# Central limit theorems for martingales-I : continuous limits

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.

When the limiting compensator of a sequence of martingales is continuous, we
obtain a weak convergence theorem for the martingales; the limiting process
can be written as a Brownian motion evaluated at the compensator and we find
sufficient conditions for both processes to be independent. Examples of
applications are provided, notably for occupation time processes and
statistical estimators of financial volatility measures.

###### 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.

##  1\. Introduction

Many central limit theorems involving a sequence of martingales are about weak
convergence to a Brownian motion, and often they are obtained by requesting
that the associated sequence of compensators converge in probability to a
deterministic function. There are also some results the sequence of
compensators converge in probability to a stochastic process, and then the
limit can be written as a mixture of a Brownian motion independent of the
limiting compensator. Such results appear naturally for occupation times of
symmetric random walks (Csáki and Földes,, 1998, 2000) through complicated
constructions of related processes.

Our goal here is to gather, under the umbrella of a single martingale central
limit theorem (CLT), a number of weak convergence results for sequences of
real-valued stochastic processes to mixtures of Brownian motions and
independent increasing processes, without relying to special properties of the
processes, nor involving complicated constructions. In fact, under weak
conditions involving sequence of martingale compensators, we will show that
the CLTs holds for general martingales, and that the limiting process is a
mixture of a Brownian motion with the limiting compensator of the sequence of
martingales, and both processes are independent. These conditions are easy to
verify and are general enough to be applicable to a wide range of situations.

The main result is stated in Section 2, while some examples involving either
independent and identically distributed (iid) sequences or Markov processes,
are found in Section 3. A longer application regarding volatility modeling in
mathematical finance is in Section 4, extending previous results of Barndorff-
Nielsen and Shephard, (2002, 2003). Some proofs are relegated to Appendix A.

##  2\. Main result

Let $D=D[0,\infty)$ be the Polish space of $\mathbb{R}^{d}$-valued càdlàg
trajectories (right continuous with left limits everywhere), equipped with the
Skorohod’s $\mathcal{J}_{1}$-topology on $D$ — consult either Ethier and
Kurtz, (1986) or Jacod and Shiryaev, (2003) for additional insight, as well as
any unexplained terminology. All processes considered here 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 $x_{t}$.

Weak convergence of a sequence of $D$-valued processes $X_{n}$ to another such
process $X$, this last with continuous trajectories, will be denoted by
$X_{n}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}X$, while the
weaker convergence of finite dimensional distributions will be denoted by
$X_{n}\stackrel{{\scriptstyle f.d.d}}{{\rightsquigarrow}}X$. In this paper the
limit $X$ always has continuous trajectories (even though the sequence of
processes $X_{n}$ may well not) so weak convergence in the
$\mathcal{J}_{1}$-topology coincides with that in the $\mathcal{C}$-topology,
induced by the supremum norm over compact time sets. We will refer to
$\mathcal{C}$-tightness etc. without further ado. All processes are written in
coordinatewise fashion such as $X_{n}=(X_{n}^{i})_{1\leq i\leq d}$. Writing
$|\cdot|$ for the Euclidean norm, square integrable processes $X$ are those
satisfying $E\\{|X(t)|^{2}\\}<\infty$ for every $t\geq 0$, while
$L_{2}$-bounded ones also satisfy $\sup_{t\geq 0}E\\{|X(t)|^{2}\\}<\infty$.

Suppose that $M_{n}$ is a sequence of $D$-valued square integrable
$\mathbb{F}$-martingales started at $M_{n}(0)=0$. Because of the discontinuity
of trajectories, the (matrix-valued or cross) quadratic variation
$[M_{n}]:=([M_{n}^{i},M_{n}^{j}])_{1\leq i,j\leq d}$ is distinct from its
matrix-valued (predictable) compensator $A_{n}:=\langle M_{n}\rangle$ —
another writing economy which means $(A_{n}^{ij})_{1\leq i,j\leq d}:=(\langle
M_{n}^{i},M_{n}^{j}\rangle)_{1\leq i,j\leq d}$ — and it is the latter that is
of interest from a practical point of view. In the literature, $A_{n}$ is also
occasionally referred to as the sharp bracket or conditional quadratic
variation. Coordinatewise we use $[M_{n}^{i}]=[M_{n}^{i},M_{n}^{i}]$ and
$\langle M_{n}^{i}\rangle=\langle M_{n}^{i},M_{n}^{i}\rangle$ as well. The
largest jump is denoted by
$J_{T}(M_{n}):=\sup_{s\in[0,T]}|M_{n}(s)-M_{n}(s-)|$. The following assumption
will be used repeatedly.

######  Hypothesis 2.1.

All of the following hold:

  1. (a)

$A_{n}^{ii}(t)=\langle M_{n}^{i}\rangle_{t}\to\infty$ as $t\to\infty$ almost
surely, for each fixed $n\geq 1$ and $i\in\\{1,2,\ldots,d\\}$;

  2. (b)

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

Writing the inverse process for $A_{n}^{ii}$ as $\tau_{n}^{i}(s)=\inf\\{t\geq
0;A_{n}^{ii}(t)>s\\}$, one defines the rescaled
$\mathbb{F}_{\tau_{n}^{i}}$-martingale $W_{n}^{i}=M_{n}^{i}\circ\tau_{n}^{i}$,
with compensator $A_{n}^{ii}\circ\tau_{n}^{i}$. Note that by definition and
using the right-continuity of $A_{n}^{ii}$,
$A_{n}^{ii}\circ\tau_{n}^{i}(t)\geq t$. Actually, $W_{n}^{i}$ is an
$\mathbb{F}_{\tau_{n}^{i}}$-Brownian motion with respect to the filtration
$\mathbb{F}_{\tau_{n}^{i}}=\\{\mathcal{F}_{\tau_{n}^{i}(t)}\\}_{t\geq 0}$,
whenever Hypothesis 2.1.a holds and $M_{n}^{i}$ is continuous everywhere, by
Dambis, (1965) or Dubins and Schwarz, (1965). In the latter case, the
continuity of $M_{n}^{i}$ implies the continuity of both $[M_{n}^{i}]$ and
$\langle M_{n}^{i}\rangle$, as well as their equality $[M_{n}^{i}]=\langle
M_{n}^{i}\rangle$. In this special case, the sequence $M_{n}$ therefore
comprises a naturally associated sequence of Brownian motions $W_{n}^{i}$
coordinatewise. This is no longer the case as soon as at least one of the
$M_{n}^{i}$’s has a discontinuity anywhere. Obtaining a CLT therefore requires
building such a Brownian motion, possibly on an enlargement of the stochastic
basis $(\Omega,\mathcal{F},\mathbb{F},P)$ with
$\mathbb{F}=(\mathcal{F}_{t})_{t\geq 0}$. Such enlargements will be used
systematically in this paper and are understood to affect some statements
implicitly, without further ado, for instance in some of the proofs.

Equality in law is denoted by $\stackrel{{\scriptstyle Law}}{{=}}$;
convergence in probability is denoted by $\stackrel{{\scriptstyle
Pr}}{{\to}}$, in law by $\stackrel{{\scriptstyle Law}}{{\to}}$, and almost
sure convergence by $\stackrel{{\scriptstyle a.s.}}{{\to}}$.

######  Theorem 2.1.

Assume that Hypothesis 2.1 holds with $A$ continuous everywhere; that
$J_{t}(M_{n})\stackrel{{\scriptstyle Law}}{{\to}}0$ 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)

$(A_{n},B_{n})\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}(A,B)$
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)

$\langle M_{n}^{i},B_{n}^{j}\rangle_{t}\stackrel{{\scriptstyle Law}}{{\to}}0$,
for any $i,j\in\\{1,\ldots,d\\}$ and $t\geq 0$.

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, $M^{i}=W^{i}\circ
A^{ii}$, $i\in\\{1,\ldots,d\\}$ holds, with $W$ a standard Brownian motion
which is independent of $B$ and $A$.

No need for $A_{n}$ to converge in probability, nor for the nested filtrations
required for stable convergence (Jacod and Shiryaev,, 2003, Section VIII.5c),
the usual way to characterize the law of $M$ uniquely. The proof is relegated
to Appendix A.

######  Remark 2.1.

An historically important prototype of Theorem 2.1 is Rebolledo’s landmark CLT
for local martingales, when restricted to sequences of square integrable
martingales (Rebolledo,, 1980) satisfying an asymptotic rarefaction of jumps
condition. Functional CLTs involving limiting mixtures with non deterministic
$A$ go back to Rebolledo, (1979, p. 92-93) for processes converging to a
diffusion in $\mathbb{R}^{d}$ and Johansson, (1994), for point process
martingales. Additional references on the early successes in the discrete case
can be found in Hall and Heyde, (1980) and in the continuous case in Jacod and
Shiryaev, (2003, Section VIII.5). More recently, Merlevède et al., (2019)
display the current state of the art of the functional CLT for rescaled non-
stationary sequences and arrays of dependent random variables when the limit
is continuous — either a Brownian motion or the stochastic integral of a
continuous function with respect to a Brownian motion.

