# Central limit theorems for martingales-II: convergence in the weak dual
topology

Bruno Rémillard  GERAD and Department of Decision Sciences, HEC Montréal
3000, chemin de la Côte-Sainte-Catherine, Montréal (Québec), Canada H3T 2A7
<EMAIL_ADDRESS>and  Jean Vaillancourt  Department of Decision
Sciences, HEC Montréal
3000, chemin de la Côte-Sainte-Catherine, Montréal (Québec), Canada H3T 2A7
<EMAIL_ADDRESS>
###### Abstract.

A convergence theorem for martingales with càdlàg trajectories (right
continuous with left limits everywhere) is obtained in the sense of the weak
dual topology on Hilbert space, under conditions that are much weaker than
those required for any of the usual Skorohod topologies. Examples are provided
to show that these conditions are also very easy to check and yield useful
asymptotic results, especially when the limit is a mixture of stochastic
processes with discontinuities.

###### Key words and phrases:

Brownian motion, stochastic processes, weak convergence, martingales,
mixtures.

###### 1991 Mathematics Subject Classification:

Primary 60G44, Secondary 60F17.

Partial funding in support of this work was provided by the Natural Sciences
and Engineering Research Council of Canada and the Fonds québécois de la
recherche sur la nature et les technologies. We would like to thank Bouchra R.
Nasri for suggesting Proposition A.2.

##  1\. Introduction

Let $D=D[0,\infty)$ be the space of real-valued càdlàg trajectories (right
continuous with left limits everywhere). The space is a natural choice for
describing phenomena exhibiting bounded jumps, hence devoid of essential
singularities (and no jump at the origin). In the literature, there are many
examples of sequences of $D$-valued martingales $M_{n}$ such that the finite
dimensional distributions (f.d.d.) converge to those of $W\circ A$, where $W$
is Brownian motion and $A$ is a non-decreasing process starting at zero and
independent of $W$. However, if $A$ is not continuous, the f.d.d. convergence
cannot be extended in general to any of the three familiar topologies on $D$:
the $\mathcal{M}_{1}$-topology, the $\mathcal{J}_{1}$-topology or the
$\mathcal{C}$-topology.

The focus of this paper is on those instances where the sequence of $D$-valued
processes of interest does not converge in any of these topologies. This
motivates the consideration of an alternative topology: the weak dual topology
on the dual of a Hilbert space, which coincides with the pointwise convergence
in the Hilbert space in question, namely that associated with locally square-
integrable trajectories. Since weak convergence of probability measures is at
the core of this paper, and in order to avoid the potential confusion brought
upon by the overuse of the term “weak”, we will from now on refer to this
weaker topology as the $\mathcal{L}^{2}_{w}$-topology (and all related terms
accordingly).

The paper is organized as follows. Section 2 presents the
$\mathcal{L}^{2}_{w}$-topology. The corresponding CLT for sequences of
$D$-valued square integrable martingales is in Section 3, along with the
required preliminaries about martingales. Section 4 provides instances where
even $\mathcal{M}_{1}$-convergence (let alone $\mathcal{J}_{1}$\- or
$\mathcal{C}$-convergence) fails for interesting sequences of $D$-valued
processes, while $\mathcal{L}^{2}_{w}$-convergence does occur. Examples of
applications are worked out in Section 4. More precisely, Section 4.1 deals
with stock price approximation and random sums, and it is also shown in Remark
4.1 that many results about the limiting behaviour of renewal counting
processes in the literature are erroneous, showing the difficulty to work with
$\mathcal{J}_{1}$ or $\mathcal{M}_{1}$ convergence when the limit is not
continuous. Next, in Section 4.2, we study the occupation time for planar
Brownian motion; in particular, we complete the proof of Theorem 3.1 of
Kasahara and Kotani, (1979) and we correct their Theorem 3.2. Ancillary
results as well as most definitions and notations are presented in Appendix A,
while any unexplained terminology can be found in either Ethier and Kurtz,
(1986) or Jacod and Shiryaev, (2003). The main proofs are collected in
Appendix B.

##  2\. Locally square integrable processes and
$\mathcal{L}^{2}_{w}$-convergence

Denote by $\mathcal{L}^{2}_{loc}$ the space of real-valued Lebesgue measurable
functions on $[0,\infty)$ which are locally square integrable with respect to
Lebesgue measure, in the sense that their restriction to $[0,T]$ belongs to
Hilbert space $\mathcal{L}^{2}[0,T]$ for every $T>0$. The (squared)
$\mathcal{L}^{2}[0,T]$-norm is denoted by
$|x|_{T}^{2}:=\int_{0}^{T}x^{2}(t)dt$.

A (deterministic) sequence $x_{n}\in\mathcal{L}^{2}_{loc}$ is said to
$\mathcal{L}^{2}_{w}$-converge to $x\in\mathcal{L}^{2}_{loc}$ if and only if
the sequence of scalar products
$\int_{0}^{T}x_{n}(t)f(t)dt\to\int_{0}^{T}x(t)f(t)dt$ converges as
$n\to\infty$, for each $f\in\mathcal{L}^{2}[0,T]$ and each $T>0$. There is a
metric on $\mathcal{L}^{2}_{loc}$ which is compatible with the
$\mathcal{L}^{2}_{w}$-topology, turning it into a complete separable metric
(hereafter Polish) space — see Appendix A.

######  Remark 2.1.

Note that $D\subset\mathcal{L}^{2}_{loc}$ since càdlàg functions are bounded
on bounded time sets. Furthermore $\mathcal{M}_{1}$-convergence of a
(deterministic) sequence $x_{n}\in D$ to a limit $x\in D$ implies
$\sup_{n}\sup_{t\in[0,T]}|x_{n}(t)-x(t)|<\infty$ for every $T>0$, since
$\\{x_{n}\\}$ is a $\mathcal{M}_{1}$-relatively compact set and hence
uniformly bounded on finite time intervals. By the dominated convergence
theorem, convergence of $x_{n}$ to the same limit $x$ ensues in the
$\mathcal{L}^{2}[0,T]$-norm topology for every $T>0$ and hence so does
$\mathcal{L}^{2}_{w}$-convergence; indeed, a (deterministic) sequence
$x_{n}\in\mathcal{L}^{2}[0,T]$ converges to $x\in\mathcal{L}^{2}[0,T]$ in the
(strong) norm topology, if and only if both $x_{n}$
$\mathcal{L}^{2}_{w}$-converges to $x$ and $|x_{n}|_{T}$ converges to
$|x|_{T}$. Thus $x\mapsto|x|_{T}$ is not $\mathcal{L}^{2}_{w}$-continuous even
though $T\mapsto|x|_{T}$ is continuous.

We next turn to weak convergence of probability measures on the Borel subsets
of $\mathcal{L}^{2}_{loc}$ for the $\mathcal{L}^{2}_{w}$-topology. All
$\mathcal{L}^{2}_{loc}$-valued $\mathbb{F}$-processes are built on the same
filtered probability space $(\Omega,\mathcal{F},\mathbb{F},P)$ and adapted to
a filtration $\mathbb{F}=(\mathcal{F}_{t})_{t\geq 0}$ satisfying the usual
conditions (notably, right continuity of the filtration and completeness).
Since $\mathcal{L}^{2}_{loc}$ is a Polish space, families of probability
measures are relatively compact if and only if they are tight, by Prohorov’s
Theorem. Here is a quick test for $\mathcal{L}^{2}_{w}$-tightness.

######  Lemma 2.1.

A sequence of $\mathcal{L}^{2}_{loc}$-valued $\mathbb{F}$-processes $X_{n}$ is
$\mathcal{L}^{2}_{w}$-tight if and only if the sequence of random variables
$\bigl{\\{}|X_{n}|_{T};n\geq 1\bigr{\\}}$ is tight, for each $T>0$.

###### Proof.

Balls in $\mathcal{L}^{2}[0,T]$ are relatively compact in the
$\mathcal{L}^{2}_{w}$-topology and because it is a Polish space, tightness is
equivalent to relative compactness. ∎

Weak convergence of a sequence of $\mathcal{L}^{2}_{loc}$-valued processes
$X_{n}$ to an $\mathcal{L}^{2}_{loc}$-valued process $X$ is denoted by
$X_{n}\stackrel{{\scriptstyle\mathcal{L}^{2}_{w}}}{{\rightsquigarrow}}X$. For
convenience, write $\mathcal{I}(f)(t)=\int_{0}^{t}f(s)ds$ for any $t\geq 0$.
This convergence is characterized by the following result proven in Appendix
B.1.

######  Theorem 2.2.

$X_{n}\stackrel{{\scriptstyle\mathcal{L}^{2}_{w}}}{{\rightsquigarrow}}X$ holds
if and only if

  1. (i)

$\bigl{\\{}|X_{n}|_{T};n\geq 1\bigr{\\}}$ is tight, for each $T>0$;

  2. (ii)

There is an $\mathcal{L}^{2}_{loc}$-valued process $X$ such that
$\mathcal{I}(X_{n})\stackrel{{\scriptstyle
f.d.d.}}{{\rightsquigarrow}}\mathcal{I}(X)$.

The Cramér-Wold device yields the equivalence of statement Theorem 2.2 $(ii)$
to $\mathcal{I}(X_{n}h)\stackrel{{\scriptstyle
f.d.d.}}{{\rightsquigarrow}}\mathcal{I}(Xh)$, for any
$h\in\mathcal{L}^{2}_{loc}$. Note that
$X_{n}\stackrel{{\scriptstyle\mathcal{L}^{2}_{w}}}{{\rightsquigarrow}}X$ does
not necessarily imply $X_{n}\stackrel{{\scriptstyle
f.d.d.}}{{\rightsquigarrow}}X$; in fact it may even happen that
$X_{n}(t)\stackrel{{\scriptstyle Law}}{{\rightsquigarrow}}X(t)$ does not hold
for any $t\in[0,T]$, even though the integrals $\mathcal{I}(X_{n})(t)$
converge and their limits are almost surely differentiable everywhere.
Moreover, if $X_{n}\in D$, $X_{n}(t)$ is arbitrarily close to
$\dfrac{\mathcal{I}(X_{n})(t+\epsilon)-\mathcal{I}(X_{n})(t)}{\epsilon}$.

######  Remark 2.2.

Extension to processes with $\mathbb{R}^{d}$-valued trajectories, of
$\mathcal{L}^{2}_{loc}$-valued processes, $\mathcal{L}^{2}_{w}$-convergence,
and so on — through writing $|X|_{t}:=\sum_{i=1}^{d}|X_{i}|_{t}$,
$X(s)f(s):=\sum_{i=1}^{d}X_{i}(s)f_{i}(s)$, etc., when
$X=(X_{1},\ldots,X_{d})$ and $f=(f_{1},\ldots,f_{d})$ — leads to immediate
confirmation of Remark 2.1, Lemma 2.1 and Theorem 2.2 for processes with
trajectories in $\mathbb{R}^{d}$ as well. We need this clarification for the
next statement.

