The signature transform of a path $\gamma$ is a series of high order iterated integrals that plays a fundamental role in the the study of controlled differential equations. Understanding how properties of $\gamma$ are encoded in its signature has been an active area of research for the last 10 years <cit.>. This article aims to contribute to the global effort in three ways:

* by advancing that high order terms in the asymptotic expansion of the signature can contain information about the original path.

* by establishing a bridge between the Hilbert-Schmidt norm of the signature and earth mover's distances, ultimately yielding a generalisation of the Hambly-Lyons limit <cit.> from $C^3$ curves to a wide class of $C^1$ ones.

* by providing a way to completely recover the asymptotic development of the signature in the Hilbert-Schmidt norm, that we illustrate by explicitly computing terms up to $O(n^{-3})$.

§ INTRODUCTION

Let $\gamma$ be a $d$-dimensional (Euclidean) path $\gamma$, i.e. a continuous function from a compact interval $[a,b]\subset \mathbb{R}$ into $\mathbb{R}^d$. Assuming $\gamma \in C^1$ and $\gamma'(t) \neq 0$ $\forall t \in [a,b]$, define its length by

\begin{equation}
L(t-a):=L(\gamma)_{[a;t]}:=\int_{a}^t |\gamma'_s|ds
\end{equation}

and denote $l:=L(b-a)$. Such $\gamma$ admits a unit-speed parametrisation $\rho:[0,l]\rightarrow[a,b]$ defined as

\begin{equation}
\rho(s)=L^{-1}(s)
\end{equation}

such that the path $\gamma \circ \rho$ is a unit speed curve. Now, assume $\gamma$ controls the following differential equation

\begin{align}
\end{align}

where $V_k:\mathbb{R}^d \rightarrow \mathbb{R}^d$ are linear vector fields. The solution $y_t$ of this controlled differential equation is known to be a linear function of the signature of $\gamma$ <cit.>.

Let $\gamma:[a,b] \rightarrow \mathbb{R}^d$ be a path such that the following iterated integrals make sense. For $(i_1,..,i_k)\in\{1,2...,d\}$, the truncated signature $X^n$ of $\gamma$ of order $n$ is the family of tensors defined by

\begin{equation}
X^n:=\{X^{n; \,I}\}_{I \in J}
\end{equation}

where $J:=\{(i_1,...,i_n) |\;a\leq i_1 < ... < i_n \leq b\}$ is a set of multi-index and where

\begin{equation}
X^{n; \,i_1,...,i_k}:=\bigg( \int_{a\leq t_1 \leq t_2 \leq .. \leq t_k \leq b} d\gamma_{t_1}^{i_1} \otimes d\gamma_{t_2}^{i_2} \otimes... \otimes d\gamma_{t_n}^{i_n}   \bigg) \in \mathbb{R}^{\otimes k}
\end{equation}

The signature $X$ of $\gamma$ is defined by

\begin{equation}
X:=\bigg(1, S(\gamma)^1_{[a,b]}, S(\gamma)^2_{[a,b]}, ...\bigg) \in T((\mathbb{R})):=\prod_{k=0}^{\infty}\mathbb{R}^{\otimes k}
\end{equation}

Representing a path using its signature has several pleasant theoretical and computational properties. A celebrated example of them is that the signature transform satisfies a universality theorem, i.e. any continuous function $f:J\rightarrow \mathbb{R}$ from a compact subset $J$ of signature features be can arbitrarily well approximated by a linear function $L:J \rightarrow \mahtbb{R}$ <cit.>. This result led to the development of several new methods and paradigms in time-series analysis <cit.>, machine learning <cit.> and more recently deep learning <cit.>. This resulting traction also made the question of recovering original path properties from its signature even more pertinent.

While there is a steam of work tackling the explicit reconstruction of the original path from its signature <cit.>, the existing methods (although successful and insightful) do not allow one to easily relate properties of the signature with its original path as the reconstruction procedures are algorithmic. In this article, we will follow another important branch of investigation: the study of the signature through its asymptotics.

The study of the asymptotics of the signature transform has known been an active area of research for the last 10 years <cit.>, thanks to Hambly and Lyons <cit.> who initially showed that the arc-length of a path can be recovered from the asymptotics of its signature under a broad-class of norms, suggesting that other properties of the original path could also be recovered in a similar fashion.