##  3\. Examples of application to occupation times

###  3.1. Occupation times for Brownian motion

Let $B$ denote a Brownian motion, $V$ continuous with compact support, and
$\mu_{V}=\int_{-\infty}^{\infty}V(y)dy=0$ but $V$ is not identically $0$. Set
$F(x)=-2\int_{-\infty}^{x}V(y)dy$ and $G(x)=\int_{-\infty}^{x}F(y)dy$ and
consider the martingale $M_{1}(t)=G(B_{t})+\int_{0}^{t}V(B_{s})ds$ and the
occupation time $\int_{0}^{t}V(B_{s})ds$. The continuous martingale
$M_{n}(t)=n^{-1/4}\int_{0}^{nt}F(B_{s})dB_{s}$ has the same asymptotic
behavior as $n^{-1/4}\int_{0}^{nt}V(B_{s})ds$ since $G$ is bounded. Using the
scaling property of Brownian motion, it follows that
$\left\\{M_{n}(t),[M_{n}]_{t}\right\\}\stackrel{{\scriptstyle
Law}}{{=}}\left\\{\tilde{M}_{n}(t),[\tilde{M}_{n}]_{t}\right\\}$ where

$\left\\{\tilde{M}_{n}(t),[\tilde{M}_{n}]_{t}\right\\}=\left(n^{1/4}\int_{0}^{t}F(\sqrt{n}\tilde{B}_{u})d\tilde{B}_{u},n^{\frac{1}{2}}\int_{0}^{t}F^{2}(\sqrt{n}\tilde{B}_{u})du\right)$

with $\tilde{B}$ another Brownian motion. Hence

$A_{n}(t)\stackrel{{\scriptstyle
Law}}{{=}}[\tilde{M}_{n}]_{t}=\sqrt{n}\int_{\mathbb{R}}F^{2}(\sqrt{n}x)\ell_{t}(x)dx\stackrel{{\scriptstyle
a.s.}}{{\to}}\|F\|^{2}\ell_{t}(0)$

as $n\to\infty$, uniformly over compact time sets, where $\ell$ is the local
time of Brownian motion $\tilde{B}$, and

$\|F\|^{2}=\int_{\mathbb{R}}F^{2}(x)dx=-2\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}|y-z|V(y)V(z)dzdy.$

Hence
$(A_{n},B)\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}(A,B)$, with
$A_{t}=\|F\|^{2}\ell_{t}(0)$, $t\geq 0$. Using Theorem 2.1, both $M_{n}$ and
$n^{-1/4}\int_{0}^{n\cdot}V(B_{s})ds$ converge weakly to $W\circ A$, where $W$
is a Brownian motion independent of $A$ and $B$. In fact, the full consequence
of Theorem 2.1 states that
$(M_{n},A_{n},B)\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}(M,A,B)$
holds. The result for $M_{n}$ was first proven in Papanicolaou et al., (1977).
The proof given here is much easier and is similar to the one in Ikeda and
Watanabe, (1989).

The argument above for handling $A_{n}$ under $\mu_{V}=0$, carries through to
yield
$n^{-1/2}\int_{0}^{n\cdot}V(B_{s})ds\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}\mu_{V}\ell_{\cdot}(0)$
when $\mu_{V}\not=0$.

######  Remark 3.1.

Recall that $\ell_{t}(0)/\sqrt{t}\stackrel{{\scriptstyle Law}}{{=}}|B_{1}|$
which has Mittag-Leffler distribution with parameter $1/2$. Next, the inverse
local time $\tau$ is known to be a Lévy process with density
$ty^{-3/2}e^{-\frac{t^{2}}{2y}}$, $y>0$, and Laplace transform
$E\left[e^{-\lambda\tau_{t}}\right]=e^{-t\sqrt{2\lambda}}$, $\lambda\geq 0$
(Borodin and Salminen,, 2002). Hence $\tau_{t}$ has a positive stable
distribution with index $1/2$.

###  3.2. Occupation times for random walks

Let $S_{n}$ be the symmetric simple random walk on the integers $\mathbb{Z}$,
$\displaystyle N_{n}(x)=\sum_{k=1}^{n}\mathbb{I}(S_{k}=x)$ the number of its
visits to $x\in\mathbb{Z}$ up to time $n$ and $V$ a real-valued function on
$\mathbb{Z}$ with compact support but $V\not\equiv 0$. Setting
$\displaystyle\mu_{V}:=\sum_{x\in\mathbb{Z}}V(x)$, Dobrušin, (1955) proved
that if $\mu_{V}\neq 0$, then
$n^{-1/2}\sum_{k=1}^{n}V(S_{k})\stackrel{{\scriptstyle
Law}}{{\to}}\mu_{V}{\mathcal{V}}$, where $\mathcal{V}\stackrel{{\scriptstyle
Law}}{{=}}|Z|$, where $Z\sim N(0,1)$; while if $\mu_{V}=0$, then
$n^{-1/4}\sum_{k=1}^{n}V(S_{k})\stackrel{{\scriptstyle
Law}}{{\to}}\mu_{v}\sqrt{\mathcal{V}}Z$, where $Z\sim N(0,1)$ is independent
of $\mathcal{V}$, and
$\displaystyle\mu_{v}=2c_{V}^{2}-\sum_{x\in\mathbb{Z}}V^{2}(x)$, where
$\displaystyle
c_{V}^{2}=-\sum_{y,z\in\mathbb{Z}}|y-z|V(y)V(z)=2\sum_{z\in\mathbb{Z}}\left\\{\sum_{y<z}V(y)\right\\}^{2}$.
Note that $\mu_{v}$ corresponds to expression $\|V\|^{2}$ in Lee and
Rémillard, (1994).

Just as in Section 3.1, we prove that
$(V_{n},B_{n})\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}(M,B)$
ensues when $\mu_{V}=0$, where $V_{n}(t):=n^{-1/4}\sum_{k=1}^{\lfloor
nt\rfloor}V(S_{k})$, $B_{n}(t):=n^{-1/2}S_{{\lfloor nt\rfloor}}$, $B$ is a
Brownian motion, $A(t)=c_{V}^{2}\ell_{t}(0)$ with $\ell$ the local time for
$B$, and $M=W\circ A$, where $W$ is a Brownian motion independent of $A$ and
$B$.

We first build the pair $(M_{n},A_{n})$. To this end, set $\displaystyle
G(x)=-\sum_{y\in\mathbb{Z}}|x-y|V(y)$. Then

$\displaystyle TG(x)$ $\displaystyle=$
$\displaystyle\frac{G(x+1)+G(x-1)}{2}=-\sum_{y}V(y)\left\\{\frac{|y-x-1|+|y-x+1|}{2}\right\\}$
$\displaystyle=$
$\displaystyle-V(x)-\sum_{y>x}V(y)(y-x)-\sum_{y<x}V(y)(x-y)=-V(x)+G(x),$

with $G$ is constant outside the support of $V$: in fact, if $V\equiv 0$ on
$[a,b]^{\complement}$, then $G(x)=\sum_{y\in[a,b]}yV(y)=c$ if $x>b$, while if
$x<a$, then $G(x)=-c$. Consequently,

$\displaystyle M_{n}(t):$ $\displaystyle=$ $\displaystyle
n^{-1/4}\sum_{k=1}^{\lfloor nt\rfloor}\\{G(S_{k})-TG(S_{k-1})\\}$
$\displaystyle=$ $\displaystyle n^{-1/4}\left\\{G(S_{\lfloor
nt\rfloor})-G(0)\right\\}+n^{-1/4}\sum_{k=1}^{\lfloor nt\rfloor}V(S_{k-1})$

is a martingale with

$\displaystyle A_{n}(t)=\langle M_{n}\rangle_{t}$ $\displaystyle=$
$\displaystyle n^{-1/2}\sum_{k=1}^{{\lfloor
nt\rfloor}}E\left[\left\\{G(S_{k})-TG(S_{k-1})\right\\}^{2}|\mathcal{F}_{k-1}\right]$
$\displaystyle=$ $\displaystyle n^{-1/2}\sum_{k=1}^{{\lfloor
nt\rfloor}}\left[TG^{2}(S_{k-1})-\\{TG(S_{k-1})\\}^{2}\right]=n^{-1/2}\sum_{k=1}^{\lfloor
nt\rfloor}v(S_{k-1}).$

Now, since $TG=G-V$, it follows that
$v=TG^{2}-(TG)^{2}=2VG-V^{2}+TG^{2}-G^{2}$ which has compact support since
$G^{2}$ is constant outside $[a,b]$. As a result,

$\displaystyle A_{n}(t)$ $\displaystyle=$ $\displaystyle
n^{-1/2}\sum_{k=1}^{\lfloor
nt\rfloor}v(S_{k-1})=n^{-1/2}\sum_{x\in\mathbb{Z}}v(x)N_{{\lfloor
nt\rfloor}-1}(x)$ $\displaystyle=$ $\displaystyle
n^{-1/2}\sum_{x\in\mathbb{Z}}v(x)\left\\{N_{{\lfloor
nt\rfloor}-1}(x)-N_{{\lfloor
nt\rfloor}-1}(0)\right\\}+\mu_{v}n^{-1/2}N_{{\lfloor nt\rfloor}-1}(0)$
$\displaystyle=$ $\displaystyle n^{-1/4}O_{P}(1)+\mu_{v}n^{-1/2}N_{{\lfloor
nt\rfloor}}(0),$

using Dobrushin’s results and the fact that $v$ has compact support, where

$\mu_{v}=\sum_{x\in\mathbb{Z}}v(x)=-2\sum_{y,x\in\mathbb{Z}}|y-x|V(y)V(x)-\sum_{x\in\mathbb{Z}}V^{2}(x)=\sum_{x\in\mathbb{Z}}\left\\{V(x)+2H(x)\right\\}^{2},$

and $H(x)=\sum_{y<x}V(y)$. These expressions are proven in Appendix A. In
particular, if $V(a)=1$, $a\neq 0$, and $V(0)=-1$, then $\mu_{v}=4|a|-2$.

It then follows that
$(A_{n},B_{n})\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}(A,B)$,
where $B$ is a Brownian motion and $A(t)=\mu_{v}\ell_{t}(0)$ is the local time
of the Brownian motion at $0$.

Next, setting $f(x)=x$, one gets that

$\displaystyle\langle M_{n},B_{n}\rangle_{t}$ $\displaystyle=$ $\displaystyle
n^{-3/4}\sum_{k=1}^{\lfloor
nt\rfloor}\\{T(fG)(S_{k-1})-Tf(S_{k-1})TG(S_{k-1})\\}$ $\displaystyle=$
$\displaystyle n^{-3/4}\sum_{k=1}^{\lfloor nt\rfloor}g(S_{k-1}),$

where $g=T(fG)-TfTG$. Since $Tf=f$, $G(x)=c$, $x>b$ and $G(x)=-c$, $x<a$, it
follows that $g$ also has compact support and, from the previous calculations
and Dobrushin’s result, that for any $t\geq 0$, $\langle
M_{n},B_{n}\rangle_{t}=n^{-1/4}O_{P}(1)$. Therefore, using Theorem 2.1,
$(M_{n},A_{n},B_{n})\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}(W\circ
A,A,B)$, where $W$ is a Brownian motion independent of $A$ and $B$; hence
$(V_{n},A_{n},B_{n})\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}(W\circ
A,A,B)$ as well, since the above calculations yield also
$\sup_{t\in[0,T]}|V_{n}(t)-M_{n}(t)|\leq n^{-1/4}O(1)$ for any $T>0$. The
corresponding result
$n^{-1/2}V_{n}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}\mu_{V}\ell$
when $\mu_{V}\not=0$ ensues from the definition of the local time for Brownian
motion, as in the proof of
$A_{n}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}A$ in the case
$\mu_{V}=0$.