######  Lemma 2.3.

If $\mathcal{L}^{2}_{loc}$-valued processes $X_{n}$, $Y_{n}$, $X$ and $Y$ are
such that
$X_{n}\stackrel{{\scriptstyle\mathcal{L}^{2}_{w}}}{{\rightsquigarrow}}X$,
$Y_{n}\stackrel{{\scriptstyle\mathcal{L}^{2}_{w}}}{{\rightsquigarrow}}Y$ and
$\left(\mathcal{I}(X_{n}),\mathcal{I}(Y_{n})\right)\stackrel{{\scriptstyle
f.d.d.}}{{\rightsquigarrow}}\left(\mathcal{I}(X),\mathcal{I}(Y)\right)$, then
$(X_{n},Y_{n})\stackrel{{\scriptstyle\mathcal{L}^{2}_{w}}}{{\rightsquigarrow}}(X,Y)$.

######  Remark 2.3.

Lemma 2.3 also extends at once to processes with $\mathbb{R}^{d}$-valued
trajectories and to $d$-tuples of processes instead of just pairs. We proceed
with $d=1$ for the rest of this section, since several of the statements
involve the $\mathcal{M}_{1}$-topology, which is not compatible with the
linear structure on $D$ and therefore cannot be treated coordinatewise, save
in some special cases — see Whitt, (2002).

######  Remark 2.4.

In the light of Proposition A.4, a sequence of $D$-valued processes $X_{n}$
satisfying both $X_{n}\stackrel{{\scriptstyle f.d.d.}}{{\rightsquigarrow}}X$
and $|X_{n}|\stackrel{{\scriptstyle f.d.d.}}{{\rightsquigarrow}}|X|$ for some
$X\in D$, must also satisfy
$|X_{n}|\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}|X|$ since the
processes $|X_{n}|_{t},|X|_{t}$ are nondecreasing in $t\geq 0$; hence the
sequence of continuous processes $\bigl{\\{}|X_{n}|;n\geq 1\bigr{\\}}$ is
$\mathcal{C}$-tight. By Remark 2.1, $D$-valued processes $X$, $X_{n}$ for
which there holds
$X_{n}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}X$
automatically verify both
$X_{n}\stackrel{{\scriptstyle\mathcal{L}^{2}_{w}}}{{\rightsquigarrow}}X$ and
$|X_{n}|\stackrel{{\scriptstyle f.d.d.}}{{\rightsquigarrow}}|X|$ (as well as
$X_{n}\stackrel{{\scriptstyle f.d.d.}}{{\rightsquigarrow}}X$, by definition of
$\mathcal{M}_{1}$-convergence); thus,
$|X_{n}|\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}|X|$ follows.
As a result, so does
$|X_{n}-X|\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}0$.

Summarizing, one has the following result.

######  Proposition 2.4.

Suppose $X$, $X_{n}$ are $D$-valued processes satisfying
$X_{n}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}X$. Then
$X_{n}\stackrel{{\scriptstyle\mathcal{L}^{2}_{w}}}{{\rightsquigarrow}}X$,
$|X_{n}|\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}|X|$ and
$|X_{n}-X|\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}0$.

One of the main reasons for choosing the $\mathcal{L}^{2}_{w}$-topology is the
following theorem stating sufficient conditions for the
$\mathcal{L}^{2}_{w}$-continuity of the composition mapping for sequences of
$D$-valued processes. The proof of the theorem is relegated to Appendix B.2.
Note that even in the $\mathcal{M}_{1}$-topology, the composition mapping is
not continuous in general, even when $X$ is a Brownian motion; see, e.g.,
Proposition A.2.

######  Theorem 2.5.

Suppose $X$, $Y$, $X_{n}$, $Y_{n}$ are $D$-valued processes such that
$X_{n}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}X$ and
$(X_{n},Y_{n})\stackrel{{\scriptstyle f.d.d.}}{{\rightsquigarrow}}(X,Y)$ hold,
with $Y_{n}$, $Y$ non-negative, non-decreasing such that $Y_{n}(t)\to\infty$
as $t\to\infty$ almost surely and either of the following hold: a)
$Y_{n}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}Y$ with
$Y_{n}^{-1}(0)=0$ for every $n$; or b)
$Y_{n}^{-1}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}Y^{-1}$
with $Y_{n}(0)=0$ for every $n$. Then $X_{n}\circ
Y_{n}\stackrel{{\scriptstyle\mathcal{L}^{2}_{w}}}{{\rightsquigarrow}}X\circ
Y$.

##  3\. $D$-valued martingales and a CLT in the $\mathcal{L}^{2}_{w}$-topology

Suppose that $M_{n}$ is a sequence of $D$-valued square integrable
$\mathbb{F}$-martingales started at $M_{n}(0)=0$. By square integrable
martingale we mean those satisfying $E\\{M^{2}(t)\\}<\infty$ for every $t\geq
0$. Note that because of possible discontinuities of trajectories, the
quadratic variation $[M_{n}]$ is can be distinct from its predictable
compensator $A_{n}:=\langle M_{n}\rangle$. The largest jump is denoted by
$J_{T}(M_{n}):=\sup_{s\in[0,T]}|M_{n}(s)-M_{n}(s-)|$. Some useful assumptions
are formulated next.

######  Hypothesis 3.1.

All of the following hold:

  1. (a)

$A_{n}(t)\to\infty$ as $t\to\infty$ almost surely, for each fixed $n\geq 1$;

  2. (b)

There is a $D$-valued process $A$ such that
(i) $A_{n}\stackrel{{\scriptstyle f.d.d.}}{{\rightsquigarrow}}A$;
(ii) For all $t\geq 0$,
$\displaystyle\lim_{n\to\infty}E\left\\{A_{n}(t)\right\\}=E\left\\{A(t)\right\\}<\infty$;
(iii) $A(t)\to\infty$ as $t\to\infty$ almost surely.

Writing the inverse process for $A_{n}$ as $\tau_{n}(s)=\inf\\{t\geq
0;A_{n}(t)>s\\}$, one defines the rescaled $\mathbb{F}_{\tau_{n}}$-martingale
$W_{n}=M_{n}\circ\tau_{n}$, with compensator $\langle
W_{n}\rangle:=A_{n}\circ\tau_{n}$.

######  Hypothesis 3.2.

There holds $\displaystyle\lim_{n\to\infty}E\langle W_{n}\rangle_{t}=t$ for
all $t\geq 0$.

######  Remark 3.1.

By the right-continuity of $A_{n}$, $A_{n}\circ\tau_{n}(t)\geq t$ holds;
therefore, $\lim_{n\to\infty}E\langle W_{n}\rangle_{t}=t$ is equivalent to
$\langle W_{n}\rangle_{t}\stackrel{{\scriptstyle
L^{1}}}{{\rightsquigarrow}}t$. Hypothesis 3.2 yields $\langle
W_{n}\rangle_{t}\stackrel{{\scriptstyle Law}}{{\rightsquigarrow}}t$ for any
$t\geq 0$; by Proposition A.4, $\langle W_{n}\rangle$ is $\mathcal{C}$-tight.

An inhomogeneous Lévy process $A$ is an $\mathbb{F}$-adapted $D$-valued
process with independent increments (with respect to $\mathbb{F}$), without
fixed times of discontinuity and such that $A_{0}=0\in\mathbb{R}$. We assume
also here that $A$ has deterministic characteristics (in the sense of the
Lévy-Khintchine formula), so all Lévy processes here are also semimartingales
— see Jacod and Shiryaev, (2003, Theorem II.4.15). The inhomogeneity (in time)
means that stationarity of the increments, usually required of Lévy processes,
is lifted. This choice reflects the existence of weak limits exhibiting
independent increments without homogeneity in time or space, in some
applications. All processes in the present section are built on $D$, ensuring
continuity in probability of the trajectories, another usual requirement.

######  Hypothesis 3.3.

There is a $D$-valued process $A$ started at $A(0)=0$, such that

  1. (a)

$\tau(s)=\inf\\{t\geq 0;A(t)>s\\}$ is an inhomogeneous Lévy process;

  2. (b)

Either of the following conditions hold:
(i) $A_{n}\stackrel{{\scriptstyle f.d.d.}}{{\rightsquigarrow}}A$ and for every
$n$, $\tau_{n}(0)=0$;
(ii) $\tau_{n}\stackrel{{\scriptstyle f.d.d.}}{{\rightsquigarrow}}\tau$ and
for every $n$, $A_{n}(0)=0$.

######  Remark 3.2.

By Proposition A.3 and Remark A.2, Hypothesis 3.3.b(i) implies both
$A_{n}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}A$ and the
$\mathcal{M}_{1}$-continuity of mapping $A_{n}\mapsto\tau_{n}$, so
$\tau_{n}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}\tau$ as
well. The reverse argument holds under Hypothesis 3.3.b(ii); therefore,
Hypothesis 3.3 automatically implies both
$\tau_{n}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}\tau$ and
$A_{n}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}A$. Invoking
Remark A.2 again confirms
$(A_{n},\tau_{n})\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}(A,\tau)$
as well, since the mapping $A\mapsto(A,\tau)$ is also
$\mathcal{M}_{1}$-continuous, under condition (i); and similarly for mapping
$\tau\mapsto(A,\tau)$, under condition (ii).

Our main result is the following CLT, proven in Appendix B.3.

######  Theorem 3.1.

Assume that both Hypotheses 3.1 and 3.2 hold. Assume that either: a)
Hypothesis 3.3 holds; or b) $A$ is an inhomogeneous Lévy process and
$\tau_{n}(0)=0$, for every $n$. Then there holds
$W_{n}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}W$ and
$(M_{n},A_{n},W_{n})\stackrel{{\scriptstyle\mathcal{L}^{2}_{w}}}{{\rightsquigarrow}}(M,A,W)$,
with $M=W\circ A$ and $W$ is a Brownian motion, independent of process $A$.