To be more precise, they proved that if $\gamma:\left[  0,l\right]  \rightarrow V$ is a continuously
differentiable unit-speed curve, then it holds that

\begin{equation}
\lim_{n\rightarrow\infty}\frac{n!\left\vert \left\vert X^{n}\right\vert
\right\vert }{l^{n}}=1,\label{strong}%
\end{equation}

for a class of norms on the tensor product $V^{\otimes n}$ which includes the
projective tensor norm, but excludes the Hilbert-Schmidt norm in the case
where $V$ is endowed with an inner-product. For the same class of norms, this
also implies the weaker statement

\begin{equation}
\lim_{n\rightarrow\infty}\left(  n!\left\vert \left\vert X^{n}\right\vert
\right\vert \right)  ^{1/n}=l.\label{weak}%
\end{equation}

When $V$ is an inner-product space and the Hilbert-Schmidt norm is considered,
<cit.> also show that (<ref>) holds, but that statement (<ref>)
fails to be true in general: indeed under the assumption that $\gamma$ is
three times continuously differentiable it is proved that

\begin{equation}
\lim_{n\rightarrow\infty}\frac{n!\left\vert \left\vert X^{n}\right\vert
\right\vert }{l^{n}}=c\left(  l\right)  :\mathbb{=E}\left[  \exp\left(
\int_{0}^{l}\left(  W_{s}^{0}\right)  ^{2}\left\langle \gamma_{s}'%
,\gamma_{s}'''\right\rangle ds\right)  \right]  ^{1/2},\label{HL id}%
\end{equation}

where $\left(  W_{s}^{0}\right)  _{s\in\lbrack0,l]}$ is a Brownian bridge
which returns to zero at time $l$. It is easily seen that $c\left(  l\right)
<1$ unless $\gamma$ is a straight line.
More recent articles have focused on proving statements similar to (<ref>)
under the fewest possible assumptions on $\gamma$ and on the tensor norms. In the article <cit.> for example, it was proved that if $\gamma$ is continuous
and of finite $1-$variation then, under any reasonable tensor norm, we have

\begin{equation}
\lim_{n\rightarrow\infty}\left(  n!\left\vert \left\vert X^{n}\right\vert
\right\vert \right)  ^{1/n}=\inf_{n\geq1}\left(  n!\left\vert \left\vert
X^{n}\right\vert \right\vert \right)  ^{1/n}>0\label{inf id}%
\end{equation}

provided that the sequence $\left\{  \left\vert |X^{n}|\right\vert
:n=1,2,...,\infty\right\}  $ does not contain an infinite subsequence of zeros. Boedhardjo and Geng <cit.> strengthened this result by proving that the existence of such a subsequence is equivalent to the underlying path being tree-like, see also <cit.>. Taken together, these articles prove that for
any continuous bounded variation path the identity in (<ref>) is true.
It is conjectured that for a tree-reduced path $\gamma$ the limit is exactly
the length of $\gamma$, see <cit.>. This question remains open at the
time of writing.

In this article we work within the framework of the above articles, but advance a somewhat different point of view, namely the recovery of the original path properties using the high-order terms of its signature asymptotic development instead of just using the limit as previously explored.
By starting from (<ref>) and
(<ref>), we notice the latter statement can be rewritten as a
statement of asymptotic equivalence, namely

\begin{equation}
\left(  n!\left\vert \left\vert X^{n}\right\vert \right\vert _{\text{HS}%
}\right)  ^{1/n}\sim l+\frac{1}{n}\log c\left(  l\right)  +o\left(  \frac
{1}{n}\right)  \text{ as }n\rightarrow\infty\text{ ,} \label{eq:original_asymptotics_hilbert_schmidt} \end{equation}

where we write $\left\vert \left\vert \cdot\right\vert \right\vert _{\text{HS
}}$ for the moment to emphasise the dependence of this expansion on the choice
of tensor norm. By contrast for the projective tensor norm it follows from
(<ref>) that we have

\[
\left(  n!\left\vert \left\vert X^{n}\right\vert \right\vert _{\text{proj}%
}\right)  ^{1/n}\sim l+o\left(  \frac{1}{n}\right)  \text{ as }n\rightarrow
\infty.
\]