###  3.3. Occupation times on random combs

Let $(\xi_{n},\zeta_{n})$ and $(0,\psi_{n})$ be two iid sequences independent
of each other, with $P(\psi_{n}=\pm 1)=1/2$, and $P\\{(\xi_{n},\zeta_{n})=(\pm
1,0)\\}=P\\{(\xi_{n},\zeta_{n})=(0,\pm 1)\\}=1/4$. The comb process
$(C_{1},C_{2})$ (Bertacchi,, 2006, Csáki et al.,, 2009) is a martingale with
values on the integer lattice $\mathbb{Z}^{2}$, as well as a Markov chain,
started at $(0,0)$ and defined by

$\displaystyle C_{1}(n+1)$ $\displaystyle=$ $\displaystyle
C_{1}(n)+\xi_{n+1}\mathbb{I}_{\\{C_{2}(n)=0\\}},$ $\displaystyle C_{2}(n+1)$
$\displaystyle=$ $\displaystyle C_{2}(n)+\psi_{n+1}\mathbb{I}_{\\{C_{2}(n)\neq
0\\}}+\zeta_{n+1}\mathbb{I}_{\\{C_{2}(n)=0\\}}.$

Note that $C_{2}$ is also a Markov chain on its own, while $C_{1}$ is not. For
all $n\geq 0$,

$\displaystyle E\left[\\{C_{1}(n+1)-C_{1}(n)\\}^{2}|\mathcal{F}_{n}\right]$
$\displaystyle=$ $\displaystyle\frac{1}{2}\mathbb{I}_{\\{C_{2}(n)=0\\}},$
$\displaystyle E\left[\\{C_{2}(n+1)-C_{2}(n)\\}^{2}|\mathcal{F}_{n}\right]$
$\displaystyle=$ $\displaystyle 1-\frac{1}{2}\mathbb{I}_{\\{C_{2}(n)=0\\}},$
$\displaystyle
E\left[\\{C_{1}(n+1)-C_{1}(n)\\}\\{C_{2}(n+1)-C_{2}(n)\\}|\mathcal{F}_{n}\right]$
$\displaystyle=$ $\displaystyle 0.$

Set $A_{1}(n)=\sum_{k=1}^{n}\mathbb{I}_{\\{C_{2}(k)=0\\}}$ and
$\tau_{k}=\inf\\{n\geq 1;A_{1}(n)\geq k\\}$, the time of the $k$-th visit to
$0$ by $C_{2}$ after the initial departure, with defaults $A_{1}(0)=0$ and
$\tau_{0}=0$. Since the increments $\sigma_{n}=\tau_{n}-\tau_{n-1}$ are iid
and
$n^{-1/2}C_{2}([n\cdot])\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}W_{2}$,
a Brownian motion, there ensues
$n^{-1/2}2^{-1}A_{1}([n\cdot])\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}\eta_{2}$,
the local time at $0$ of $W_{2}$. Build the $\mathbb{R}^{2}$-valued martingale
$M_{n}(t):=\\{n^{-1/4}C_{1}({\lfloor nt\rfloor}),n^{-1/2}C_{2}({\lfloor
nt\rfloor})\\}$ with predictable quadratic variations
$n^{-1/2}2^{-1}A_{1}({\lfloor nt\rfloor})$ and $n^{-1}{\lfloor
nt\rfloor}-(2n)^{-1}A_{1}({\lfloor nt\rfloor})$ componentwise. Since
$J_{T}(M_{n})n^{-1/4}\to 0$ as $n\to\infty$ almost surely, the only possible
weak limits of $M_{n}$ have continuous trajectories. Further,
$\\{n^{-1/2}2^{-1}A_{1}([n\cdot]),n^{-1}[n\cdot]-(2n)^{-1}A_{1}([n\cdot])\\}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}\\{t\mapsto(\eta_{2}(t),t)\\}$,
a process with continuous trajectories. Since $M_{n}$ has uncorrelated
components, all the conditions of Theorem 2.1 are met and
$M_{n}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}(W_{1}\circ\eta_{2},W_{2})$.
Notice that if we write $X_{n}=C_{1}(\tau_{n})-C_{1}(\tau_{n-1})$, then
$(X_{n},\sigma_{n})_{n\geq 1}$ are iid.

##  4\. Volatility modeling and estimation

The financial return $R(t)$ at time $t$ for some investment under
consideration is modelled as $R(t)=Z\circ\tau(t)$, with $Z(s):=\gamma
B(s)+\beta s$, where $\tau$ is a strictly increasing process, independent of
the Brownian motion $B$. This model was first proposed by Ané and Geman,
(2000). Time scale $\tau$ is meant to reflect business cycles and other
features known collectively in economics as business time; $Z$ is thus the
financial return adjusted accordingly. Constant $\gamma$ is a scaling
parameter and constant $\beta$ the trend after the correction for business
time. Note that a different model with the same distributional features is
$R(t)=\int_{0}^{t}\sigma(s)dW(s)+\beta\tau(t)$, where $\sigma$ is a continuous
process independent of the Brownian motion $W$ and
$\tau(t)=\int_{0}^{t}\sigma^{2}(s)ds$. The latter was considered by Barndorff-
Nielsen and Shephard, (2002).

Rigorous treatment of these models requires some technical results concerning
$D$-valued square integrable $\mathbb{F}$-martingales $M$ and their
compensator $A=\langle M\rangle$. We gather these next.

Consider the filter of partitions $\\{0=s_{0,\delta}<s_{1,\delta}<\cdots\\}$,
independent of $t$, where $s_{K(t,\delta),\delta}\leq
t<s_{K(t,\delta)+1,\delta}$, with a mesh at most $\delta$, that is,
$\displaystyle\max_{1\leq k\leq
K(t,\delta)}(s_{k,\delta}-s_{k-1,\delta})\leq\delta$. Here of course
$K(t,\delta)\to\infty$ as $\delta\downarrow 0$. For each such filter,
construct

$V_{p,\delta}(t):=\delta^{1-\frac{p}{2}}\sum_{1\leq k\leq
K(t,\delta)}|M(s_{k,\delta})-M(s_{k-1,\delta})|^{p}$

and

$U_{p,\delta}(t):=\delta^{1-\frac{p}{2}}\sum_{1\leq k\leq
K(t,\delta)}\\{A(s_{k,\delta})-A(s_{k-1,\delta})\\}^{\frac{p}{2}}.$

Before stating the next result, set
$\displaystyle\mu_{p}=E(|Z|^{p})=2^{\frac{p}{2}}\dfrac{\Gamma\left(\frac{p+1}{2}\right)}{\Gamma\left(\frac{1}{2}\right)},$
where $Z\sim N(0,1)$. So $\mu_{2n}=\prod_{k=1}^{n}(2k-1)$ and
$\mu_{2n+1}=\sqrt{\frac{2}{\pi}}\cdot\prod_{k=1}^{n}(2k)$. Typically useful
values are $\mu_{2}=1$, $\mu_{4}=3$ and $\mu_{8}=105$.

######  Lemma 4.1.

Given is a real-valued martingale $M$ started at $M_{0}=0$ with finite
$p^{th}$ moment for some integer $p\geq 2$ and compensator $A$. Assume the
existence of a Brownian motion $B$ such that $M=B\circ A$, with $A$
independent of $B$. For $0\leq s\leq t<\infty$, there holds

(4.1)
$E\left\\{|M_{t}-M_{s}|^{p}|\mathcal{F}_{s}\right\\}=\mu_{p}E\left\\{(A_{t}-A_{s})^{\frac{p}{2}}|\mathcal{F}_{s}\right\\}.$

Assume that $A_{t}=\int_{0}^{t}a_{s}ds$, for some measurable non-negative $a$
which is Riemann integrable. The random process
$\mathcal{U}_{p}:=\lim_{\delta\downarrow 0}U_{p,\delta}$ is well defined in
the almost sure sense, with value given by
$\mathcal{U}_{p}(t)=\int_{0}^{t}a_{s}^{\frac{p}{2}}ds$, which is almost surely
finite and independent of the choice of filter, provided

(4.2) $\lim_{\delta\downarrow 0}\min_{1\leq k\leq
K(t,\delta)}\left(\frac{s_{k,\delta}-s_{k-1,\delta}}{\delta}\right)=1,$

for any $t>0$. If $a$ is continuous everywhere, then $\mathcal{U}_{p}$ is well
defined even if the choice of filter is expanded to the weaker requirement

(4.3) $\lim_{\delta\downarrow
0}\delta\sum_{k=1}^{K(t,\delta)}\left(\frac{s_{k,\delta}-s_{k-1,\delta}}{\delta}\right)^{\frac{p}{2}}=t,$

for any $t>0$. Furthermore, under the additional assumption that $M$ has
finite $(2p)^{th}$ moment,
$\displaystyle\mathcal{V}_{p}:=\lim_{\delta\downarrow 0}V_{p,\delta}$ is also
a random process well defined in the almost sure sense, almost surely finite
and such that the value
$\mathcal{V}_{p}(t)=\mu_{p}\mathcal{U}_{p}(t)=\mu_{p}\int_{0}^{t}a_{s}^{\frac{p}{2}}ds$
does not depend on the choice for the filter of finite partitions. In
particular, if $M_{t}=\int_{0}^{t}\sigma(s)dW(s)$, then
$\mathcal{V}_{p}(t)=\mu_{p}\int_{0}^{t}\sigma^{p}(s)ds$.

######  Remark 4.1.

This result is an extension of Barndorff-Nielsen and Shephard, (2003). They
only prove their result for a fixed $t$ with convergence in probability but
claimed it could also be true as a process. The case $p=2$ is just the
definition of quadratic variation $[M]$ — see the proof of Ethier and Kurtz,
(1986, Proposition 2.3.4), where the existence of
$\mathcal{V}_{2}:=\lim_{\delta\downarrow 0}V_{2,\delta}$ is shown to hold for
any right continuous local martingale and without any additional restriction,
neither on $A$ nor on the filter. Their proof reduces the choice of filters to
the dyadic one by first showing that any sub-filter will yield the same limit
as the original filter. This will not work here since we need a very strong
restriction on the small gaps in the mesh of the filter. Note that, when
$p>2$, the limit $\mathcal{V}_{p}$ does not exist if $M$ is not continuous
everywhere. The asymptotics for $V_{p,\delta}$ under $p\in(0,2)$ are covered
extensively by Jacod, (2007) and Jacod, (2008), for a large class of processes
with jumps, based on equally spaced observations, so the only filter involved
consists in multiples of $\delta$ — in that case $s_{k}=k\delta$, $0\leq k\leq
K(t,\delta)=\left\lfloor\frac{t}{\delta}\right\rfloor$, and (4.2) is
satisfied. The cases $p\geq 3$ are also examined in Jacod, (2008, Theorems
2.11(i)) — the presence of jumps in the limit yields a CLT with
$\delta^{-1/2}$ instead of $\delta^{1-\frac{p}{2}}$. For our continuous
limits, the larger fluctuations observed there disappear. See his comments in
Remarks 2.14, 2.15 and 2.16.