##  4\. Examples of application

###  4.1. Price approximation and random sums

Following Chávez-Casillas et al., (2019), Swishchuk et al., (2019), Guo et
al., (2020) who studied limit order books, the price structure on markets can
be written as $X_{t}=\sum_{k=1}^{N_{t}}V_{0}(\xi_{k})$, where
$\xi=(\xi_{k})_{k\geq 1}$ is a finite Markov chain independent of a counting
process $N$. Here, $N_{t}$ can be seen as the number of orders executed on a
stock up to time $t$. If $\mu$ is the mean of $V(\xi)$ under the stationary
measure $\pi$, then $X_{t}=\sum_{k=1}^{N_{t}}V(\xi_{k})+\mu N_{t}$, where
$V(x)=V_{0}(x)-\mu$. Next, according to (Durrett,, 1996, Theorem 7.7.2 and
Example 7.7.2), one can find $f$ bounded so that $Tf-f=Lf=-V$ and
$V(\xi_{k})=Y_{k}+Z_{k}-Z_{k+1}$, with $Z_{k}=Tf(\xi_{k-1})$ and
$Y_{k}=f(\xi_{k})-Tf(\xi_{k-1})$ is a $L^{2}$ ergodic martingale difference.
As a result,
$\displaystyle\sum_{k=1}^{n}V(\xi_{k})=\sum_{k=1}^{n}Y_{k}+Z_{1}-Z_{n+1}$.
Then, if $\sigma^{2}=E(Y_{k}^{2})$, it follows that
$\mathcal{W}_{n}(t)=\dfrac{1}{\sigma\sqrt{n}\;}\sum_{k=1}^{\lfloor
nt\rfloor}Y_{k}$ converges in $\mathcal{C}$ to a Brownian motion
$\mathcal{W}$. Next, assume that $A_{n}(t)=N_{a_{n}t}/n$ converges in
$\mathcal{M}_{1}$ to $A(t)$. Then,
$M_{n}(t)=\dfrac{1}{n^{1/2}}\sum_{k=1}^{N_{a_{n}t}}Y_{k}=\sigma\mathcal{W}_{n}\circ
A_{n}(t)$ converges to $\sigma\mathcal{W}\circ A_{t}$ f.d.d., but the
convergence in not necessarily with respect to $\mathcal{M}_{1}$. It then
follows that $\displaystyle\frac{X_{a_{n}t}-\mu
nA_{n}(t)}{n^{1/2}}=\sigma\mathcal{W}_{n}\circ A_{n}(t)+o_{P}(1)$. This can be
written as

(4.1) $X_{a_{n}t}=n^{1/2}\sigma\mathcal{W}_{n}\circ A_{n}(t)+n\mu
A_{n}(t)+o_{P}(n^{1/2}).$

Processes of the form $Y=\sigma W\circ A+\mu A$, where $A$ is a non-negative
stochastic process independent of the Brownian motion $W$ have been proposed
by Ané and Geman, (2000) to model the behaviour of stock returns. A particular
case is the famous Variance Gamma model (Madan et al.,, 1998), where $A$ is a
Gamma process. It follows that these processes can appear as limiting cases of
(4.1). If in addition, $A$ is deterministic, continuous, and
$\displaystyle\mathbb{A}_{n}(t)=\sqrt{n}\;\left\\{A_{n}(t)-A(t)\right\\}=\dfrac{N_{a_{n}t}-nA(t)}{n^{1/2}}\stackrel{{\scriptstyle\mathcal{J}_{1}}}{{\rightsquigarrow}}\mathbb{A}(t)$,
then

$n^{1/2}\left\\{\dfrac{X_{a_{n}t}}{n}-\mu
A(t)\right\\}=\sigma\mathcal{W}_{n}\circ
A_{n}(t)+\mu\mathbb{A}_{n}(t)+o_{P}(1).$

In most cases, $\mathbb{A}$ will be a Brownian motion (up to a constant),
independent of $\mathcal{W}$. In what follows, we assume that $N_{t}$ be a
renewal counting process defined by $\\{N_{t}\geq k\\}=\\{S_{k}\leq t\\}$,
where $S_{k}=\sum_{j=1}^{k}\tau_{k}$, and the $\tau_{k}$s are iid positive
random variables. The asymptotic behaviour of $N$ is determined by the
asymptotic behaviour of $S$.

####  4.1.1. First case: $S_{n}$ has finite variance

Suppose $\tau_{k}$ has mean $c_{1}$ and variance $\sigma_{1}^{2}$. Then,
taking $a_{n}=n$, one gets that $A(t)=\dfrac{t}{c_{1}}$,
$\mathbb{A}=-\dfrac{\sigma_{1}}{c_{1}^{3/2}}\mathbb{W}$, where $\mathbb{W}$ is
a Brownian motion independent of $\mathcal{W}$. It then follows from Rémillard
and Vaillancourt, (2023, Theorem 2.1) (stated as Theorem A.6 here) that
$n^{1/2}\left\\{\dfrac{X_{a_{n}t}}{n}-t\dfrac{\mu}{c_{1}}\right\\}$ converges
in $\mathcal{C}$ to $\tilde{\sigma}\tilde{}\mathcal{W}(t)$, where
$\tilde{}\mathcal{W}$ is a Brownian motion and
$\tilde{\sigma}=\left(\dfrac{\sigma^{2}}{c_{1}}+\dfrac{\mu^{2}\sigma_{1}^{2}}{c_{1}^{3}}\right)^{1/2}$.
This corrects formula (16) in Chávez-Casillas et al., (2019), where the factor
$\sigma_{1}^{2}$ is missing. This is also Theorem 7.4.1 in Whitt, (2002).

####  4.1.2. Second case: $S_{n}$ is in the domain of attraction of a stable
process of order $\alpha<1$

Setting $a_{n}=n^{1/\alpha}$, assume that $\mathcal{S}_{n}(t)=S_{\lfloor
nt\rfloor}/a_{n}$ converges in $\mathcal{J}_{1}$ to $\mathcal{S}(t)$. It
follows that $A_{n}(t)=\dfrac{N_{n^{1/\alpha}t}}{n}$ converges in
$\mathcal{M}_{1}$ to $A(t)=S^{-1}(t)$ since
$S_{n}^{-1}(t)=\dfrac{1+N_{n^{1/\alpha}t}}{n}$. Then,
$M_{n}(t)=\dfrac{1}{\sigma
n^{1/2}}\sum_{k=1}^{N_{a_{n}t}}V(\xi_{k})=\mathcal{W}_{n}\left(N_{a_{n}t}/n\right)$
converges to $\mathcal{W}\circ A_{t}$ f.d.d., but the convergence in not with
respect to $\mathcal{M}_{1}$, according to Proposition A.2. However, it
follows from Theorem 2.5 that $\mathcal{W}_{n}\circ
A_{n}\stackrel{{\scriptstyle\mathcal{L}^{2}_{w}}}{{\rightsquigarrow}}\mathcal{W}\circ
A$. Note that it also follows that
$\dfrac{X_{a_{n}t}}{n}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}\mu
A(t)$.

######  Remark 4.1.

There have been many incorrect statements about convergence of $A_{n}$ in the
literature. For example, Corollary 3.4 of Meerschaert and Scheffler, (2004)
about the $\mathcal{J}_{1}$-convergence of $A_{n}$ is not true. The
convergence is in $\mathcal{M}_{1}$, not $\mathcal{J}_{1}$. The authors quoted
an erroneous result in Theorem 3 of Bingham, (1971), where the error in the
proof of the latter result comes from the incorrect inequality in Equation (9)
in Bingham, (1971). As stated in that article, while it is true that
$\frac{1}{2}\omega_{\mathbb{J}}\left(x,\frac{T}{2k},T\right)\leq D(x,k,T)$,
where $\displaystyle D(x,k,T)=\max_{1\leq r\leq k}|x(rT/k)-x((r-1)T/k)|$, it
is not true that $D(x,k,T)\leq\omega_{\mathbb{J}}\left(x,\frac{T}{k},T\right)$
as claimed, even if $x$ is monotone. If $X\in D$, then
$\limsup_{k\to\infty}P(D(k,x,T)>\epsilon)>0$ unless $P(X\in C)=1$. In fact,
for any $x\in D$, $\displaystyle\limsup_{k\to\infty}D(k,x,T)=\sup_{0<t\leq
T}|\Delta x(t)|$, and if $x$ is nondecreasing, then

$\omega_{\mathbb{J}}\left(x,\delta,T\right)\leq\sup_{\delta\leq t\leq
T-\delta}\min\\{x(t)-x(t-\delta),x(t+\delta)-x(t)\\}.$

Next, Theorem 4.2 in Meerschaert and Scheffler, (2004), also quoted in Scalas
and Viles, (2012, 2014), is also incorrect. The authors concluded that since
the jumps of $\mathcal{W}$ and $\mathcal{S}$ are not simultaneous by
independence, it follows that $A$ is strictly increasing at points on
discontinuity of $\mathcal{W}$, then $\mathcal{W}_{n}\circ
A_{n}\stackrel{{\scriptstyle\mathcal{J}_{1}}}{{\rightsquigarrow}}\mathcal{W}\circ
A$. However, this is an incorrect use of (Whitt,, 2002, Theorem 13.2.4). One
cannot even conclude that $\mathcal{W}_{n}\circ
A_{n}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}\mathcal{W}\circ
A$.

####  4.1.3. Third case: $S_{n}$ is in the domain of attraction of a stable
process of order $\alpha=1$

In this case, if $\displaystyle\lim_{x\to\infty}xP(\tau_{1}>x)\to
c_{1}\in(0,\infty)$, then $S_{\lfloor nt\rfloor}/n\log{n}\to c_{1}$ and
$N_{nt}/\left(n/\log{n}\right)\to t/c_{1}$, using
$E\left(e^{-s\tau_{1}}\right)=1+c_{1}s\log{s}+o(s\log{s})$, as $s\to 0$, from
the proof of (Chávez-Casillas et al.,, 2019, Proposition A.1). In particular,
if $a_{n}=n\log{n}$, then $N_{a_{n}t}/n$ converges in probability to
$\frac{t}{c_{1}}$. If in addition
$E\left(e^{-s\tau_{1}}\right)=1+c_{1}s\log{s}+c_{2}s+o(s)$, as $s\to 0$, then
$S_{n}$ is in the domain of attraction of a stable law of order 1, since
$\frac{S_{\lfloor nt\rfloor}}{n}-c_{1}t\log{n}\rightsquigarrow
c_{1}\mathcal{S}_{t}-c_{2}t$ in $\mathcal{J}_{1}$, where $\mathcal{S}$ is a
stable process of index $1$. This clarifies conditions under which the last
case in (Chávez-Casillas et al.,, 2019, Section 4.2.2) holds true. As a
result, as $n\to\infty$, $E\left\\{e^{-s\mathcal{S}_{n}(t)}\right\\}\to
e^{t\left(c_{1}s\log{s}+c_{2}s\right)}$, so $\displaystyle
E\left\\{e^{is\mathcal{S}(t)}\right\\}=e^{-ic_{2}st+tc_{1}\psi(s)}$, where
$\psi(s)=-|s|\left\\{\dfrac{\pi}{2}+i{\rm sgn}(s)\log{|s|})\right\\}$. See
Formula (3.19) in Feller, (1971). Hence,
$\displaystyle\log{n}\left\\{\dfrac{X_{a_{n}t}}{n}-\mu
t/c_{1}\right\\}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}-\mu\left(\mathcal{S}_{t}-t\dfrac{c_{2}}{c_{1}}\right)$,
since
$\log{n}\left\\{A_{n}(t)-t/c_{1}\right\\}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}t\dfrac{c_{2}}{c_{1}}-\mathcal{S}_{t}$.

