{\\_}{i^{\prime}=0}^{i-1}\sum_{\\_}{m,m^{\prime}=i}^{n-2}\alpha_{\\_}{i,m}\alpha_{\\_}{i^{\prime},m^{\prime}}\right)^{2}=$
$\displaystyle\sum_{\\_}{i=0}^{n-1-l_{\\_}{n}}\sum_{\\_}{i^{\prime}=0}^{i-1}\sum_{\\_}{m,m^{\prime}=i}^{n-2}\mathbb{E}[\alpha_{\\_}{i,m}^{2}]\mathbb{E}[\alpha_{\\_}{i^{\prime},m^{\prime}}](\theta_{\\_}{i,m,m^{\prime}}^{(3)})^{2}$
(62) $\displaystyle\leq
C\sum_{\\_}{i=0}^{n}\sum_{\\_}{m=i}^{n}\sum_{\\_}{i^{\prime}=0}^{i}\sum_{\\_}{m^{\prime}=i}^{n}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,m))$
$\displaystyle\times\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i^{\prime},m^{\prime}))$
$\displaystyle\leq
C\sum_{\\_}{i=0}^{n}\sum_{\\_}{m=i}^{n}\mathrm{Leb}(\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,m))$
$\displaystyle\leq C\Delta_{\\_}{n}n.$

(58) is obtained as a simple application of the previous previous estimates
along with Lemma 1 in [25].

$p=5$:

Since in this situation

$\displaystyle\frac{1}{n\Delta_{\\_}{n}}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\mathbb{E}((\xi_{\\_}{j,n}^{(5)})^{2}\mid\mathscr{G}_{\\_}{j-1}^{n})=$
$\displaystyle\frac{1}{n\Delta_{\\_}{n}}\sum_{\\_}{j=l_{\\_}{n}+1}^{n-1}\mathbb{E}((\xi_{\\_}{j,n}^{(5)})^{2})$
$\displaystyle+\frac{1}{n\Delta_{\\_}{n}}\sum_{\\_}{k=l_{\\_}{n}+1}^{n-2}\sum_{\\_}{m=k}^{n-2}\left(\alpha_{\\_}{k,m}^{2}-\mathbb{E}[\alpha_{\\_}{k,m}^{2}]\right)\theta_{\\_}{k,m}^{(4)}$
$\displaystyle+\frac{2}{n\Delta_{\\_}{n}}\sum_{\\_}{k=l_{\\_}{n}+1}^{n-3}\sum_{\\_}{m=k+1}^{n-2}\sum_{\\_}{m^{\prime}=k}^{m-1}\alpha_{\\_}{k,m}\alpha_{\\_}{k,m^{\prime}}\theta_{\\_}{k,m}^{(4)}$
$\displaystyle+\frac{2}{n\Delta_{\\_}{n}}\sum_{\\_}{k=l_{\\_}{n}+2}^{n-2}\sum_{\\_}{m=k}^{n-2}\sum_{\\_}{k^{\prime}=l_{\\_}{n}+1}^{k-1}\sum_{\\_}{m^{\prime}=k}^{n-2}\alpha_{\\_}{k,m}\alpha_{\\_}{k^{\prime},m^{\prime}}\theta_{\\_}{k,m,m^{\prime}}^{(5)},$

with the error term being once again sums of
$(\mathscr{F}_{\\_}{k}^{n})$-martingale differences, and where

$\theta_{\\_}{k,m}^{(4)}=\mathrm{Leb}(\cup_{\\_}{i=0}^{k-1-l_{\\_}{n}}\cup_{\\_}{j=m+1}^{n-1}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,j));\,\,\theta_{\\_}{k,m,m^{\prime}}^{(5)}=\mathrm{Leb}(\cup_{\\_}{i=0}^{k-1-l_{\\_}{n}}\cup_{\\_}{j=m\lor
m^{\prime}+1}^{n-1}\mathcal{\mathcal{P}}_{\\_}{A}^{n}(i,j)),$ (63)