When written in this way, the constant identified in (<ref>) has the
interpretation of being the second term in the asymptotic expansion of
$(n!\left\vert \left\vert X^{n}\right\vert \right\vert _{\text{HS}})^{1/n}$ as
$n\rightarrow\infty.$ Given that the leading order term is length $l$ of
$\gamma,$ a natural question is to ask whether the higher order terms have
some interpretation in terms of the geometry of the underlying $\gamma.$ The
following lemma shows that this term is related to the curvature of $\gamma.$

Let $V$ be a finite dimensional inner product space. Suppose that
$\gamma:\left[  0,l\right]  \rightarrow V$ is parameterised at unit speed and
is three-times continuously differentiable (under this parameterisation). Let
$\mu$ denote the finite Borel measure on $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}$ which is absolutely continuous with respect to the Lebesgue measure
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}}$ with density given by $\frac{d\mu}{d\lambda}\left(  t\right)
=l^{2}\left\vert \gamma_{tl}^{\prime\prime}\right\vert ^{2}1_{[0,1]}\left(
t\right)  $ a.e. There exists a unique continuous function $\phi:\left[
0,1\right]  \rightarrow\mathbb{R}$ which solves, in the distributional sense,
the second-order differential equation

\begin{equation}
\psi^{\prime\prime}=2\psi\mu,\text{with }\psi_{0}=0,\psi_{0}^{\prime
\end{equation}

Moreover the constant $c\left(  l\right)  $ in (<ref>) equals $\psi

Define the unit-speed curve $\tilde{\gamma}_{t}:=\frac{1}{l}\gamma_{lt}$ over
$\left[  0,1\right]  $ and let the terms in its signature be given by $\left(
1,\tilde{X}^{1},..,\tilde{X}^{n},..\right).$ It suffices to prove the result
under the assumption that $l=1$ since the general result can then be recovered
by noticing

\[
\left(  n!\left\vert \left\vert X^{n}\right\vert \right\vert _{\text{HS}%
}\right)  ^{1/n}=l\left(  n!||\tilde{X}^{n}
|| _{\text{HS}}\right)  ^{1/n}\sim l+\frac{l}{n}\log c\left(
1\right)  +o\left(  \frac{1}{n}\right)  ,
\]

so that $c\left(  l\right)  =c\left(  1\right)  ^{l}.$ We therefore assume
that $l=1,$ and seek to prove that $c\left(  1\right)  =\psi_{1}^{-1/4}.$ To
this end, we first observe that the unit-speed parameterisation of $\gamma$
gives that $\left\langle \gamma_{t}^{\prime},\gamma_{t}^{\prime\prime\prime
}\right\rangle =-\left\vert \gamma_{t}^{\prime\prime}\right\vert ^{2}$ for
every $t$ in $\left[  0,1\right]  .$ When used together with
(<ref>) this gives that

\[
c\left(  1\right)  =\mathbb{E}\left[  \exp\left(  -\int_{0}^{1}\left(
W_{t}^{0}\right)  ^{2}\mu\left(  dt\right)  \right)  \right]  ^{1/2}.
\]

By noticing that $\left(  W_{\cdot}^{0}\right)  ^{2}$ is a squared Bessel
bridge of dimension $1$ which returns to zero at time $1$, we can then use
Theorem 3.2 of <cit.> to see that

\begin{equation}
c\left(  1\right)  =\left(  \phi_{1}\int_{0}^{1}\frac{1}{\phi_{t}^{2}%
}dt\right)  ^{-1/4},\label{c1}%
\end{equation}

where $\phi$ is the unique continuous positive function solving the
distributional equation $\phi^{\prime\prime}=2\phi\mu,$ with $\phi_{0}%
=1,\;\phi_{0}^{\prime}=0.$. Using
Exercise 1.34 in <cit.>, we know that the function $\psi$ in the
statement and $\phi$ are related by

\[
\psi_{t}=\phi_{t}\int_{0}^{t}\frac{1}{\phi_{s}^{2}}ds,
\]

and therefore (<ref>) $c\left(  1\right)  =\psi_{1}^{-1/2}.$

Note that this result provides an alternative way to compute the limit.
[The planar circle]
Let $\gamma_{t}=\frac{1}{2\pi}\left(  \sin(2\pi
t),\cos(2\pi t)\right)  $ for $t$ in $\left[  0,1\right]  $, then $\gamma$ is a
smooth unit-speed curve of length $l=1$ and $\left\vert \gamma_{t}%
^{\prime\prime}\right\vert =2\pi$ so that $\mu\left(  dt\right)
=2\pi1_{[0,1]}\left(  t\right)  dt$. By solving the differential equation
(<ref>) we find that

\[
c\left(  1\right)  =\left(  \frac{2\sqrt{\pi}}{\sinh2\sqrt{\pi}}\right)
\]