####  4.1.4. Fourth case: $S_{n}$ is in the domain of attraction of a stable
process of order $\alpha\in(1,2)$

Finally, if $1<\alpha<2$, then we have the following result (Whitt,, 2002,
Theorem 7.4.3): setting $a_{n}=n$, one gets

$n^{1-1/\alpha}\left(\dfrac{X_{nt}}{n}-t\dfrac{\mu}{\mu_{1}}\right)=n^{-\left(\frac{1}{\alpha}-\frac{1}{2}\right)}\mathcal{W}_{n}\circ
A_{n}(t)+\mu
n^{1-1/\alpha}\left(A_{n}(t)-\frac{t}{\mu_{1}}\right)\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}-\mu\dfrac{\mathcal{S}_{t}}{\mu_{1}^{1+1/\alpha}},$

since $\mathcal{S}_{n}(t)=n^{-1/\alpha}\left(S_{\lfloor
nt\rfloor}-nt\mu_{1}\right)\stackrel{{\scriptstyle\mathcal{J}_{1}}}{{\rightsquigarrow}}\mathcal{S}_{t}$,
where $\mu_{1}=E(\tau_{1})$. Note that $\mathcal{S}_{t}$ has a stable
distribution normalized so that
$\displaystyle\lim_{x\to\infty}x^{\alpha}P(\tau_{1}>x)=\lim_{x\to\infty}x^{\alpha}P(\mathcal{S}_{1}>x)=c_{1}$.

###  4.2. Occupation times for planar Brownian motion

The main results of this section are: a complete proof of Theorem 3.1 in
Kasahara and Kotani, (1979) and a correction to their Theorem 3.2.

Let $B$ be a planar Brownian motion and let
$V:\mathbb{R}^{2}\mapsto\mathbb{R}$ be continuous with compact support and set
$\bar{V}=\int_{\mathbb{R}^{2}}V(x)dx$. Then
$F(x)=\pi^{-1}\int_{\mathbb{R}^{2}}V(y)\log|y-x|dy$ is bounded and
$M_{t}=F(B_{t})-\int_{0}^{t}V(B_{s})ds$ is a martingale with
$[M]_{t}=\int_{0}^{t}|\nabla F(B_{s})|^{2}ds$ by Itô’s formula, since
$\dfrac{1}{2}\Delta F=V$ holds (Lieb and Loss,, 2001, Theorems 6.21 and 10.2).
So, under rescaling, the martingale $M$ and the occupation time
$\int_{0}^{t}V(B_{s})ds$ have the same asymptotic behavior. Hypothesis 3.1.a
holds, provided $V$ is not identically null, since $\nabla F$ is continuous
and also not everywhere null, entailing $[M]_{\infty}=\infty$. Since $M$ is
continuous, Hypothesis 3.2 is also satisfied for all rescalings. Let $m(t)$
denote a numerical function that goes to $\infty$ with $t$, and define the
continuous martingale
$\tilde{M}_{n}(t)=\\{\log{m(n)}\\}^{-1/2}\int_{0}^{m(nt)}\nabla F(B_{s})\cdot
dB_{s}$ with quadratic variation
$\tilde{A}_{n}(t)=[\tilde{M}_{n}]_{t}=\\{\log{m(n)}\\}^{-1}\int_{0}^{m(nt)}|\nabla
F(B_{s})|^{2}ds$. The common formulations involve rescalings $m(t)=t$ and
$m(t)=te^{2t}$ (ensuring that $\tilde{A}_{n}(0)=0$).

####  4.2.1. First case: $m(t)=t$

Here, removing the superscript on $\tilde{A}_{n}$ in order to avoid confusion
for the reader between the treatment of the two rescaled sequences,
$A_{n}(t)=\displaystyle\\{\log{n}\\}^{-1}\int_{0}^{nt}|\nabla F(B_{s})|^{2}ds$
converges f.d.d. to $A$, where $A(0)=0$ and $A(t)=A(1)\stackrel{{\scriptstyle
Law}}{{=}}\displaystyle\dfrac{\mathcal{E}}{2\pi}\int_{\mathbb{R}^{2}}|\nabla
F(x)|^{2}dx=\dfrac{\mathcal{E}}{2\pi}c_{V}$ for every $t>0$, where
${\mathcal{E}}$ is an Exponential random variable with mean $1$, and where

(4.2)
$c_{V}:=-\frac{2}{\pi}\int\int_{\mathbb{R}^{4}}V(x)V(y)\log|x-y|dxdy=\int_{\mathbb{R}^{2}}|\nabla
F(x)|^{2}dx.$

The proof of (4.2) is given in B.4. The proof of $A_{n}\stackrel{{\scriptstyle
f.d.d.}}{{\rightsquigarrow}}A$ follows from the fact that for any $0<s\leq t$,
$A_{n}(t)-A_{n}(s)\stackrel{{\scriptstyle Law}}{{\rightsquigarrow}}0$, since
$0\leq EA_{n}(t)-EA_{n}(s)\to 0$, using Darling and Kac, (1957). As a result,
$A$ is clearly not right-continuous at $0$ and therefore does not belong to
$D$. It also fails to verify another requirement of Theorem 3.1, namely
$A(t)\to\infty$ as $t\to\infty$ in Hypothesis 3.1.

######  Remark 4.2.

Proving the convergence in distribution of $A_{n}(t)$ was achieved in stages.
Kallianpur and Robbins, (1953) claimed that $[M]_{t}/\log{t}$ converges in law
to an Exponential distribution. In fact they never proved this result. They
refer to Robbins, (1953), where a result on sums of independent random
variables on the plane is claimed but not proved. Kallianpur and Robbins,
(1953) actually appeared before Robbins, (1953). It seems that the result
claimed by Robbins, (1953) was in fact proven in Darling and Kac, (1957),
while Erdős and Taylor, (1960) proved a special case involving the number of
visits at $0$ for symmetric random walks. In fact, Darling and Kac, (1957),
when $V$ is a non-negative function, showed that for any given $t>0$, not only
does $A_{n}(t)$ converge in law, but all moments of $A_{n}(t)$ also converge
to those of the limiting Exponential distribution. The general case appeared
in Yamazaki, (1992) and Le Gall, (1992, Lemma II.6 and ensuing remarks), after
some preliminary work by others, notably Erdős and Taylor, (1960) and Kasahara
and Kotani, (1979).

####  4.2.2. Second case: $m(t)=te^{2t}$

This scaling was proposed by Kasahara and Kotani, (1979). In fact, Kasahara
and Kotani, (1979) showed that, if $G$ is a continuous positive function with
$\bar{G}=\int_{\mathbb{R}^{2}}|x|^{\alpha}G(x)dx<\infty$, for some $\alpha>0$,
then $\dfrac{1}{n}\int_{0}^{m(nt)}G(B_{s})ds$ is equivalent in law to
$\displaystyle
H_{n}(t)=\frac{1}{n}\int_{0}^{S^{-1}\\{m(nt)\\}}f(\beta_{s},\theta_{s})ds$,
where $\beta$ and $\theta$ are two independent Brownian motions,
$S_{t}=\int_{0}^{t}e^{2\beta_{u}}du$, and $f$ is chosen so that
$\bar{f}=\int_{-\infty}^{\infty}\int_{0}^{2\pi}f(x,u)dxdu=\bar{G}$. Then,
setting $T_{n}(t)=\dfrac{1}{n}m^{-1}\left\\{S\left(n^{2}t\right)\right\\}$,
one gets $H_{n}\circ
T_{n}(t)=\dfrac{1}{n}\int_{0}^{n^{2}t}f(\beta_{s},\theta_{s})ds$. As a result,
they proved that

(4.3) $\left(H_{n}\circ
T_{n}(t),\beta(n^{2}t)/n,T_{n}(t)\right)\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}\left(2\bar{G}\ell,b,\mathbb{M}\right),$

where $b$ is a Brownian motion, $\mathbb{M}_{t}=\displaystyle\sup_{s\leq
t}b(s)$, and $\ell_{t}$ is the local time at $0$ of $b$. It then follows that
$T_{n}^{-1}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}\mathbb{M}^{-1}$,
which is a Lévy process. In their Theorem 3.1, Kasahara and Kotani, (1979)
stated that
$H_{n}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}H=2\bar{G}\ell\circ
M^{-1}$ without proving it, stopping at (4.3). To complete their proof, one
may apply Proposition A.2 with $x_{n}=H_{n}\circ T_{n}$, $y_{n}=T_{n}^{-1}$,
$x=2\bar{G}\ell$, and $y=\mathbb{M}^{-1}$ to conclude that
$H_{n}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}H=x\circ
y=2\bar{G}\ell\circ\mathbb{M}^{-1}$, since $\ell$ is monotone.

######  Remark 4.3.

It does not seem that such a result on the convergence of composition was
available to Kasahara and Kotani, (1979), since its first version appeared in
(Yamazaki,, 1992, Proposition 3.1).

Next, in their Theorem 3.2, Kasahara and Kotani, (1979) stated, without proof,
that if $\bar{V}=0$, then
$\tilde{M}_{n}(t)=\frac{1}{\sqrt{n}\;}\int_{0}^{m(nt)}V(B_{s})ds$ converges in
$\mathcal{M}_{1}$ to $c_{V}\beta_{2}\circ H$, where
$c_{V}=\dfrac{\bar{f}}{2\pi}$, $H=2\bar{G}\ell\circ M^{-1}$ and $\beta_{2}$ is
a Brownian motion independent of $b$ and $H$. To try to prove this result,
note that $\frac{1}{\sqrt{n}\;}\int_{0}^{m(nt)}V(B_{s})ds$ is equivalent in
law to $\displaystyle
M_{n}(t)=\frac{1}{\sqrt{n}\;}\int_{0}^{S^{-1}\\{m(nt)\\}}f(\beta_{s},\theta_{s})ds$,
where $\bar{f}=\bar{V}=0$. The latter is almost a martingale due to the
decomposition described previously. Applying Theorem A.6, one gets that, for
some $\gamma>0$,

$\left(M_{n}\circ T_{n}(t),H_{n}\circ
T_{n}(t),\beta(n^{2}t)/n,T_{n}(t)\right)\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}\left(\gamma\beta_{2}\circ\ell,\bar{G}\ell,b,\mathbb{M}\right),$

where $\beta_{2}$ is a Brownian motion independent of $\ell$. Because of
Proposition A.2, $M_{n}=M_{n}\circ T_{n}\circ T_{n}^{-1}$ does not
$\mathcal{M}_{1}$-converge to $\gamma\beta_{2}\circ\ell\circ\mathbb{M}^{-1}$
in $D$, since $\beta_{2}$ is not monotone with probability one on the sets
$[\ell\circ\mathbb{M}^{-1}(t-),\ell\circ\mathbb{M}^{-1}(t)]$. Therefore,
Theorem 3.2 in Kasahara and Kotani, (1979) is incorrect. However, since the
points of discontinuity of $T^{-1}$ are at most countable and not fixed, it
follows from Whitt, (2002, Theorem 12.4.1) that $M_{n}\stackrel{{\scriptstyle
f.d.d.}}{{\rightsquigarrow}}M_{\infty}=\gamma\beta_{2}\circ\ell\circ\mathbb{M}^{-1}$.
This result was also proven in Csáki et al., (2004), using strong
approximations. Next, the f.d.d. convergence implies that
$\int_{0}^{t}M_{n}(s)ds\stackrel{{\scriptstyle
f.d.d.}}{{\rightsquigarrow}}\int_{0}^{t}M_{\infty}(s)ds$, and hence this
completes the proof of
$M_{n}\stackrel{{\scriptstyle\mathcal{L}^{2}_{w}}}{{\rightsquigarrow}}M_{\infty}$,
in this second case where $m(t)=te^{2t}$, by an argument similar to the one
used to prove Theorem 2.5. In fact, for any choice of finite partition
$0=t_{0}<t_{1}<t_{2}\cdots<t_{m}=t$, $m\geq 1$ and $n=\infty,1,2,\dots$, put