###### Proof.

Equation (4.1) proceeds from $E\left\\{|B\circ A_{t}-B\circ
A_{s}|^{p}|\sigma\\{A\\}\right\\}=\mu_{p}(A_{t}-A_{s})^{\frac{p}{2}}$ which
itself ensues from the independence of $A$ and $B$, plus the moments of a
standard normal distribution. When $p>2$, note that
$\displaystyle\int_{0}^{t}a_{s}^{\frac{p}{2}}ds\geq U_{p,\delta}$ and
$\displaystyle\int_{0}^{t}a_{s}^{\frac{p}{2}}ds=\lim_{\delta\downarrow
0}U_{p,\delta}$ by expanding

$\delta^{1-\frac{p}{2}}\\{A(s_{k,\delta})-A(s_{k-1,\delta})\\}^{\frac{p}{2}}=\delta\\{a(w_{k,\delta})\\}^{\frac{p}{2}}\left(\frac{s_{k,\delta}-s_{k-1,\delta}}{\delta}\right)^{\frac{p}{2}}\\\
\leq\left(\frac{s_{k,\delta}-s_{k-1,\delta}}{\delta}\right)^{\frac{p}{2}-1}\int_{s_{k-1,\delta}}^{s_{k,\delta}}a_{s}^{\frac{p}{2}}ds,$

for some $w_{k,\delta}\in[s_{k-1,\delta},s_{k,\delta}]$. If $a$ is continuous,
convergence is assured through the condition (4.3) on filter. If $a$ is only
Riemann integrable, then $a$ is bounded on $[0,t]$ for any $t$, and the
stronger condition (4.2) guarantees $\lim_{\delta\downarrow 0}\delta
K(t,\delta)=t$ and

$0\leq\delta\sum_{k=1}^{K(t,\delta)}\\{a(w_{k,\delta})\\}^{\frac{p}{2}}\left\\{\left(\frac{s_{k,\delta}-s_{k-1,\delta}}{\delta}\right)-\left(\frac{s_{k,\delta}-s_{k-1,\delta}}{\delta}\right)^{\frac{p}{2}}\right\\}\to
0,$

using the fact that function $f(x)=x-x^{\frac{p}{2}}$ is continuous on $[0,1]$
and decreasing on a small interval of the form $[1-\epsilon,1]$, with
$\epsilon$ small enough and $f(1)=0$. Finally,
$\displaystyle\lim_{\delta\downarrow
0}\sum_{k=1}^{K(t,\delta)}\int_{s_{k-1,\delta}}^{s_{k,\delta}}\left[a_{u}^{\frac{p}{2}}-\\{a(w_{k,\delta})\\}^{\frac{p}{2}}\right]du=0$,
since $a^{\frac{p}{2}}$ is Riemann integrable. It remains to show that the
filter of martingales $Z_{\delta}:=V_{p,\delta}-\mu_{p}U_{p,\delta}$ converges
in probability to $0$, uniformly on compact time sets — the martingale
property proceeds at once, just as in the proof of Equation (4.1). Assuming
$E\\{|M|^{2p}\\}<\infty$, the representation $M=B\circ A$ yields

$E\left\\{|M(s_{k,\delta})-M(s_{k-1,\delta})|^{p}\\{A(s_{k,\delta})-A(s_{k-1,\delta})\\}^{\frac{p}{2}}\Big{|}\mathcal{F}_{s_{k-1,\delta}}\right\\}\\\
=\mu_{p}E\left\\{\\{A(s_{k,\delta})-A(s_{k-1,\delta})\\}^{p}\Big{|}\mathcal{F}_{s_{k-1,\delta}}\right\\}.$

Expanding the square in

$E\left\\{\left(|M(s_{k,\delta})-M(s_{k-1,\delta})|^{p}-\mu_{p}\\{A(s_{k,\delta})-A(s_{k-1,\delta})\\}^{\frac{p}{2}}\right)^{2}\Big{|}\mathcal{F}_{s_{k-1,\delta}}\right\\}\\\
=(\mu_{2p}-\mu_{p}^{2})E\left\\{\\{A(s_{k,\delta})-A(s_{k-1,\delta})\\}^{p}\Big{|}\mathcal{F}_{s_{k-1,\delta}}\right\\}$

implies that $Z_{\delta}$ is square integrable with compensator $\langle
Z_{\delta}\rangle$ given by

$\langle Z_{\delta}\rangle_{t}=\delta^{2-p}(\mu_{2p}-\mu_{p}^{2})\sum_{1\leq
k\leq
K(t,\delta)}E\left\\{\\{A(s_{k,\delta})-A(s_{k-1,\delta})\\}^{p}\Big{|}\mathcal{F}_{s_{k-1,\delta}}\right\\},$

so there ensues $E\\{\langle
Z_{\delta}\rangle_{t}\\}=\delta(\mu_{2p}-\mu_{p}^{2})E\\{U_{2p,\delta}(t)\\}$
which goes to $0$ with $\delta$. By Lemma A.1, $\displaystyle\sup_{0\leq s\leq
t}|Z_{\delta}(s)|$ converges in probability to $0$ with $\delta$ as well. ∎

We can now state a first consistency result for an estimator of the realized
volatility error in investment returns, when data is collected at regular
intervals. This is an extension of the results presented in Barndorff-Nielsen
and Shephard, (2002), where convergence was limited to a fixed $t$, while here
we obtain the convergence of the whole process.

Before stating the first theorem, define

$X_{n}(t):=n^{\frac{1}{2}}\sum_{j=1}^{\lfloor
nt\rfloor}\left[\left\\{\Delta_{n}M(j/n)\right\\}^{2}-\Delta_{n}A(j/n)\right],$
$\langle X_{n}\rangle_{t}:=2n\sum_{j=1}^{\lfloor
nt\rfloor}E\left[\left\\{\Delta_{n}A(j/n)\right\\}^{2}|\mathcal{F}_{(j-1)/n}\right],$
$V_{n}(t):=n\sum_{j=1}^{\lfloor
nt\rfloor}\left\\{\Delta_{n}M(j/n)\right\\}^{4},$

with $\Delta_{n}f(s)=f(s)-f(s-1/n)$.

######  Theorem 4.2 (Numerical scheme).

Assume that $A(t)\to\infty$ and $\mathcal{V}_{4}(t)\to\infty$ as $t\to\infty$,
with $\mathcal{V}_{4}$ continuous. Under all the conditions of Lemma 4.1 with
$p=4$, including the finiteness of $E\\{|M|^{8}\\}<\infty$, there is a
standard Brownian motion $W$ independent of $\mathcal{V}_{4}$ such that

$(X_{n},\langle
X_{n}\rangle,V_{n})\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}\left\\{W\left(\dfrac{2}{3}\mathcal{V}_{4}\right),\dfrac{2}{3}\mathcal{V}_{4},\mathcal{V}_{4}\right\\}.$

Furthermore, for any adapted $D$-valued process $N$ such that $\displaystyle
n^{\frac{1}{2}}\sum_{j=1}^{\lfloor
nt\rfloor}\left\\{\Delta_{n}N(j/n)\right\\}^{2}\stackrel{{\scriptstyle
Pr}}{{\to}}0$ and $n\sum_{j=1}^{\lfloor
nt\rfloor}\\{\Delta_{n}N(j/n)\\}^{2}\Delta_{n}A(j/n)\stackrel{{\scriptstyle
Pr}}{{\to}}0$ both hold for all $t>0$, there also comes
$Y_{n}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}W\left(\dfrac{2}{3}\mathcal{V}_{4}\right)$,
where

$Y_{n}(t):=n^{\frac{1}{2}}\sum_{j=1}^{\lfloor
nt\rfloor}\left[\left\\{\Delta_{n}(M+N)(j/n)\right\\}^{2}-\Delta_{n}A(j/n)\right].$

###### Proof.

By Lemma 4.1, $V_{n}$ converges to $\mathcal{V}_{4}$ almost surely and hence
$V_{n}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}\mathcal{V}_{4}$
holds, by Proposition A.4. The expression for $\langle X_{n}\rangle$ is a
direct consequence of Lemma 4.1, which entails
$E\left[\left\\{\Delta_{n}M(j/n)\right]^{4}|\mathcal{F}_{(j-1)/n}\right]=3E\left[\left\\{\Delta_{n}A(j/n)\right]^{2}|\mathcal{F}_{(j-1)/n}\right]$,
while the representation $M=B\circ A$ yields

$E\left[\left\\{\Delta_{n}M(j/n)\right]^{2}\Delta_{n}A(j/n)|\mathcal{F}_{(j-1)/n}\right]=E\left[\left\\{\Delta_{n}A(j/n)\right]^{2}|\mathcal{F}_{(j-1)/n}\right].$

Also, $V_{n}=\frac{3}{2}\langle X_{n}\rangle+\mathcal{Z}_{n}$, where

$\mathcal{Z}_{n}(t)=n\sum_{j=1}^{\lfloor
nt\rfloor}\Bigl{[}\\{\Delta_{n}M(j/n)\\}^{4}-E\left[\\{\Delta_{n}M(j/n)\\}^{4}|\mathcal{F}_{(j-1)/n}\right]\Bigr{]}$

is a martingale with

$\langle\mathcal{Z}_{n}\rangle_{t}=\mu_{8}n^{2}\sum_{k=1}^{\lfloor
nt\rfloor}E\left[\left\\{\Delta_{n}A(j/n)\right]^{4}|\mathcal{F}_{(j-1)/n}\right]-\mu_{4}^{2}n^{2}\sum_{k=1}^{\lfloor
nt\rfloor}E^{2}\left[\left\\{\Delta_{n}A(j/n)\right]^{2}|\mathcal{F}_{(j-1)/n}\right].$