§ MAIN RESULTS

The main contributions of this paper are Theorem <ref> and Proposition <ref>. The first is a generalisation of the Hambly-Lyons characterisation of limit (<ref>) from $C^3$ curves to a broad class of $C^1$ ones, that we call $C^{1+}$.
def:c1+_curves[$C^{1+}$ curves]
Let $\gamma:[0,1]\rightarrow \mathbb{R}^d$ be a unit-speed $C^1$ curve such that $|\gamma'_s|>0$ $\forall s \in [0,1]$. We say that $\gamma$ is a $\mathbf{C^{1+}}$ curve if

* its second derivative exists,

* the map $V:a\mapsto|\gamma_{a}''|^{-1}$
is a $L$-Lipschitz continuous function from $\left[  0,1\right]$ into $\mathbb{R}_{\geq0}$.

thm:charac_limit_c2_generalisation[Hambly-Lyons' limit]
Let $\gamma:[0,1] \rightarrow \mathbb{R}$ be a $\mathcal{C}^{1+}$ curve (Definition <ref>). Then,

\begin{equation}
\lim_{n\rightarrow \infty} n!||X^n||^2=\mathbb{E}\bigg[  \exp\bigg(-\int_{0}^1 (W_{s}^{0})^{2}\;|\gamma_s''|^2ds\bigg) \bigg]^{1/2}
\end{equation}

where $\left(  W_{s}^{0}\right)  _{s\in\lbrack0,l]}$ is a Brownian bridge
which returns to zero at time $1$.

The second is the explicit expression of the asymptotics of the above limit up to $O(n^{-3})$ terms.
prop:asymptotic_dev_augmented[Asymptotics in $||\cdot||_{\text{HS}}$]
Let $\gamma:[0,1] \rightarrow \mathbb{R}$ be a unit-speed $\mathcal{C}^{1+}$ curve (Definition <ref>). Then, as $n \rightarrow \infty$,

\begin{align}
(n!||X^n||_{\text{HS}})^{\frac{1}{n}} \sim  1 + \frac{1}{n}\log(c(1)) -\frac{2}{n^4}\sum_{i=1}^{n-1} \sum_{s=1}^{i} s \Gamma_s + \frac{1}{1n^6}\sum_{i=1}^{n-1}\xi_2^i + o(n^{-3}) \label{eq:final_asymptotics}
\end{align}

\begin{align}
c(1)&:=\mathbb{E}\bigg[  \exp\bigg(-\int_{0}^1 (W_{s}^{0})^{2}\;|\gamma_s''|^2 ds \bigg) \bigg]^{1/2} \\
\xi_i^j &:= \sum_{1\leq s_1 < ... < s_i \leq j} \bigg( \sum_{s=1}^{s_1}s\Gamma_s \bigg)\cdot \Gamma_{s_2}\Gamma_{s_3} \ldots \Gamma_{s_i}, \quad i,j\in\{2,...,n\} \\
\Gamma_i&:=||\gamma_{i/n}''||^2
\end{align}

The proof of this result illustrates a more general methodology than can be used to recover higher order terms. Review this remark once section 5 fixed

The rest of this article is structured as follows:

Section <ref> We work with a simple class of curves (lattice paths) to illustrate two points: (i) the strong dependence of the signature asymptotics on the choice of norm and path regularity and (ii) the fact that local limit theorems seem to be the natural tools to recover the full signature asymptotics.

Section <ref> Motivated by the above point (ii), we revisit a lemma in Hambly-Lyons expressing signature norm in terms of order statistics to derive a development of the signature in terms of earth mover's distances for $C^1$ curves. In order to use this development to recover the Hambly-Lyons limit <ref>, some conditions are required, leading us to work with a subfamily that we call $C^{1+}$. We conclude by deriving an upper and lower bound on $n!||X^n||$ for such curves.

Section <ref> The above bounds are then reworked using local limit theorems and ultimately used to recover the Hambly-Lyons limit <ref>. We rework them such that they both consist in the sum of two terms depending on $n$: the first one being a discretization of the Hambly-Lyons integral (and converging to it when $n \rightarrow \infty$) while the second one vanishes when $n \rightarrow \infty$.