$Z_{n,m}(t):=\sum_{k=1}^{m}M_{n}(t_{k})\left\\{(t_{k}\wedge t)-(t_{k-1}\wedge
t)\right\\}.$

As $n\to\infty$, $Z_{n,m}(t)\stackrel{{\scriptstyle
Law}}{{\rightsquigarrow}}Z_{\infty,m}(t)$, while, as $m\to\infty$,

$E\left\\{(Z_{n,m}(t)-\hbox{$\int_{0}^{t}$}M_{n}(s)ds)^{2}\right\\}\\\
\leq\sum_{k=1}^{m}\left\\{(t_{k}\wedge t)-(t_{k-1}\wedge
t)\right\\}^{2}\left\\{E[M_{n}]_{t_{k}\wedge t}-E[M_{n}]_{t_{k-1}\wedge
t}\right\\}\\\ \leq\sup_{k}(t_{k}-t_{k-1})^{2}\sup_{n}E[M_{n}]_{t}\to 0,$

through the selection of any sequence of partitions $\\{t_{k,m};k\leq m\\}$
with vanishing mesh, i.e., such that $\sup_{k}(t_{k,m}-t_{k-1,m})\to 0$ as
$m\to\infty$.

##  Appendix A Miscellanea

Let $D=D[0,\infty)$ be the space of $\mathbb{B}$-valued càdlàg trajectories
(right continuous with left limits everywhere), for some separable Banach
space $\mathbb{B}$, i.e., $\mathbb{B}$ is a complete separable normed linear
space with norm $\|\cdot\|$. $\mathbb{B}$ will always be specified implicitly
by the context at hand (usually $d$-dimensional Euclidean space
$\mathbb{R}^{d}$) so the subscript $\mathbb{B}$ is omitted. All processes
considered in this section have their trajectories in $D$ and are adapted to a
filtration $\mathbb{F}=(\mathcal{F}_{t})_{t\geq 0}$ on a probability space
$(\Omega,\mathcal{F},P)$ satisfying the usual conditions (notably, right
continuity of the filtration and completeness). Trajectories in $D$ are
usually noted $x(t)$ but occasionally $x_{t}$.

On $\mathbb{B}^{3}$, set $\mathbb{C}(x_{1},x_{2},x_{3})=\|x_{3}-x_{1}\|$,
$\mathbb{J}(x_{1},x_{2},x_{3})=\|x_{2}-x_{1}\|\wedge\|x_{2}-x_{3}\|$, where
$k\wedge\ell=\min(k,\ell)$ and let $\mathbb{M}(x_{1},x_{2},x_{3})$ be the
minimum distance between $x_{2}$ and the Banach space segment
$[x_{1},x_{3}]:=\\{\lambda
x_{1}+(1-\lambda)x_{3}\in\mathbb{B};\lambda\in[0,1]\\}$. Then
$\mathbb{M}(x_{1},x_{2},x_{3})=0$ if $x_{2}\in[x_{1},x_{3}]$ and otherwise
$\mathbb{M}(x_{1},x_{2},x_{3})\leq\mathbb{J}(x_{1},x_{2},x_{3})$.

For $\mathbb{H}=\mathbb{M}$, $\mathbb{H}=\mathbb{J}$ or
$\mathbb{H}=\mathbb{C}$, $T>0$, and $x\in D$, set

$\omega_{\mathbb{H}}(x,\delta,T)=\sup_{0\leq t_{1}<t_{2}<t_{3}\leq
T,\;t_{3}-t_{1}<\delta}\mathbb{H}\\{x(t_{1}),x(t_{2}),x(t_{3})\\}.$

Using the terminology in Skorohod, (1956, Theorem 3.2.2), for each
$\mathbb{H}=\mathbb{M}$, $\mathbb{H}=\mathbb{J}$ or $\mathbb{H}=\mathbb{C}$, a
sequence of $D$-valued processes $X_{n}$ is $\mathcal{H}_{1}$-tight
(respectively $\mathcal{M}_{1}$-tight, $\mathcal{J}_{1}$-tight or simply
$\mathcal{C}$-tight) if and only if

  1. (i)

for every $t$ in an everywhere dense subset of $[0,\infty)$ that includes the
origin, the marginal distributions of $X_{n}(t)$ are tight,

  2. (ii)

for any $\epsilon>0$, and $T>0$,

(A.1) $\lim_{\delta\to
0}\limsup_{n\to\infty}P\\{\omega_{\mathbb{H}}(X_{n},\delta,T)>\epsilon\\}=0.$

A sequence of processes $X_{n}$ converges weakly under the
$\mathcal{H}_{1}$-topology on $D$ to $X$, denoted
$X_{n}\stackrel{{\scriptstyle\mathcal{H}_{1}}}{{\rightsquigarrow}}X$ when
$\mathbb{H}=\mathbb{J}$ or $\mathbb{H}=\mathbb{M}$, if and only if it is
$\mathcal{H}_{1}$-tight and the finite dimensional distributions of $X_{n}$
converge to those of $X$, denoted $X_{n}\stackrel{{\scriptstyle
f.d.d.}}{{\rightsquigarrow}}X$, over a dense set of times in $[0,\infty)$
containing $0$. Similarly write
$X_{n}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}X$ for weak
convergence under the $\mathcal{C}$-topology on $D$, that of uniform
convergence over compact time sets, defined by taking $\mathbb{H}=\mathbb{C}$.
While we are on the subject of notation, equality in law or equidistribution
is denoted by $\stackrel{{\scriptstyle Law}}{{=}}$, convergence in the
$p^{th}$ mean by $\stackrel{{\scriptstyle L^{p}}}{{\rightsquigarrow}}$, in
probability by $\stackrel{{\scriptstyle Pr}}{{\rightsquigarrow}}$, in law by
$\stackrel{{\scriptstyle Law}}{{\rightsquigarrow}}$ and almost sure
convergence by $\stackrel{{\scriptstyle a.s.}}{{\rightsquigarrow}}$.

Complete coverage of the $\mathcal{J}_{1}$-topology can be found in Ethier and
Kurtz, (1986) and Jacod and Shiryaev, (2003), where $\mathbb{B}$ is actually
allowed be a Polish space; while Whitt, (2002) is our main reference on the
$\mathcal{M}_{1}$-topology. Neither topology turns $D$ into a topological
vector space, that is to say, neither is compatible with the linear structure
inherited on $D$ from the Banach space $\mathbb{B}$, which would have made the
sum a continuous operator. Consequently, when considering vectors of processes
valued in a product $\prod_{i=1}^{d}\mathbb{B}_{i}$ of Banach spaces, neither
topology offers the equivalence between $\mathcal{H}_{1}$-convergence of
sequences of probability measures on $D$ and coordinatewise
$\mathcal{H}_{1}$-convergence, an equivalence which holds for
$\mathcal{C}$-convergence. Similarly $\mathcal{H}_{1}$-tightness cannot be
treated coordinatewise; it does however ensue from the
$\mathcal{H}_{1}$-tightness of each coordinate plus that of each pairwise sum
of coordinates — $d^{2}$ verifications are thus required. This fact precludes
immediate extensions of some results from the real line to higher dimensional
spaces, but there are exceptions, as will be seen next.

######  Remark A.1.

All topological statements in this paper involving multiple coordinate
functions or processes with $\mathbb{B}$-valued càdlàg trajectories, are
stated and proved for the space $D=D([0,\infty):\mathbb{B})$, with Banach
space product $\mathbb{B}:=\prod_{i=1}^{d}\mathbb{B}_{i}$ equipped with the
norm $|\cdot|_{\mathbb{B}}:=\vee_{i=1}^{d}|\cdot|_{\mathbb{B}_{i}}$. Any
conclusion thus stated remains valid for the weaker product topology for the
space $\prod_{i=1}^{d}D([0,\infty):\mathbb{B}_{i})$ without further ado. This
is the case whether the topology involved is $\mathcal{C}$,
$\mathcal{L}^{2}_{w}$, $\mathcal{M}_{1}$ or $\mathcal{J}_{1}$. On those rare
occasions when the latter product topology is used, it is done explicitly by
displaying the product space, such as in Proposition A.2.

The following inequalities link the moduli associated with
$\mathcal{H}_{1}$-topologies, for both $\mathbb{H}=\mathbb{J}$ and
$\mathbb{H}=\mathbb{M}$, with that of the $\mathcal{C}$-topology, thus
providing a first situation when coordinatewise arguments do work.

######  Lemma A.1.

For any pair of functions $X,Y\in D$ there holds, for every choice of
$\delta>0$ and $T>0$,
$\omega_{\mathbb{H}}(X+Y,\delta,T)\leq\omega_{\mathbb{H}}(X,\delta,T)+\omega_{\mathbb{C}}(Y,\delta,T)$.
For sequences of $D$-valued processes, if $X_{n}$ is $\mathcal{H}_{1}$-tight
and $Y_{n}$ is $\mathcal{C}$-tight, then $X_{n}+Y_{n}$ and $(X_{n},Y_{n})$ are
also $\mathcal{H}_{1}$-tight. Similarly if
$X_{n}\stackrel{{\scriptstyle\mathcal{H}_{1}}}{{\rightsquigarrow}}X$,
$Y_{n}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}Y$, and
$(X_{n},Y_{n})\stackrel{{\scriptstyle f.d.d.}}{{\rightsquigarrow}}(X,Y)$ then
$X_{n}+Y_{n}\stackrel{{\scriptstyle\mathcal{H}_{1}}}{{\rightsquigarrow}}X+Y$
and
$(X_{n},Y_{n})\stackrel{{\scriptstyle\mathcal{H}_{1}}}{{\rightsquigarrow}}(X,Y)$.