By Lemma 4.1 with $p=8$, $nE\\{\langle\mathcal{Z}_{n}\rangle_{t}\\}$ is
bounded above by sequence $E\\{U_{p,1/n}(t)\\}$ which converges to
$E\\{\mathcal{V}_{8}(t)\\}$ and hence
$\langle\mathcal{Z}_{n}\rangle\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}0$
holds, by Proposition A.4. By Lemma A.1, $\displaystyle\sup_{0\leq t\leq
T}|\mathcal{Z}_{n}(t)|$ converges in probability to $0$. This implies $\langle
X_{n}\rangle\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}\dfrac{2}{3}\mathcal{V}_{4}$.
Since $\langle X_{n}\rangle$ satisfies Hypothesis 2.1, Theorem 2.1 can be
applied to martingale $X_{n}$, with $B_{n}\equiv M\circ A^{-1}$ and therefore
$A$ is $\mathbb{F}_{B}$-measurable. Finally, writing

$Z_{n}(t):=n^{\frac{1}{2}}\sum_{j=1}^{\lfloor
nt\rfloor}\Delta_{n}N(j/n)\cdot\mathbb{I}\\{(j-1)<nt\leq j\\}$

yields square integrable martingale

$\int_{0}^{t}Z_{n}(s)dM(s)=n^{\frac{1}{2}}\sum_{j=1}^{\lfloor
nt\rfloor}\Delta_{n}M(j/n)\Delta_{n}N(j/n)$

with

$\int_{0}^{t}Z_{n}^{2}(s)ds=\sum_{j=1}^{\lfloor
nt\rfloor}\\{\Delta_{n}N(j/n)\\}^{2}$

and quadratic variation

$\left\langle\int_{0}^{t}Z_{n}(s)dM(s)\right\rangle=\int_{0}^{t}Z_{n}^{2}(s)d\langle
M\rangle_{t}=n\sum_{j=1}^{\lfloor
nt\rfloor}\\{\Delta_{n}N(j/n)\\}^{2}\Delta_{n}A(j/n).$

Therefore
$Y_{n}(t)-X_{n}(t)=n^{\frac{1}{2}}\int_{0}^{t}Z_{n}^{2}(s)ds+2\int_{0}^{t}Z_{n}(s)dM(s)\stackrel{{\scriptstyle
Pr}}{{\to}}0$ holds for every $t>0$, since $\int_{0}^{t}Z_{n}^{2}(s)d\langle
M\rangle_{t}\stackrel{{\scriptstyle Pr}}{{\to}}0$. Hence
$Y_{n}-X_{n}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}0$ holds
as well, yielding the last statement for $Y_{n}$ via Lemma A.1, Theorem A.3
and Proposition A.4. ∎

A frequent choice for perturbation process $N$ is a linear function of the
volatility. Recall the modulus of continuity of $A$, defined by

$\omega_{\mathbb{C}}(A,\delta,T)=\sup_{0\leq t_{1}<t_{2}\leq
T,\;t_{2}-t_{1}<\delta}\|A(t_{2})-A(t_{1})\|.$

######  Corollary 4.3.

Suppose that $N(t)=\mu t+\beta A(t)$, for some constants $\mu\in\mathbb{R}$
and $\beta\geq 0$, and that the modulus of continuity of $A$ satisfies
$\delta^{-\alpha}\omega_{\mathbb{C}}(A,\delta,t)\stackrel{{\scriptstyle
Pr}}{{\to}}0$ as $\delta\to 0$, for some $\alpha>3/4$ and all $t>0$. The
conclusions of Theorem 4.2 hold.

###### Proof.

For the terms in $A$, the two conditions on $N$ are a consequence of

$\displaystyle n^{\frac{1}{2}}\sum_{j=1}^{\lfloor
nt\rfloor}\\{\Delta_{n}A(j/n)\\}^{2}+n\sum_{j=1}^{\lfloor
nt\rfloor}\\{\Delta_{n}A(j/n)\\}^{3}\leq
tn^{3/2}\omega_{\mathbb{C}}^{2}(A,1/n,t)+tn^{2}\omega_{\mathbb{C}}^{3}(A,1/n,t).$

The terms in $\mu$ are treated similarly. ∎

This numerical scheme extends readily to higher powers. We state it without
proof, leaving the details to the reader.

######  Theorem 4.4 (New numerical scheme).

Assume that $A(t)\to\infty$ and $\mathcal{V}_{p}(t)\to\infty$ as $t\to\infty$,
with $\mathcal{V}_{p}$ continuous, for some $p>4$. Under all the conditions of
Lemma 4.1, including the finiteness of the $(2p)^{th}$ moment of $M$, there is
a standard Brownian motion $W$ independent of $\mathcal{V}_{p}$ such that

$(X_{n,p},\langle
X_{n,p}\rangle,V_{n,p})\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}\left\\{W\left(\dfrac{(\mu_{p}-\mu_{\frac{p}{2}}^{2})}{\mu_{p}}\mathcal{V}_{p}\right),\dfrac{(\mu_{p}-\mu_{\frac{p}{2}}^{2})}{\mu_{p}}\mathcal{V}_{p},\mathcal{V}_{p}\right\\},$

for martingale $X_{n,p}$, where $a=\frac{p}{2}-1$

$\displaystyle X_{n,p}(t)$ $\displaystyle:=$ $\displaystyle
n^{\frac{p-2}{4}}\sum_{j=1}^{\lfloor
nt\rfloor}\left[\left|\Delta_{n}M(j/n)\right|^{\frac{p}{2}}-\mu_{\frac{p}{2}}\\{\Delta_{n}A(j/n)\\}^{p/4}\right],$
$\displaystyle\langle X_{n,p}\rangle_{t}$ $\displaystyle:=$
$\displaystyle\left(\mu_{p}-\mu_{\frac{p}{2}}^{2}\right)n^{\frac{p}{2}-1}\sum_{j=1}^{\lfloor
nt\rfloor}E\left[\left\\{\Delta_{n}A(j/n)\right\\}^{\frac{p}{2}}|\mathcal{F}_{(j-1)/n}\right],$
$\displaystyle V_{n,p}(t)$ $\displaystyle:=$ $\displaystyle
n^{\frac{p}{2}-1}\sum_{j=1}^{\lfloor
nt\rfloor}\left|\Delta_{n}M(j/n)\right|^{p}.$

######  Remark 4.2.

In their analysis of the asymptotic properties of realized volatility error in
investment returns, Barndorff-Nielsen and Shephard, (2002) assumed that
$M_{t}=\int_{0}^{t}\sigma_{u}dW_{u}$, with square integrable $\sigma$
independent of Brownian motion $W$ — $\sigma_{u}^{2}$ is known as the spot
volatility or instantaneous volatility at time $u$. Since
$A_{t}=\int_{0}^{t}\sigma_{u}^{2}du$ is continuous, it follows that $B=M\circ
A^{-1}$ is a Brownian motion. If $\mathcal{A}=\sigma\\{A_{t};t\geq 0\\}$, and
$t_{0}=0\leq t_{1}\cdots\leq t_{m}$, then

$E\left\\{\left.e^{i\sum_{k=1}^{m}\lambda_{k}(B_{t_{k}}-B_{t_{k-1}})}\right|\mathcal{A}\right\\}=e^{-\frac{1}{2}\sum_{k=1}^{m}\lambda_{k}^{2}\left(A\circ
A_{t_{k}}^{-1}-A\circ
A_{t_{k-1}}^{-1}\right)}=e^{-\frac{1}{2}\sum_{k=1}^{m}\lambda_{k}^{2}\left(t_{k}-t_{k-1}\right)},$

because $W$ and $\sigma$ are independent. As a result, $M=B\circ A$ with $B$
independent of $A$. They also said that one could consider $N(t)=\mu t+\beta
A(t)$ but mentioned that it could be difficult to obtain $p$-variation
convergence, specially if $\beta\neq 0$. From our Corollary 4.3, we see that
having this extra term does not influence the value of the limit
$W\left(\dfrac{2}{3}\mathcal{V}_{4}\right)$ for $Y_{n}$, confirming that they
can be ignored when estimating realized volatility, at least below the order
of third moments. Note that their setting is a modification of the models
introduced in Ané and Geman, (2000) by adding the drift term $\mu$, while
setting business time to $\tau=A$ and scaling parameter to $\gamma=1$. Note
also that from Theorem 4.2, Barndorff-Nielsen and Shephard, (2002, Theorem 1)
can be restated as follows: as $n\to\infty$,
$\displaystyle\dfrac{X_{n}(1)}{\left\\{2n\sum_{j=1}^{n}\\{\Delta_{n}A(j/n)\\}^{2}\right\\}^{1/2}}\stackrel{{\scriptstyle
Law}}{{\to}}N(0,1)$ and $\displaystyle
n\sum_{j=1}^{n}\\{\Delta_{n}A(j/n)\\}^{2}\stackrel{{\scriptstyle
a.s.}}{{\to}}\int_{0}^{1}\sigma^{4}(s)ds:=\mathcal{U}_{4}(1)$. Furthermore,
$\displaystyle 2n\sum_{j=1}^{\lfloor
nt\rfloor}\\{\Delta_{n}A(j/n)\\}^{2}-\langle X_{n}\rangle_{t}$ is a martingale
that converges to $0$, due to the continuity of $A$ and the fact that
$\sigma^{4}$ is locally integrable. As a result,
$\mathcal{V}_{4}=3\mathcal{U}_{4}$. Their result is therefore an upshot of
Theorem 4.2.

More generally, Barndorff-Nielsen et al., (2006) prove a CLT for sequences of
continuous $\mathbb{R}^{d}$-valued semimartingales $Y$ of the following
general form — we stick to the case $d=1$ here for the sake of simplicity:

$Y(t)=Y(0)+\int_{0}^{t}a_{s}ds+\int_{0}^{t}\sigma_{s-}dW(s)$

with $W$ a standard Brownian motion, $a$ bounded predictable and $\sigma$
$D$-valued. For any pair $G$ and $H$ of continuous real-valued functions with
at most polynomial growth, the sequence of approximations

$X_{n}(G,H)_{t}=n^{-1}\sum_{j=1}^{\lfloor
nt\rfloor}G\left\\{\sqrt{n}\Delta_{n}Y(j/n)\right\\}\cdot
H\left\\{\sqrt{n}\Delta_{n}Y((j+1)/n)\right\\}$

are first shown to obey a Law of Large Numbers :

$X_{n}(G,H)_{t}\stackrel{{\scriptstyle
Pr}}{{\to}}X(G,H)_{t}:=\int_{0}^{t}\rho_{\sigma_{s}}(G)\rho_{\sigma_{s}}(H)ds$

where $\rho_{\sigma_{s}}(G):=E\\{G(Z)\\}$ where $Z\sim N(0,\sigma_{s})$. Under
some additional restrictions on both the stochastic structure of $\sigma_{s}$
and the smoothness of $G$ and $H$, a CLT also ensues — actually in the sense
of stable convergence:

$\sqrt{n}\\{X_{n}(G,H)_{t}-X(G,H)_{t}\\}\stackrel{{\scriptstyle
Law}}{{\to}}U(G,H)_{t}$

where $U(G,H)_{t}$ is a stochastic integral with respect to another Brownian
motion independent of the ambient filtration. A functional version of this
result should ensue from our Theorem 4.4 through the same type of arguments,
when $G$ and $H$ have at most polynomial growth. We do not pursue this here.