Section <ref> Finally, we present a way to recover the full asymptotic development and conclude by the explicit computation of high-order terms of that limit up to $\mathcal{O}(n^{-3})$.

§ FULL ASYMPTOTICS FOR LATTICE PATH

In this section, we work with a family of lattice paths to illustrate the role of the choice of tensor norm and path regularity on the signature asymptotics. Once done, we show how their full asymptotic developments can be recovered using local limit theorems.

First, the asymptotic behaviour of the terms in the signature depend on the choice of tensor norm, as already evident by the difference between (<ref>) and
(<ref>), but it will also depend on the smoothness of the underlying
path $\gamma.$ We can illustrate this very simply in the case where
$V=\mathbb{R}^{d},$ with the Euclidean inner-product, by considering a special
class of piecewise linear paths, called isotropic lattice paths.

Let $\gamma_{v_i}:\left[  0,1\right]  \rightarrow\mathbb{R}^{d}$ denote the
straight line

\begin{equation}
\gamma_{v_i}\left(  t\right)  =tv_i
\end{equation}

for some non-zero $v_i \in \mathbb{R}^d$. Let $\gamma:[0,1]\rightarrow \mathbb{R}^d$. We say that $\gamma$ is an isotropic $n$-lattice path with directions $v_1,...,v_n$ if

\begin{equation}
\gamma = \gamma_{v_1} * \gamma_{v_2} * ... * \gamma_{v_n}
\end{equation}

where $v_i\in\mathbb{R}^d$ are non-zero vectors with norm $\sum_{i=1}^n|v_i|=1$ $\forall i \in \{1,...,n\}$ and such that $v_i$ and $v_j$ are orthogonal $\forall i\neq j$.

Henceforth we will work with the
Hilbert-Schmidt norm on the tensor product and will write $\left\vert
\left\vert \cdot\right\vert \right\vert $ without further qualification. Let $\gamma:\left[  0,1\right]  \rightarrow\mathbb{R}^{d}$ be a isotropic 2-lattice paths with directions $v$ and $w$ such that $|v|=|w|=\frac{1}{2}$. As above, we let the signature of $\gamma$ be denoted
by $X  =\left(  1,X^{1},X^{2},...\right)  $ then Chen's
identity gives that

\[
X^{n}=\sum_{k=0}^{n}\frac{v^{k}w^{n-k}}{k!\left(  n-k\right)  !},
\]

from which it follows that

\[
{2^{2n}}\sum_{k=0}^{n}\binom{n}{k}\binom{n}{n-k}=\frac{\left(  2n\right)
!}{2^{2n}\left(  n!\right)  ^{2}}.
\]

Stirling's approximation then gives the asymptotic relation

\begin{equation}
||n!X^{n}||^{2}\sim\frac{1}{\sqrt{\pi n}}\text{ as }n\rightarrow
\infty\label{case1}%
\end{equation}

so that in particular $\lim_{n\rightarrow\infty}||n!X^n||^{2}=0,$ which is different to the non-zero limit obtained in
(<ref>) under the assumption that $\gamma$ is three-times continuously differentiable.

The above calculation can be generalised, and more can be said. If we now
assume that $\gamma:[0,1]\rightarrow \mathbb{R}^d$ is an isotropic $m$-lattice path with directions $\left\{  v_{1},..,v_{m}\right\}$ for $m\leq d$, then

\begin{equation}
X^{n}=\sum_{\substack{k_{1},...,k_{m}\geq0, \\k_{1},+...+k_{m}=n}}\frac
\end{equation}

The following proposition expresses an identity between the norm of terms in
the signature and the $2$-Renyi entropy of the law of a multinomial random variable.

Given a discrete random variable $Y\in\left\{  y_{1},...,y_{r}\right\}  $ with
$\mu\left(  \left\{  y_{r}\right\}  \right)  :=p\left(  y_{r}\right)
:=\mathbb{P}\left(  Y=y_{r}\right)  $ we recall that the $\alpha$-Renyi
entropy of $Y$ (or of its distribution $p=\left(  p_{1},...,p_{r}\right)  $)
is defined by

\begin{equation}
H_{\alpha}\left(  \mu\right)  =\frac{1}{1-\alpha}\log\left(  \sum_{k=1}%
^{r}p\left(  y_{k}\right)  ^{\alpha}\right)  \text{ for }\alpha>0,\alpha
\neq1.\label{Renyi}%
\end{equation}

It holds that $H_{\alpha}\left(  \mu\right)  \rightarrow H\left(  \mu\right)
$, the classical Shannon entropy, as $\alpha\rightarrow1.$ If $\alpha=2$ then
the following identities hold

\[
\sum_{k=1}^{r}p\left(  y_{k}\right)  ^{2}=\mathbb{E}\left[  p\left(  Y\right)
\right]  =\mathbb{P}\left(  Y=Y^{\prime}\right)  ,
\]

where $Y^{\prime}$ is any independent copy of $Y$ defined on the same
probability space.