###### Proof.

Given two triplets $(x_{1},x_{2},x_{3}),(y_{1},y_{2},y_{3})\in\mathbb{B}^{3}$,
there is a $\lambda\in[0,1]$ achieving the minimum for
$\mathbb{M}(x_{1},x_{2},x_{3})=\|x_{2}-\\{\lambda x_{1}+(1-\lambda)x_{3}\\}\|$
and therefore

$\displaystyle\mathbb{M}(x_{1}+y_{1},x_{2}+y_{2},x_{3}+y_{3})$
$\displaystyle\leq$
$\displaystyle\|x_{2}+y_{2}-\\{\lambda(x_{1}+y_{1})+(1-\lambda)(x_{3}+y_{3})\\}\|$
$\displaystyle\leq$ $\displaystyle\|x_{2}-\\{\lambda
x_{1}+(1-\lambda)x_{3}\\}|+|y_{2}-\\{\lambda y_{1}+(1-\lambda)y_{3}\\}\|$
$\displaystyle=$
$\displaystyle\mathbb{M}(x_{1},x_{2},x_{3})+\|y_{2}-\\{\lambda
y_{1}+(1-\lambda)y_{3}\\}\|$ $\displaystyle\leq$
$\displaystyle\mathbb{M}(x_{1},x_{2},x_{3})+\|y_{2}-y_{1}\|\vee\|y_{2}-y_{3}\|,$

this last inequality using convexity of the norm and $k\vee\ell=\max(k,\ell)$.
Restricting to $\lambda\in\\{0,1\\}$ instead of $[0,1]$ yields the same
inequalities with $\mathbb{J}$ in place of $\mathbb{M}$. ∎

The following result provides necessary and sufficient conditions under which
the composition mapping $\circ$ is $\mathcal{M}_{1}$-continuous, when
$B=\mathbb{R}^{k}$. Let $D_{\uparrow}\subset D$ be the subspace of those non-
decreasing non-negative $y$ such that $y(t)\to\infty$ as $t\to\infty$.

######  Proposition A.2.

Suppose that
$(x_{n},y_{n})\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\to}}(x,y)$ in
$D\left([0,\infty):\mathbb{R}^{k}\right)\times D_{\uparrow}$, where $x$ is
continuous. Then $x_{n}\circ
y_{n}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\to}}x\circ y$ in
$D\left([0,\infty):\mathbb{R}^{k}\right)$ if and only if $x$ is monotone on
$[y(t-),y(t)]$, for any $t\in{\rm Disc}(y)$.

###### Proof.

Sufficiency follows from Whitt, (2002, Theorem 13.2.4) so we only prove
necessity. Suppose $y_{n}\to y$ in $\mathcal{M}_{1}$, $y_{n}$ continuous and
strictly increasing. Assume that $t\in{\rm Disc}(y)$, and let $t_{1,m}$ be
sequence increasing to $t$ with $t_{1,m}\not\in{\rm Disc}(y)$. Further let
$t_{m,3}$ be sequence decreasing to $t$ with $t_{m,3}\not\in{\rm Disc}(y)$.
Finally, for a given $\lambda\in(0,1)$, let $t_{n,2,m}$ be the unique point
such that $y_{n}(t_{n,2,m})=\lambda y_{n}(t_{1,m})+(1-\lambda)y_{n}(t_{3,m})$.
For a given $\delta>0$, $t-\delta<t_{1,m}<t_{m,3}<t+\delta$ if $m$ is large
enough. By (Whitt,, 2002, Theorem 12.5.1 (v)), since $t_{1,m},t_{3,m}$ are
continuity point of $y$, it follows that $\displaystyle\lim_{\delta_{1}\to
0}\limsup_{n\to\infty}\omega_{\mathbb{C}}(y_{n}-y,t_{j,m},\delta_{1})=0$,
$j\in\\{1,3\\}$; in particular, $y_{n}(t_{j,m})\to s_{j,m}=y(t_{j,m})$ as
$n\to\infty$. Thus, $y_{n}(t_{n,2,m})\to s_{2,m}=\lambda
y(t_{1,m})+(1-\lambda)y(t_{3,m})$. If $x_{n}\to x$ in $\mathcal{M}_{1}$ with
$x$ continuous, it follows from (Whitt,, 2002, Theorem 12.5.1 (v)) that
$x_{n}\circ y_{n}(t_{n,j,m})\to x(s_{j,m})$, $j\in\\{1,2,3\\}$. Therefore,

$\limsup_{n\to\infty}\omega_{\mathbb{M}}(x_{n}\circ
y_{n},\delta,T)\geq\mathbb{M}\left\\{x(s_{1,m}),x(s_{2,m}),x(s_{3,m})\right\\}$

if $m$ is large enough. By letting $m\to\infty$, one gets $s_{1,m}\to
s_{1}=y(t-)$, $s_{3,m}\to s_{3}=y(t)$ and $s_{2,m}\to\lambda
y(t-)+(1-\lambda)y(t)$. As a result,

$\lim_{\delta\to 0}\limsup_{n\to\infty}\omega_{\mathbb{M}}(x_{n}\circ
y_{n},\delta,T)\\\ \geq\sup_{t\in{\rm Disc}(y),t<T}\sup_{y(t-)\leq s\leq
y(t)}\mathbb{M}\left[x\\{y(t-)\\},x(s),x\\{y(t)\\}\right]>0,$

unless $x$ is monotone on each interval $[y(t-),y(t)]$. ∎

From hereon we focus exclusively on the real-valued functions and processes so
$\mathbb{B}=\mathbb{R}$. Some observations for sequences of $D$-valued non-
decreasing processes come next. For any non-decreasing non-negative function
$A\in D$, denote its inverse by $\tau(s)=\inf\\{t\geq 0;A(t)>s\\}$. Let
$D_{\uparrow}^{0}$ be the subspace of those trajectories in
$D_{\uparrow}\subset D$ where $\tau(0)=0$ as well.

######  Proposition A.3.

The inverse map $A\mapsto\tau$ is a well defined bijective mapping from
$D_{\uparrow}$ into itself, such that there holds $A\circ\tau\circ A=A$
provided $A\in D_{\uparrow}$ is either continuous everywhere or strictly
increasing everywhere. Both $A\mapsto\tau$ and the reverse map $\tau\mapsto A$
are $\mathcal{M}_{1}$-continuous when restricted to $D_{\uparrow}^{0}$; this
is not necessarily the case without this additional restriction, not even on
$D_{\uparrow}$.

###### Proof.

The first statement combines Whitt, (2002, Corollary 13.6.1) together with
Whitt, (2002, Lemma 13.6.5); the second one is Whitt, (2002, Theorem 13.6.3)
essentially. ∎

######  Remark A.2.

A sequence of $D$-valued non-decreasing processes verify
$A_{n}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}A$ if and
only if $A_{n}\stackrel{{\scriptstyle f.d.d.}}{{\rightsquigarrow}}A$, since
they all satisfy $\omega_{\mathbb{M}}(A_{n},\delta)=0$ for any $\delta>0$.
Similarly, the $A_{n}$ are $\mathcal{M}_{1}$-tight if and only if their finite
dimensional distributions are tight; since each $A_{n}$ is non-decreasing,
this last condition is equivalent to the tightness of $(A_{n}(T))_{n\geq 1}$
for every $T>0$, provided the $A_{n}$ are non-negative (or at least uniformly
bounded below). Furthermore, the sum $A_{n}+A^{\prime}_{n}$ of two
$\mathcal{M}_{1}$-tight sequences of non-negative, non-decreasing processes is
also $\mathcal{M}_{1}$-tight, hence so are $(A_{n},0)$, $(0,A^{\prime}_{n})$
and finally $(A_{n},A^{\prime}_{n})$ as well. If in addition there holds
$A_{n}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}A$ and
$A^{\prime}_{n}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}A^{\prime}$,
then there ensues
$A_{n}+A^{\prime}_{n}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}A+A^{\prime}$
and
$(A_{n},A^{\prime}_{n})\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}(A,A^{\prime})$.

More can be said when the prospective limit is continuous. The following
result follows from Rémillard and Vaillancourt, (2023).

######  Proposition A.4.

Let $D$-valued non-decreasing processes $A_{n}$ and some continuous process
$A$ be such that $A_{n}\stackrel{{\scriptstyle f.d.d.}}{{\rightsquigarrow}}A$.
Then $A_{n}$ is $\mathcal{C}$-tight and
$A_{n}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}A$.

The following technical result ensues. Once again $M_{n}$ is a sequence of
$D$-valued square integrable $\mathbb{F}$-martingales started at $M_{n}(0)=0$
with quadratic variation $[M_{n}]$ and predictable compensator $\langle
M_{n}\rangle$.

######  Proposition A.5.

$M_{n}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}0$,
$[M_{n}]\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}0$ and
$\langle
M_{n}\rangle\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}0$ are
equivalent.

###### Proof.

The operator on $D$ defined by mapping $x\in D$ to $t\mapsto\sup_{0\leq s\leq
t}x(s)$, is a continuous operator in the $\mathcal{C}$-topology. The three
statements $M_{n}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}0$,
$M_{n}^{2}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}0$ and
$\sup_{0\leq
s\leq\\{\cdot\\}}M_{n}^{2}(s)\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}0$
are therefore equivalent. By Proposition A.4, it suffices to prove the
proposition with each of
$M_{n}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}0$,
$[M_{n}]\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}0$ and
$\langle
M_{n}\rangle\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}0$,
respectively replaced by $\sup_{0\leq s\leq
t}M_{n}^{2}(s)\stackrel{{\scriptstyle Law}}{{\rightsquigarrow}}0$,
$[M_{n}]_{t}\stackrel{{\scriptstyle Law}}{{\rightsquigarrow}}0$ and $\langle
M_{n}\rangle_{t}\stackrel{{\scriptstyle Law}}{{\rightsquigarrow}}0$, for all
$t\geq 0$ in all three cases. Doob’s inequality yields

$E\\{[M_{n}]_{t}\\}\leq E\left\\{\sup_{0\leq s\leq t}M_{n}^{2}(s)\right\\}\leq
4E\\{[M_{n}]_{t}\\}.$

Applying Lemma A.8 twice, interchanging the roles of $X$ and $Y$, yields the
first equivalence for $M_{n}$ and $[M_{n}]$. Equality $E\\{\langle
M_{n}\rangle_{t}\\}=E\\{[M_{n}]_{t}\\}$ and another double application of
Lemma A.8 yields the last one, this time with $\langle M_{n}\rangle$ and
$[M_{n}]$. ∎

Next is a miscellanea of useful results. First comes a CLT in the
$\mathcal{C}$-topology, taken from Rémillard and Vaillancourt, (2023). We only
state it for $d=1$ here.

######  Theorem A.6.

Assume that Hypothesis 3.1 holds with $A$ continuous everywhere; that
$J_{t}(M_{n})\stackrel{{\scriptstyle Law}}{{\rightsquigarrow}}0$ for any
$t>0$; that there exists an $\mathbb{F}$-adapted sequence of $D$-valued square
integrable martingales $B_{n}$ started at $B_{n}(0)=0$ so that

  1. (1)

$(B_{n},A_{n})\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}(B,A)$
holds, where $B$ is a Brownian motion with respect to its natural filtration
$\mathbb{F}_{B}=\\{\mathcal{F}_{B,t}:\;t\geq 0\\}$ and $A$ is
$\mathbb{F}_{B}$-measurable;

  2. (2)

$\langle M_{n},B_{n}\rangle_{t}\stackrel{{\scriptstyle
Law}}{{\rightsquigarrow}}0$, for any $t\geq 0$.