##  5\. Conclusion

We have shown that under weak conditions involving compensators, one can get a
CLT for general martingales, and the limiting process is a mixture of a
Brownian motion with the limiting compensator of the sequence of martingales.
These conditions are easy to verify and are general enough to be applicable to
a wide range of situations.

##  Appendix A Auxiliary results and main proofs

###  A.1. Expression for $\mu_{v}$

Since $\displaystyle\sum_{a\leq x\leq b}\\{TG^{2}(x)-G^{2}(x)\\}=0$, it
follows that

$\mu_{v}=\sum_{x\in\mathbb{Z}}v(x)=\sum_{x\in\mathbb{Z}}\left\\{2V(x)G(x)-V^{2}(x)\right\\}.$

One can check that
$\displaystyle\sum_{y}\sum_{z}|z-y|V(y)V(z)=4\sum_{y}yV(y)H(y)+2\sum_{y}yV^{2}(y)$,
where $\displaystyle H(x)=\sum_{y<x}V(y)$. Note that $H\equiv 0$ outside
$[a,b]$. Now, $G(x)=-2xH(x)+2\sum_{y<x}yV(y)-c$. As a result,

$\displaystyle\mu_{v}$ $\displaystyle=$
$\displaystyle-4\sum_{x}xV(x)H(x)+4\sum_{x}\sum_{y<x}yV(y)V(x)-\sum_{x}V^{2}(x)$
$\displaystyle=$
$\displaystyle-8\sum_{x}xV(x)H(x)-4\sum_{x}xV^{2}(x)-\sum_{x}V^{2}(x),$

which is the expression of Equation 10 in Dobrušin, (1955). Furthermore, since
$\displaystyle
0=\sum_{x}V^{2}(x)+2\sum_{x}\sum_{y<x}V(y)V(x)=\sum_{x}V^{2}(x)+2\sum_{x}V(x)H(x)$,
one also gets

$2\sum_{x}H^{2}(x)=2\sum_{a\leq x\leq b}\sum_{a\leq y<x}\sum_{a\leq
z<x}V(y)V(z)\\\ =2\sum_{a\leq y\leq b}\sum_{a\leq z\leq b}V(y)V(z)\sum_{b\geq
x>\max(y,z)}=2\sum_{a\leq y\leq b}\sum_{a\leq z\leq
b}V(y)V(z)\\{b-\max(y,z)\\}\\\
=4\sum_{z}\sum_{y<z}V(y)V(z)(b-z)+2\sum_{y}V^{2}(y)(b-y)=-4\sum_{z}zH(z)V(z)-2\sum_{y}yV^{2}(y),$

proving the expressions $c_{V}^{2}=2\sum_{x}H^{2}(x)$ and $\displaystyle
2c_{V}^{2}=\mu_{v}+\sum_{x}V^{2}(x)$. Note that $\mu_{v}=2B_{V}$, where
$\displaystyle
B_{V}=\frac{1}{2}\sum_{x\in\mathbb{Z}}\left\\{2H(x)+V(x)\right\\}^{2}$
(Rémillard,, 1990).

###  A.2. Some useful results

######  Lemma A.1 (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.1)
$P(\sup_{s\leq\tau}|X(s)|\geq\varepsilon)\leq\frac{\eta}{\varepsilon}+P(Y(\tau)\geq\eta).$

  * b)

if $Y$ is adapted,

(A.2)
$P(\sup_{s\leq\tau}|X(s)|\geq\varepsilon)\leq\frac{1}{\varepsilon}\left[\eta+E\left\\{J_{\tau}(Y)\right\\}\right]+P(Y(\tau)\geq\eta).$

###### Proof.

See Jacod and Shiryaev, (2003, Lemma I.3.30). ∎

Proving $\mathcal{J}_{1}$-tightness generally involves the following lemma.

######  Lemma A.2 (Aldous’s criterion).

Let $\\{X_{n}\\}_{n\geq 1}$ be a sequence of $D$-valued processes. Suppose
that for any sequence of bounded discrete stopping times
$\left\\{{\tau_{n}}\right\\}_{n\geq 1}$ and for any sequence
$\left\\{{\delta_{n}}\right\\}_{n\geq 1}$ in $[0,1]$ converging to $0$, the
following condition holds, for every $T>0$: (A)
$X_{n}((\tau_{n}+\delta_{n})\wedge T)-X_{n}(\tau_{n})\stackrel{{\scriptstyle
Law}}{{\to}}0$. Then, $\left\\{{X_{n}}\right\\}_{n\geq 1}$ is
$\mathcal{J}_{1}$-tight, if either of the two following conditions holds:

  1. (1)

$\\{X_{n}(0)\\}_{n\geq 1}$ and $(J_{T}(X_{n}))_{n\geq 1}$ are tight;

  2. (2)

$\\{X_{n}(t)\\}_{n\geq 1}$ is tight for any $t\in[0,T]$.

###### Proof.

See Aldous, (1978), Jacod and Shiryaev, (2003, Theorem VI.4.5) or for several
variants see Ethier and Kurtz, (1986, Theorem 3.8.6). Note that condition (1)
or condition (2) are necessary for $\mathcal{J}_{1}$-tightness, but not
condition (A). ∎

Now comes the main result about tightness, stated for real-valued processes.

######  Theorem A.3.

Let $M_{n}$ be a sequence of $D$-valued square integrable
$\mathbb{F}$-martingales with predictable quadratic variation $A_{n}=\langle
M_{n}\rangle$, such that $(M_{n}(0))_{n\geq 1}$ is tight.

  * a)

If every $M_{n}$ is continuous, then $M_{n}$ is $\mathcal{C}$-tight if and
only if $A_{n}$ is $\mathcal{C}$-tight.

  * b)

In general, $M_{n}$ is $\mathcal{C}$-tight if and only if $[M_{n}]$ is
$\mathcal{C}$-tight.

  * c)

Assume that for any $t>0$, $\limsup_{n\to\infty}E\\{A_{n}(t)\\}<\infty$ and
$J_{t}(M_{n})\stackrel{{\scriptstyle Law}}{{\to}}0$ hold. If either $A_{n}$ or
$[M_{n}]$ is $\mathcal{C}$-tight, then they both are.

###### Proof.

Part a) goes back to Rebolledo, (1979). For the sake of completeness, we
supply a very short and elegant proof here. For statement a) the continuity of
$M_{n}$ implies that of $M_{n}^{2}$, $[M_{n}]$ and $A_{n}$, so that
$E\\{M_{n}^{2}(T)\\}=E[M_{n}]_{T}=E\\{A_{n}(T)\\}$ for every $T\geq 0$.
Suppose that $M_{n}$ is $\mathcal{C}$-tight, so that $M_{n}^{2}$ also is.
Apply Lemma A.1 with $X(t)=\langle M_{n}\rangle_{t+\tau_{n}}-\langle
M_{n}\rangle_{\tau_{n}}$ and $Y(t)={\sup_{0\leq s\leq
t}}\\{M_{n}(s+\tau_{n})-M_{n}(\tau_{n})\\}^{2}$, where $\tau_{n}$ is a
stopping time uniformly bounded by $T$ for any $n$. For any bounded stopping
time $\tau$, $E\\{X(\tau)\\}\leq E\\{Y(\tau)\\}$. Let $\delta_{n}$ be a
sequence of positive numbers bounded by $1$ and converging to zero. According
to Lemma A.1, we have, for any $\eta>0$,

$\displaystyle P(\langle M_{n}\rangle_{\tau_{n}+\delta_{n}}-\langle
M_{n}\rangle_{\tau_{n}}\geq\varepsilon)$ $\displaystyle\leq$
$\displaystyle\frac{\eta}{\varepsilon^{2}}+P\left(|M_{n}(\tau_{n}+\delta_{n})-M_{n}(\tau_{n})|\geq\sqrt{\eta}\right)$
$\displaystyle\leq$
$\displaystyle\frac{\eta}{\varepsilon^{2}}+P\left\\{\omega_{\mathbb{C}}(M_{n},\delta_{n},T+1)>\sqrt{\eta}\right\\}.$

Since $M_{n}$ is $\mathcal{C}$-tight, it follows that
$P\left\\{\omega_{\mathbb{C}}(M_{n},\delta_{n},T+1)>\sqrt{\eta}\right\\}\to 0$
as $n\to\infty$. Set $\eta=\varepsilon^{3}$. Then
$\limsup_{n\to\infty}P\left\\{\langle
M_{n}\rangle_{\tau_{n}+\delta_{n}}-\langle
M_{n}\rangle_{\tau_{n}}\geq\varepsilon\right\\}\leq\varepsilon$, showing that
the conditions of Lemma A.2 are met. Hence $\langle M_{n}\rangle$ is
$\mathcal{J}_{1}$-tight and, since the trajectories are continuous, also
$\mathcal{C}$-tight.

On the other hand, suppose that $\langle M_{n}\rangle$ is $\mathcal{C}$-tight.
Since $M_{n}$ is continuous, condition (1) of Lemma A.2 is met. Setting
$X=\left\\{M_{n}(s+\tau_{n})-M_{n}(\tau_{n})\right\\}^{2}$ and $Y=\langle
M_{n}\rangle_{s+\tau_{n}}-\langle M_{n}\rangle_{\tau_{n}}$, we have
$E\\{X(\tau)\\}\leq E\\{Y(\tau)\\}$, for any bounded stopping time $\tau$.