Then
$(M_{n},A_{n},B_{n})\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}(M,A,B)$
holds, where $M$ is a continuous square integrable
${\mathcal{F}}_{t}$-martingale with respect to (enlarged) filtration with
predictable quadratic variation process $A$. Moreover, $M=W\circ A$ holds,
with $W$ a standard Brownian motion which is independent of $B$ and $A$.

Next, we have the following observations about Lévy processes.

The characteristic function of an inhomogeneous Lévy process $A$ is given by

$E\left[\left.e^{i\theta(A_{t}-A_{s})}\right|\mathcal{F}_{s}\right]=e^{\Psi_{t}(\theta)-\Psi_{s}(\theta)},\quad
0\leq s<t,$

where $A$ is $\mathbb{F}$-adapted with $\mathbb{F}=(\mathcal{F}_{t})_{t\geq
0}$,

$\Psi_{t}(\theta)=i\theta
B_{t}-\frac{\theta^{2}C_{t}}{2}+\left\\{\prod_{0<r\leq t}e^{-ir\Delta
B_{r}}\right\\}\int_{0}^{t}\int\left(e^{i\theta x}-1-i\theta
x\mathbb{I}_{\\{|x|\leq 1\\}}\right)\nu(du,dx),$

$\Delta B_{r}=B_{r}-B_{r-}$, and $\nu$ is a Lévy measure. For details see
Jacod and Shiryaev, (2003, Theorem II.4.15, Theorem II.5.2 and Corollaries
II.5.11,12,13). The characteristics of $A$ are defined to be the deterministic
functions $(B,C,\nu)$.

######  Lemma A.7.

Suppose $A$ is an $\mathbb{F}$-adapted inhomogeneous Lévy process with finite
variation on any finite interval $[0,T]$ and no Brownian component, i.e.,
$C_{t}=0$ for all $t\geq 0$. Then $A$ is independent of any
$\mathbb{F}$-Brownian motion $W$.

###### Proof.

If $W$ is an $\mathbb{F}$-Brownian motion, then the martingale
$N_{1}(t)=e^{i\theta_{1}W_{t}+\frac{\theta^{2}t}{2}}$ is continuous and square
integrable on any finite interval $[0,T]$ and the martingale
$N_{2}(t)=e^{i\theta_{2}A_{t}-\Psi_{t}(\theta_{2})}$ has finite variation
$V_{T}$ on any finite interval $[0,T]$, and $V_{T}$ is square integrable.
Then, from Applebaum, (2004), one obtains that for any $t\geq 0$, with $\Delta
N(u)=N(u)-N(u-)$,

$E[N_{1}(t)N_{2}(t)]=1+\sum_{0<u\leq t}E[\Delta N_{1}(u)\Delta N_{2}(u)]=1.$

Hence
$E\left[e^{i\theta_{1}W(t)+i\theta_{2}A(t)}\right]=e^{-\frac{\theta_{1}^{2}t}{2}}e^{\Psi_{t}(\theta_{2})}$,
proving that $A(t)$ and $W(t)$ are independent. Using this and the
independence of the increments of $A$ and $W$, one may conclude that $A$ and
$W$ are independent. ∎

The proof of the following is in Jacod and Shiryaev, (2003, Lemma I.3.30).

######  Lemma A.8 (Lenglart’s inequality).

Let $X$ be an $\mathbb{F}$-adapted $D$-valued process. Suppose that $Y$ is
optional, non-decreasing, and that, for any bounded stopping time $\tau$,
$E|X(\tau)|\leq E\\{Y(\tau)\\}$. Then for any stopping time $\tau$ and all
$\varepsilon,\eta>0$,

  * a)

if $Y$ is predictable,

(A.2)
$P(\sup_{s\leq\tau}|X(s)|\geq\varepsilon)\leq\frac{\eta}{\varepsilon}+P(Y(\tau)\geq\eta).$

  * b)

if $Y$ is adapted,

(A.3)
$P(\sup_{s\leq\tau}|X(s)|\geq\varepsilon)\leq\frac{1}{\varepsilon}\left[\eta+E\left\\{J_{\tau}(Y)\right\\}\right]+P(Y(\tau)\geq\eta).$

We end with a few comments about $\mathcal{L}^{2}_{loc}$ which are used in the
proofs herein.

A (deterministic) sequence $x_{n}\in\mathcal{L}^{2}_{loc}$ is said to
$\mathcal{L}^{2}_{w}$-converge to $x\in\mathcal{L}^{2}_{loc}$ if and only if
the sequence of scalar products
$\int_{0}^{T}x_{n}(t)f(t)dt\to\int_{0}^{T}x(t)f(t)dt$ converges as
$n\to\infty$, for each $f\in\mathcal{L}^{2}_{loc}$ and each $T>0$. This holds
if and only if both the following statements hold for every $T>0$: the
sequence $\bigl{\\{}|x_{n}|_{T};n\geq 1\bigr{\\}}$ is bounded and the
requirement for convergence of the above scalar products is achieved for a
norm dense subset of $\mathcal{L}^{2}[0,T]$. In fact, given any complete
orthonormal set $\\{f_{k}\\}$ for $\mathcal{L}^{2}[0,1]$, the countable family
$\\{f_{k}(\ell^{-1}\cdot);k,\ell=1,2,\ldots\\}$, with $f_{k}(\ell^{-1}t):=0$
when $t>\ell$, will suffice, since it has a norm dense linear span in each
vector space $\mathcal{L}^{2}[0,T]$. Alternatively, given any complete
orthonormal set $\\{g_{k}\\}$ for Hilbert space $\mathcal{L}^{2}[0,\infty)$ or
$\mathcal{L}^{2}(-\infty,\infty)$, the same holds with the countable family
$\\{g_{k}\\}$ — for instance, in $\mathcal{L}^{2}(-\infty,\infty)$, take the
Hermite functions $\\{h_{k};k\geq 0\\}$, defined by
$h_{k}(t):=(\pi^{1/2}2^{k}k!)^{-1/2}(-1)^{k}e^{t^{2}/2}D^{k}_{t}(e^{-t^{2}})$,
where $D^{k}_{t}$ is the $k^{\hbox{th}}$ derivative with respect to $t$, which
also satisfy the uniform bound $\sup_{k,t}|h_{k}(t)|=h_{0}(0)=\pi^{-1/4}$, due
to Szász, (1951). The metric $d$ on $\mathcal{L}^{2}_{loc}$, defined by

$d(x,y):=\sum_{k=0}^{\infty}\sum_{\ell=1}^{\infty}2^{-(k+\ell+1)}\left(1\wedge\left|\int_{0}^{\ell}\left\\{x(t)-y(t)\right\\}h_{k}(t)dt\right|\right),$

is compatible with the $\mathcal{L}^{2}_{w}$-topology on all norm bounded
subsets $B$ of either $\mathcal{L}^{2}[0,\infty)$ ($\sup_{x\in
B}|x|_{\infty}<\infty$) or $\mathcal{L}^{2}[0,T]$ ($\sup_{x\in
B}|x|_{T}<\infty$), turning all $\mathcal{L}^{2}_{w}$-closed and norm bounded
subsets of either $\mathcal{L}^{2}[0,\infty)$ or $\mathcal{L}^{2}[0,T]$ into
complete separable metric (hereafter Polish) spaces, when equipped with this
metric. In fact, they are compact Polish spaces since all balls in either
$\mathcal{L}^{2}[0,\infty)$ or $\mathcal{L}^{2}[0,T]$ are relatively compact
in the $\mathcal{L}^{2}_{w}$-topology — for a proof see Lieb and Loss, (2001).

##  Appendix B Proofs

###  B.1. Proof of Theorem 2.2

The projections
$(\pi_{\ell_{1},k_{1}},\pi_{\ell_{2},k_{2}},\ldots,\pi_{\ell_{m},k_{m}}):\mathcal{L}^{2}_{loc}\to\mathbb{R}^{m}$,
given by setting $\pi_{\ell,k}(f):=\int_{0}^{\ell}f(t)h_{k}(t)dt$ for all
$k\geq 0$ and $\ell,m\geq 1$, generate the Borel $\sigma$-algebra for metric
$d$, and they characterize probability measures on $\mathcal{L}^{2}_{loc}$,
just as in the proof of Kolmogorov’s existence theorem for probability
measures on $\mathbb{R}^{\infty}$. In the light of Lemma 2.1, it suffices to
prove that $(i)$ and $(ii)$ together imply, jointly for all
$(k_{1},k_{2},\ldots,k_{m})\in\\{0,1,2,\ldots\\}^{m}$ and all $m\geq 1$,

$\int_{0}^{\cdot}X_{n}(s)\\{h_{k_{1}},h_{k_{2}},\ldots,h_{k_{m}}\\}(s)ds\stackrel{{\scriptstyle
f.d.d.}}{{\rightsquigarrow}}\int_{0}^{\cdot}X(s)\\{h_{k_{1}},h_{k_{2}},\ldots,h_{k_{m}}\\}(s)ds$

as $n\to\infty$. The Cramér-Wold device, applied to linear combinations of
finitely many indicators of finite time intervals, plus a density argument
gives sufficiency. Necessity follows the same argument applied to
$\\{h_{k}\\}$ to approximate $1\in\mathcal{L}^{2}_{loc}$. ∎

###  B.2. Proof of Theorem 2.5

Setting $|X_{n}|_{\infty,t}=\sup_{0\leq s\leq t}|X_{n}(s)|$, $|X_{n}\circ
Y_{n}|_{T}\leq\sqrt{T}|X_{n}|_{\infty,Y_{n}(T)}$ follows. Since $Y_{n}(T)$ is
tight for every $T>0$, so is $|X_{n}\circ Y_{n}|_{T}$. Next, under
alternatives a) or b),
$V_{n}=Y_{n}^{-1}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}V=Y^{-1}$,
by Proposition A.3. According to Theorem 2.2, to complete the proof it
suffices to prove the convergence of the finite dimensional distributions of
the sequence of processes $\int_{0}^{t}X_{n}\circ Y_{n}(s)ds=Z_{n}\circ
Y_{n}(t)$ to the process $\int_{0}^{t}X\circ Y(s)ds=Z\circ Y(t)$ , where
$Z_{n}(t)=\int_{0}^{t}X_{n}(s)dV_{n}(s)$ and $Z(t)=\int_{0}^{t}X(s)dV(s)$.
Because $X_{n}$ $\mathcal{C}$-converges, for $M>0$, there is a finite
partition $0=t_{0}<t_{1}<t_{2}\cdots<t_{m}=M$ of $[0,M]$ such that
$\delta_{n,m}=|X_{n}-X_{n,m}|_{\infty,M}$ and
$\delta_{\infty,m}=|X-X_{\infty,m}|_{\infty,M}$ are as small as one wants with
large probability, where
$X_{n,m}(t)=\sum_{k=1}^{m}X_{n}(t_{k})\mathbb{I}_{t_{k-1}\leq t<t_{k}}$ and
$X_{\infty,m}(t)=\sum_{k=1}^{m}X(t_{k})\mathbb{I}_{t_{k-1}\leq t<t_{k}}$.
Explicitly, for every $M>0$ and $\epsilon>0$, there are integers
$m=m(M,\epsilon)\geq 1$ and $N=N(M,m,\epsilon)\geq 1$ such that $P(\sup_{n\geq
N}\delta_{n,m}\geq\epsilon)\leq 3P(\delta_{\infty,m}\geq\epsilon)<\epsilon$.
Similarly $|Z_{n}-Z_{n,m}|_{\infty,M}\leq\sqrt{V_{n}(M)}\delta_{n,m}$ and
$|Z-Z_{\infty,m}|_{\infty,M}\leq\sqrt{V(M)}\delta_{\infty,m}$ hold, where

$Z_{n,m}(t)=\int_{0}^{t}X_{n,m}(s)dV_{n}(s)=\sum_{k=1}^{m}X_{n}(t_{k})\left\\{V_{n}(t_{k}\wedge
t)-V_{n}(t_{k-1}\wedge t)\right\\},$

and $Z_{\infty,m}(t)=\int_{0}^{t}X_{\infty,m}(s)dV(s)$.