Using Lemma A.1, we have, for any $\delta_{n},\varepsilon,\eta>0$,

$P\left\\{|M_{n}(\tau_{n}+\delta_{n})-M_{n}(\tau_{n})|\geq\varepsilon\right\\}\leq
P\left\\{\langle M_{n}\rangle_{\tau_{n}+\delta_{n}}-\langle
M_{n}\rangle_{\tau_{n}}>\eta\right\\}+\frac{\eta}{\varepsilon^{2}}.$

Now $\langle M_{n}\rangle_{\tau_{n}+\delta_{n}}-\langle
M_{n}\rangle_{\tau_{n}}\leq\omega_{\mathbb{C}}\left(\langle
M_{n}\rangle,\delta_{n},T+1\right)$ tends to $0$ as $n\to\infty$ since
$\langle M_{n}\rangle$ is $\mathcal{C}$-tight. Choosing
$\eta=\varepsilon^{3}$,
$\limsup_{n\to\infty}P\left\\{|M_{n}(\tau_{n}+\delta_{n})-M_{n}(\tau_{n})|\geq\varepsilon\right\\}\leq\varepsilon$
follows, showing that condition (2) of Lemma A.2 is met. Hence $M_{n}$ is
$\mathcal{J}_{1}$-tight and, since the trajectories are continuous, also
$\mathcal{C}$-tight.

Part b) is a consequence of Ethier and Kurtz, (1986, Theorem 3.10.2) together
with
$\omega_{\mathbb{C}}(M_{n},\delta,T)^{2}\leq\omega_{\mathbb{C}}([M_{n}],\delta,T)=\sup_{0\leq
t\leq T}\\{[M_{n}]_{T\wedge(t+\delta)}-[M_{n}]_{t}\\}$, which holds for any
$T>0$ and $\delta>0$. The reverse implication is done just as in Part a).

Part c) was first proved by Rebolledo, (1979, Theorem II.3.2). When $A_{n}$ is
assumed to be $\mathcal{C}$-tight, any subsequence possesses a further
subsequence (which we also write as $A_{n}$) such that
$A_{n}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}A$ as well for
some continuous process $A$. Next, set
$X_{n}(t)=[M_{n}]_{\tau_{n}+t}-[M_{n}]_{\tau_{n}}$ and
$Y_{n}(t)=A_{n}(\tau_{n}+t)-A_{n}(\tau_{n})$, where $\tau_{n}$ is a sequence
of bounded stopping times. For any $\epsilon>0$, $\eta>0$ and any numerical
sequence $\delta_{n}\in(0,1)\to 0$,
$E\\{X_{n}(\delta_{n})\\}=E\\{Y_{n}(\delta_{n})\\}$ implies

$P([M_{n}]_{\tau_{n}+\delta_{n}}-[M_{n}]_{\tau_{n}}\geq\varepsilon)\leq\frac{1}{\varepsilon}\left[\eta+E\left\\{J_{\delta_{n}}(Y_{n})\right\\}\right]+P(Y_{n}(\delta_{n}))\geq\eta),$

using (A.2) of Lemma A.1, with $J_{\delta_{n}}(Y_{n})\leq
Y_{n}(\delta_{n})\leq\omega_{\mathbb{C}}(A_{n},\delta_{n},T)$, since $A_{n}$
is nondecreasing, provided $\tau_{n}+\delta_{n}\leq T$ holds for all $n$.
Therefore, the continuity of the limit $A$ implies both
$Y_{n}(\delta_{n})\stackrel{{\scriptstyle Law}}{{\to}}0$ and
$\limsup_{n\to\infty}E\\{Y_{n}(\delta_{n})\\}=0$, this last since
$\limsup_{n\to\infty}E\\{A_{n}(t)\\}<\infty$. For this subsequence $A_{n}$, by
Lemma A.2, the corresponding subsequence $[M_{n}]$ is $\mathcal{J}_{1}$-tight.
Adding $J_{t}(M_{n})\stackrel{{\scriptstyle Law}}{{\to}}0$ yields
$\mathcal{C}$-tightness. Role reversal of $A_{n}$ and $[M_{n}]$ proceeds
similarly. ∎

######  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
$A_{n}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}A$.

###### Proof.

For any fixed $T>0$ and $\delta>0$ there holds, with $i$ running only through
the non-negative integers,

$\limsup_{n\to\infty}P(\omega_{\mathbb{C}}(A_{n},\delta,T)\geq\epsilon)\\\
\leq\limsup_{n\to\infty}P\left(3\max_{0\leq i\leq T/\delta}\sup_{i\delta\leq
s\leq T\wedge(i+1)\delta}|A_{n}(s)-A_{n}(i\delta)|\geq\epsilon\right)\\\
\leq\limsup_{n\to\infty}P\left(\bigcup_{0\leq i\leq
T/\delta}\left\\{|A_{n}(T\wedge(i+1)\delta)-A_{n}(i\delta)|\geq\epsilon/3\right\\}\right)\\\
\leq P\left(\bigcup_{0\leq i\leq
T/\delta}\left\\{|A(T\wedge(i+1)\delta)-A(i\delta)|\geq\epsilon/3\right\\}\right)\\\
\leq P(\omega_{\mathbb{C}}(A,\delta,T)\geq\epsilon/3),$

using convergence in law. Hence $A_{n}$ is $\mathcal{C}$-tight since $A$ is
continuous. ∎

###  A.3. Proof of Theorem 2.1

By Proposition A.4, the continuity of $A$ implies that
$A_{n}^{ii}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}A^{ii}$
holds for every $i$. By Jacod and Shiryaev, (2003, Theorem I.4.2), one
concludes
$(A_{n}^{ij}(t)-A_{n}^{ij}(s))^{2}\leq(A_{n}^{ii}(t)-A_{n}^{ii}(s))(A_{n}^{jj}(t)-A_{n}^{jj}(s))$
almost surely, for every choice of $i$ and $j$, so each $A_{n}^{ij}$ is
$\mathcal{C}$-tight and hence
$A_{n}\stackrel{{\scriptstyle\mathcal{C}}}{{\rightsquigarrow}}A$ as well.
Next, set $X_{n}^{i}(t)=[M_{n}^{i}]_{\tau_{n}+t}-[M_{n}^{i}]_{\tau_{n}}$ and
$Y_{n}^{i}(t)=A_{n}^{ii}(\tau_{n}+t)-A_{n}^{ii}(\tau_{n})$, where $\tau_{n}$
is a sequence of bounded stopping times. The filtration of interest here is
$\\{\mathcal{F}_{\tau_{n}+t}\\}_{t\geq 0}$. For any $\epsilon>0$, $\eta>0$ and
any numerical sequence $\delta_{n}\in(0,1)\to 0$,
$EX_{n}^{i}(\delta_{n})=EY_{n}^{i}(\delta_{n})$ implies

(A.3)
$P([M_{n}^{i}]_{\tau_{n}+\delta_{n}}-[M_{n}^{i}]_{\tau_{n}}\geq\varepsilon)\leq\frac{1}{\varepsilon}\left[\eta+E\left\\{J_{\delta_{n}}(Y_{n}^{i})\right\\}\right]+P(Y_{n}^{i}(\delta_{n}))\geq\eta),$

using (A.2) of Lemma A.1, with $J_{\delta_{n}}(Y_{n}^{i})\leq
Y_{n}^{i}(\delta_{n})\leq\omega_{\mathbb{C}}(A_{n}^{ii},\delta_{n},T)$, since
$A_{n}^{ii}$ is nondecreasing. Therefore, the continuity of the limit $A^{ii}$
implies $Y_{n}^{i}(\delta_{n})\stackrel{{\scriptstyle Law}}{{\to}}0$ while
$E\left\\{Y_{n}^{i}(\delta_{n})\right\\}\to 0$ follows from Hypothesis
2.1.b(ii) by the dominated convergence theorem Ethier and Kurtz, (1986,
Proposition App1.2). By Theorem A.3, $[M_{n}^{i}]$ is $\mathcal{J}_{1}$-tight.
Since $J_{t}([M_{n}^{i}])\stackrel{{\scriptstyle Law}}{{\to}}0$ is assumed,
$[M_{n}^{i}]$ is actually $\mathcal{C}$-tight and hence so are $M_{n}^{i}$ and
$(M_{n}^{i},A_{n}^{ii},[M_{n}^{i}])$ successively. For every choice of $i$ and
$j$, inequality
$([M_{n}^{i},M_{n}^{j}]_{t}-[M_{n}^{i},M_{n}^{j}]_{s})^{2}\leq([M_{n}^{i}]_{t}-[M_{n}^{i}]_{s})([M_{n}^{j}]_{t}-[M_{n}^{j}]_{s})$
ensures the individual $\mathcal{C}$-tightness of each
$[M_{n}^{i},M_{n}^{j}]$. The $\mathcal{C}$-tightness of matrix $[M_{n}]$
follows, implying that of $M_{n}$ and $(M_{n},A_{n},[M_{n}])$.

By Skorohod’s theorem, e.g., Ethier and Kurtz, (1986, Theorem 3.1.8), there
exists a subsequence $\\{n_{k}\\}$, a probability space
$(\Omega^{\prime},{\mathcal{F}}^{\prime},P^{\prime})$, and $D$-valued
processes
$Z^{\prime}_{n_{k}}:=(M^{\prime}_{n_{k}},B^{\prime}_{n_{k}},A^{\prime}_{n_{k}})$
and $Z^{\prime}:=(M^{\prime},B^{\prime},A^{\prime})$ defined on
$(\Omega^{\prime},{\mathcal{F}}^{\prime},P^{\prime})$ which are such that
$Z^{\prime}_{n_{k}}$ and $Z_{n_{k}}:=(M_{n_{k}},B_{n_{k}},\langle
M_{n_{k}}\rangle)$ are identical in law for all $k\geq 1$, and
$Z^{\prime}_{n_{k}}$ converges almost surely to $Z^{\prime}$ uniformly on
compact time sets. We next prove that, for any such limit point
$Z^{\prime}=(M^{\prime},B^{\prime},A^{\prime})$, $M^{\prime}$ and $B^{\prime}$
are both martingales with respect to the natural filtration
$\mathcal{F}_{t}^{\prime}=\sigma\\{M^{\prime}(s),B^{\prime}(s),A^{\prime}(s);s\leq
t\\}$, such that $\langle M^{\prime}\rangle=\langle
B^{\prime}\rangle=A^{\prime}$, and $\langle
M^{\prime},B^{\prime}\rangle_{t}\equiv 0$.

For every $\ell\geq 1$, every bounded open set
$O\subset\mathbb{R}^{\ell(d+d+d^{2})}$ and each selection of $0\leq
s_{1}\leq\cdots\leq s_{\ell}\leq s\leq t$, write

$\displaystyle E_{n_{k}}$ $\displaystyle:=$
$\displaystyle\\{Z_{n_{k}}(s_{1}),\ldots,Z_{n_{k}}(s_{\ell})\\}^{-1}O,$
$\displaystyle E^{\prime}_{n_{k}}$ $\displaystyle:=$
$\displaystyle\\{Z^{\prime}_{n_{k}}(s_{1}),\ldots,Z^{\prime}_{n_{k}}(s_{\ell})\\}^{-1}O,$
$\displaystyle E^{\prime}$ $\displaystyle:=$
$\displaystyle\\{Z^{\prime}(s_{1}),\ldots,Z^{\prime}(s_{\ell})\\}^{-1}O.$

For $M^{\prime}$ to be a $\mathcal{F}^{\prime}$-martingale, it suffices to
prove
$E\left\\{\left(M^{\prime}(t)-M^{\prime}(s)\right)\mathbb{I}_{E^{\prime}}\right\\}=0$.
Without loss of generality, we do so coordinatewise (hence $d=1$) and drop the
superscript $i$ for the rest of the proof, in order to keep notation to a
minimum and avoid writing things like $(M_{n_{k}}^{\prime})^{i}(t)$. Note that
$\left(M_{n_{k}}^{\prime}(t)-M_{n_{k}}^{\prime}(s)\right)\mathbb{I}_{E^{\prime}_{n_{k}}}$
converges almost surely to
$\left(M^{\prime}(t)-M^{\prime}(s)\right)\mathbb{I}_{E^{\prime}}$ and it is
uniformly integrable since

$E\left\\{\left(M^{\prime}_{n_{k}}(t)\right)^{2}\right\\}=EA^{\prime}_{n_{k}}(t)=EA_{n_{k}}(t)\to
EA(t).$

Since $A_{n_{k}}(u)$ is $\mathcal{F}_{n_{k}}(s)$-measurable for all $0\leq
u\leq s$, $E_{n_{k}}$ is $\mathcal{F}_{n_{k}}(s)$-measurable and

$\displaystyle
0=E\left\\{\left(M_{n_{k}}(t)-M_{n_{k}}(s)\right)\mathbb{I}_{E_{n_{k}}}\right\\}$
$\displaystyle=$ $\displaystyle
E\left\\{\left(M_{n_{k}}^{\prime}(t)-M_{n_{k}}^{\prime}(s)\right)\mathbb{I}_{E^{\prime}_{n_{k}}}\right\\}$
$\displaystyle\stackrel{{\scriptstyle
n_{k}\to\infty}}{{\rightarrow}}E\left\\{\left(M^{\prime}(t)-M^{\prime}(s)\right)\mathbb{I}_{E^{\prime}}\right\\}.$

Moreover, using Ethier and Kurtz, (1986, Proposition App2.3) and Hypothesis
2.1.b, $\\{M_{n_{k}}^{\prime}(t)\\}^{2}$ is uniformly integrable for each
$t\geq 0$. Next, for all $0\leq s\leq t$,

(A.4) $\displaystyle
E\left\\{\left\\{M^{\prime}(t)-M^{\prime}(s)\right\\}^{2}\right\\}$
$\displaystyle=$
$\displaystyle\lim_{k\to\infty}E\left\\{\left\\{M^{\prime}_{n_{k}}(t)-M^{\prime}_{n_{k}}(s)\right\\}^{2}\right\\}$
$\displaystyle=$
$\displaystyle\lim_{k\to\infty}E\left\\{A^{\prime}_{n_{k}}(t)-A^{\prime}_{n_{k}}(s)\right\\}=E\left\\{A(t)-A(s)\right\\}.$

Therefore $M^{\prime}$ is a square integrable martingale with respect to
filtration $\\{\mathcal{F}^{\prime}(t)\\}_{t\geq 0}$ because the cylinder sets
$E^{\prime}$ defined above generate the $\sigma$-algebra
$\mathcal{F}^{\prime}(s)$. The same argument also applies to $B^{\prime}$.
Next,

$\left\\{\left(M_{n_{k}}^{\prime}(t)-M_{n_{k}}^{\prime}(s)\right)^{2}-A^{\prime}_{n_{k}}(t)+A^{\prime}_{n_{k}}(s)\right\\}\mathbb{I}_{E^{\prime}_{n_{k}}}$

converges almost surely to
$\left\\{\left(M^{\prime}(t)-M^{\prime}(s)\right)^{2}-A^{\prime}(t)+A^{\prime}(s)\right\\}\mathbb{I}_{E^{\prime}}$
and its absolute value is bounded by
$\left(M_{n_{k}}^{\prime}(t)-M_{n_{k}}^{\prime}(s)\right)^{2}+A^{\prime}_{n_{k}}(t)$,
which converges almost surely to
$\left(M^{\prime}(t)-M^{\prime}(s)\right)^{2}+A^{\prime}(t)$. Using (A.4),

$E\left\\{\left(M_{n_{k}}^{\prime}(t)-M_{n_{k}}^{\prime}(s)\right)^{2}+A^{\prime}_{n_{k}}(t)\right\\}=2EA^{\prime}_{n_{k}}(t)-EA^{\prime}_{n_{k}}(s)$

converges to
$E\left\\{\left(M^{\prime}(t)-M^{\prime}(s)\right)^{2}+A^{\prime}(t)\right\\}$.
By dominated convergence Ethier and Kurtz, (1986, Proposition App1.2), one can
conclude that

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
E\left\\{\left\\{\left(M_{n_{k}}^{\prime}(t)-M_{n_{k}}^{\prime}(s)\right)^{2}-A^{\prime}_{n_{k}}(t)+A^{\prime}_{n_{k}}(s)\right\\}\mathbb{I}_{E^{\prime}_{n_{k}}}\right\\}$
$\displaystyle\to$ $\displaystyle
E\left\\{\left\\{\left(M^{\prime}(t)-M^{\prime}(s)\right)^{2}-A^{\prime}(t)+A^{\prime}(s)\right\\}\mathbb{I}_{E^{\prime}}\right\\}.$

It follows that $A^{\prime}$ is the quadratic variation process of the
martingale $M^{\prime}$ since, by construction, $A^{\prime}(t)$ is
$\mathcal{F}^{\prime}(t)$-measurable. Again, the same argument holds true for
$B^{\prime}$, with quadratic variation $\langle B\rangle_{t}=t$, $t\geq 0$.
Finally, both the within and cross off-diagonal terms in
$\langle\tilde{M}_{n_{k}},B_{n_{k}}\rangle$, are taken care of coordinatewise,
as in the preceding argumentation, setting $d=1$. For instance, the copy of
the cross term

$\left\\{M_{n_{k}}^{\prime}(t)B_{n_{k}}^{\prime}(t)-M_{n_{k}}^{\prime}(s)B_{n_{k}}^{\prime}(s)-\langle\tilde{M}_{n_{k}}^{\prime},B_{n_{k}}^{\prime}\rangle_{t}+\langle\tilde{M}_{n_{k}}^{\prime},B_{n_{k}}^{\prime}\rangle_{s}\right\\}\mathbb{I}_{E^{\prime}_{n_{k}}}$

converges almost surely to
$\left\\{M^{\prime}(t)B^{\prime}(t)-M^{\prime}(s)B^{\prime}(s)\right\\}\mathbb{I}_{E^{\prime}}$
and its absolute value, using Kunita-Watanabe’s inequality, is bounded by

$g_{n_{k}}=\frac{1}{2}\left(M_{n_{k}}^{\prime}(t)\right)^{2}+\frac{1}{2}\left(M_{n_{k}}^{\prime}(s)\right)^{2}+\frac{1}{2}\left(B_{n_{k}}^{\prime}(t)\right)^{2}+\frac{1}{2}\left(B_{n_{k}}^{\prime}(s)\right)^{2}+\frac{1}{2}A^{\prime}_{n_{k}}(t)+\frac{1}{2}\langle
B^{\prime}_{n_{k}}\rangle_{t},$

which converges almost surely to

$g=\frac{1}{2}\left(M^{\prime}(t)\right)^{2}+\frac{1}{2}\left(M^{\prime}(s)\right)^{2}+\frac{1}{2}\left(B^{\prime}(t)\right)^{2}+\frac{1}{2}\left(B^{\prime}(s)\right)^{2}+\frac{1}{2}t+\frac{1}{2}A^{\prime}(t).$

Hypothesis (b) implies $E(g_{n_{k}})\to E(g)$, so

$E\left\\{\left\\{M_{n_{k}}^{\prime}(t)B_{n_{k}}^{\prime}(t)-M_{n_{k}}^{\prime}(s)B_{n_{k}}^{\prime}(s)-\langle\tilde{M}_{n_{k}}^{\prime},B_{n_{k}}^{\prime}\rangle_{t}+\langle\tilde{M}_{n_{k}}^{\prime},B_{n_{k}}^{\prime}\rangle_{s}\right\\}\mathbb{I}_{E^{\prime}_{n_{k}}}\right\\}\equiv
0$

converges to
$E\left\\{\left\\{M^{\prime}(t)B^{\prime}(t)-M^{\prime}(s)B^{\prime}(s)\right\\}\mathbb{I}_{E^{\prime}}\right\\}$
and hence $\langle M^{\prime},B^{\prime}\rangle=0$. Thus any limit point
$Z^{\prime}=(M^{\prime},B^{\prime},A^{\prime})$ has the property that
$M^{\prime}$, $B^{\prime}$, and $M^{\prime}B^{\prime}$ are martingales with
respect to the natural filtration $\mathcal{F}_{t}^{\prime}$, with $\langle
M^{\prime}\rangle=A^{\prime}$, $B^{\prime}$ is a Brownian motion, and most
importantly that $\langle M^{\prime},B^{\prime}\rangle_{t}\equiv 0$. Since we
already know that the trajectories of $M^{\prime}$ and $B^{\prime}$ are
continuous, it follows from Ikeda and Watanabe, (1989, Theorem II.7.3) that
$M^{\prime}=W^{\prime}\circ A^{\prime}$ with independent Brownian motions
$W^{\prime}$ and $B^{\prime}$ on probability space
$(\Omega^{\prime},{\mathcal{F}}^{\prime},P^{\prime})$. Since, $A^{\prime}$ is
$\mathbb{F}_{B^{\prime}}$-measurable by hypothesis, $W^{\prime}$ is also
independent of $A^{\prime}$. Therefore all limit points
$(M^{\prime},B^{\prime},A^{\prime})$ have the same law since the law of
$A^{\prime}$ is the same has the one of $A$. ∎

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