By Proposition A.3 and Remark A.2, $(X_{n},Y_{n},V_{n})\stackrel{{\scriptstyle
f.d.d.}}{{\rightsquigarrow}}(X,Y,V)$ holds, hence so does
$Z_{n,m}\stackrel{{\scriptstyle f.d.d.}}{{\rightsquigarrow}}Z_{\infty,m}$ for
each fixed $m\geq 1$. Since $|Z_{n}-Z_{n,m}|_{\infty,M}\stackrel{{\scriptstyle
Pr}}{{\rightsquigarrow}}0$ and
$|Z-Z_{\infty,m}|_{\infty,M}\stackrel{{\scriptstyle Pr}}{{\rightsquigarrow}}0$
hold for each fixed $n\geq 1$ and $M>0$, plus $V_{n}(M)$ is tight for every
$M>0$, $Z_{n,m}\stackrel{{\scriptstyle
f.d.d.}}{{\rightsquigarrow}}Z_{\infty,m}$ for each fixed $m\geq 1$ yields
$Z_{n}\stackrel{{\scriptstyle f.d.d.}}{{\rightsquigarrow}}Z$. All that is left
to show is that $Z_{n}\circ Y_{n}\stackrel{{\scriptstyle
f.d.d.}}{{\rightsquigarrow}}Z\circ Y$. Since $Y_{n}(T)$ is tight, choosing
$\epsilon>0$, one can find $M=M_{\epsilon}$ so that for all $n\in\mathbb{N}$,
$P(|Y_{n}(T)|>M)\leq\epsilon$ and $P(|Y(T)|>M)\leq\epsilon$. Suppose in
addition that $0=t_{0}<t_{1}<\cdots<t_{m}=M$ are points of continuity of the
distribution of $Y(T)$. Then, on $\\{Y_{n}(T)\leq M\\}$,

$\displaystyle Z_{n,m}\circ Y_{n}(T)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{m}\mathbb{I}\\{t_{j-1}<Y_{n}(T)\leq
t_{j}\\}X_{n}(t_{j})\\{V_{n}\circ Y_{n}(T)-V_{n}(t_{j-1})\\}$
$\displaystyle\quad+\sum_{j=1}^{m}\mathbb{I}\\{t_{j-1}<Y_{n}(T)\leq
t_{j}\\}\left[\sum_{k=1}^{j-1}X_{n}(t_{k})\\{V_{n}(t_{k})-V_{n}(t_{k-1})\\}\right]$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{m}\mathbb{I}\\{t_{j-1}<Y_{n}(T)\leq
t_{j}\\}X_{n}(t_{j})\\{V_{n}\circ Y_{n}(T)-V_{n}(t_{j-1})\\}$
$\displaystyle\quad+\sum_{k=1}^{m-1}X_{n}(t_{k})\\{V_{n}(t_{k})-V_{n}(t_{k-1})\\}\mathbb{I}\\{t_{k}<Y_{n}(T)\leq
M\\}.$

Since $(X_{n},Y_{n},V_{n})\stackrel{{\scriptstyle
f.d.d.}}{{\rightsquigarrow}}(X,Y,V)$, it follows that $Z_{n,m}\circ Y_{n}(T)$
converges in law to $Z_{\infty,m}\circ Y(T)$. The rest of the proof follows
similarly by considering common points of continuity of the marginal
distributions of $Y(T_{1}),Y(T_{2}),\ldots,Y(T_{i})$ for any choice of
$0=T_{0}<T_{1}<\cdots<T_{i}=T$, first yielding $Z_{n,m}\circ
Y_{n}\stackrel{{\scriptstyle f.d.d.}}{{\rightsquigarrow}}Z_{\infty,m}\circ Y$
for each fixed $m\geq 1$ and then $Z_{n}\circ Y_{n}\stackrel{{\scriptstyle
f.d.d.}}{{\rightsquigarrow}}Z\circ Y$. ∎

###  B.3. Proof of Theorem 3.1

Hypothesis 3.2 implies that $\langle W_{n}\rangle$ is $\mathcal{C}$-tight;
applying Theorem A.6 to martingale $W_{n}=M_{n}\circ\tau_{n}$ yields
$(W_{n},\langle
W_{n}\rangle)\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}(W,\langle
W\rangle)$, with $W$ a standard Brownian motion since $\langle
W\rangle_{t}=t$, a deterministic trajectory. Hypothesis 3.1 implies
$A_{n}\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}A$ (in the
light of Remark A.2), which yields
$M_{n}\stackrel{{\scriptstyle\mathcal{L}^{2}_{w}}}{{\rightsquigarrow}}M$, by
Theorem 2.5. The independence of $W$ and $A$ stems from Lemma A.7 in the case
where $A$ is an inhomogeneous Lévy process. In the case where Hypothesis 3.3
is valid instead, Remark 3.2 yields
$(A_{n},\tau_{n})\stackrel{{\scriptstyle\mathcal{M}_{1}}}{{\rightsquigarrow}}(A,\tau)$.
Since the pair $(\tau,W)$ is $\mathbb{F}_{\tau}$-adapted and $\tau$ is a non-
decreasing inhomogeneous Lévy process, it has no Brownian component;
therefore, it is independent of $W$ by Lemma A.7. Since $A(0)=0$ guarantees
the continuity of the reverse mapping $\tau\mapsto A$, $A$ is also independent
of $W$ and so is the pair $(A,\tau)$ as the image of Borel measurable mapping
$\tau\mapsto(A,\tau)$. The independence of $W$ and $A$ implies
$(A_{n},W_{n})\stackrel{{\scriptstyle f.d.d.}}{{\rightsquigarrow}}(A,W)$ from
which there follows $(W_{n},A_{n},\tau_{n})\stackrel{{\scriptstyle
f.d.d.}}{{\rightsquigarrow}}(W,A,\tau)$ under either of the alternate
conditions in the statement. Going back to the end of the proof of Theorem 2.5
which yielded $\int_{0}^{\cdot}M_{n}(s)ds\stackrel{{\scriptstyle
f.d.d.}}{{\rightsquigarrow}}\int_{0}^{\cdot}M(s)ds$, note that common points
of continuity of the marginal distributions of
$A(T_{1}),A(T_{2}),\ldots,A(T_{i})$ also allow for the discretization of the
integrals $\mathcal{I}(A_{n})$ and $\mathcal{I}(W_{n})$ along the same pattern
as for $\mathcal{I}(M_{n})$, since $D$-valued $A_{n}$ and $W_{n}$ have bounded
trajectories on compact time sets. One gets, through the same reasoning as for
$\mathcal{I}(M_{n})$,
$\left(\mathcal{I}(M_{n}),\mathcal{I}(A_{n}),\mathcal{I}(W_{n})\right)\stackrel{{\scriptstyle
f.d.d.}}{{\rightsquigarrow}}\left(\mathcal{I}(M),\mathcal{I}(A),\mathcal{I}(W)\right)$.
By Lemma 2.3 and Remark 2.3,
$(M_{n},A_{n},W_{n})\stackrel{{\scriptstyle\mathcal{L}^{2}_{w}}}{{\rightsquigarrow}}(M,A,W)$
ensues. ∎

###  B.4. Proof of (4.2)

We only do the case where $V(x)=v(|x|)$ is radial as an illustration. Then
$F(x)=\pi^{-1}\int_{\mathbb{R}^{2}}V(y)\log|y-x|dy$ is also radial,
$\int_{0}^{\infty}(rv(r)dr=0$ since $\bar{V}=0$, and $\displaystyle
F(x)=\Phi(|x|)=\frac{1}{2\pi}\int_{0}^{\infty}\int_{0}^{2\pi}rv(r)\log\left(|x|^{2}+r^{2}-2r|x|\cos\theta\right)d\theta
dr$. Setting $\gamma=2ra/(r^{2}+a^{2})=2\sin\alpha\cos\alpha=\sin(2\alpha)$,
then successively

$\frac{1}{2\pi}\int_{0}^{2\pi}\log\left(1-\gamma\cos\theta\right)d\theta=\log\left(\frac{1+\sqrt{1-\gamma^{2}}}{2}\right)=\log\left\\{\frac{\max\left(r^{2},a^{2}\right)}{r^{2}+a^{2}}\right\\},$
$\displaystyle\Phi(a)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}rv(r)\log\left(a^{2}+r^{2}\right)dr+\frac{1}{2\pi}\int_{0}^{\infty}\int_{0}^{2\pi}rv(r)\log\left(1-\gamma\cos\theta\right)d\theta
dr$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}rv(r)\log\left(a^{2}+r^{2}\right)dr+\int_{0}^{\infty}rv(r)\log\left(\frac{\max(r^{2},a^{2})}{r^{2}+a^{2}}\right)dr$
$\displaystyle=$ $\displaystyle
2\int_{0}^{\infty}rv(r)\log{\max\left(r,a\right)}dr.$

As a result, $\displaystyle
F(x)=2\int_{0}^{\infty}rv(r)\log{\max\left(r,|x|\right)}dr=2\int_{|x|}^{\infty}rv(r)\log\left(\frac{r}{|x|}\right)dr$
and $|\nabla
F(x)|^{2}=\left\\{\frac{2}{|x|}\int_{|x|}^{\infty}rv(r)dr\right\\}^{2}$, so
that

$\displaystyle\int|\nabla F(x)|^{2}dx$ $\displaystyle=$ $\displaystyle
8\pi\int_{0}^{\infty}\frac{1}{r}\left\\{\int_{r}^{\infty}sv(s)ds\right\\}^{2}dr$
$\displaystyle=$
$\displaystyle-8\pi\int_{0}^{\infty}\int_{0}^{\infty}rav(r)v(a)\log{\max(r,a)}drda$
$\displaystyle=$
$\displaystyle-2\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{2\pi}rav(r)v(a)\log{\left(r^{2}+a^{2}-2ra\cos\theta\right)}d\theta
drda$

which is precisely $c_{V}$. ∎

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