Let $m\leq d$ and suppose that $\left\{
v_{1},..,v_{m}\right\}  \subset\mathbb{R}^{d}$ is a set of vectors which are orthogonal with respect to the Euclidean inner-product on $\mathbb{R}^{d}$and
are such that $\left\vert v_{i}\right\vert =l_{i}>0$ with $\sum_{i=1}^{m}%
l_{i}=1$. Let $n\in%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ be fixed and define a discrete probability measure on the set of integer

\[
\mathcal{K}_{n}=\left\{  \mathbf{k=}\left(  k_{1},...,k_{m}\right)
\]

\[
\mu_{n}\left(  \left\{  \mathbf{k}\right\}  \right)  =\frac{n!}{k_{1}%
\]

i.e. $\mu$ is the probability mass function of a multinomial random variable
with parameters $\left(  n;l_{1},...,l_{m}\right)  .$ Then the following
identity holds

\[
||n!X^n||^{2}=\exp\left(  -H_{2}\left(  \mu
_{n}\right)  \right)  .
\]

Using equation (<ref>) and the orthogonality of $v_{1},..,v_{m}$ we have

\[
\\k_{1},+...+k_{n}=n}}\left(  \frac{n!}{k_{1}!k_{2}!..k_{m}!}\right)
_{n}\left(  \left\{  \mathbf{k}\right\}  \right)  ^{2},
\]

and the claim follows from (<ref>).

Using this we can immediately generalise (<ref>).

As $n\rightarrow\infty$ we have that

\[
||n!X^n||^{2}\sim2\sqrt{\pi n}\left(
%TCIMACRO{\tprod _{j=1}^{m}}%
\frac{1}{4\pi nl_{j}}\right)  ^{1/2}%
\]

Fix $n\in%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ and suppose that $X_{n}=\sum_{j=i}^{n}E_{j}$ where $\left(  E_{j}\right)
_{j=1}^{n}$ are i.i.d. with distribution $\mathbb{P}\left(  E_{1}%
=e_{k}\right)  =l_{k}$, $k=1,..,m.$ Let $X_{n}^{\prime}=\sum_{j=1}^{n}%
E_{j}^{\prime}$ be an independent copy of $X_{n},$ then it is easy to see that

\begin{equation}
\sum_{\mathbf{k\in}\mathcal{K}_{n}}\mu_{n}\left(  \left\{  \mathbf{k}\right\}
\right)  ^{2}=\mathbb{P}\left(  X_{n}=X_{n}^{\prime}\right)  =\mathbb{P}%
\left(  \sum_{j=1}^{n}Z_{j}=0\right)  ,\label{LLT}%
\end{equation}

where $Z_{j}:=E_{j}-E_{j}^{\prime}.$ We obtain from this identity

\[
\mathbb{P}\left(  \sum_{i=1}^{n}Z_{i}=0\right)  =\frac{1}{\left(  2\pi\right)
^{m}}\int_{[-\pi,\pi]^{m}}\mathbb{E}\left[  e^{i\left\langle t,Z_{1}%
\right\rangle }\right]  ^{n}dt=\frac{1}{\left(  2\pi\right)  ^{m}}\int%
_{[-\pi,\pi]^{m}}\left\vert \sum_{j=1}^{m}l_{j}e^{it_{j}}\right\vert ^{2n}dt.
\]

Let $p\left(  t_{1},...t_{m}\right)  :=\left\vert \sum_{j=1}^{m}l_{j}%
e^{it_{j}}\right\vert ^{2}.$ Then using Fubini's theorem we have that

\[
||n!S\left(  \gamma\right)  _{n}||^{2}=\frac{1}{\left(  2\pi\right)  ^{m}}%
\int_{-\pi}^{\pi}\int_{[-\pi,\pi]^{m-1}}\exp\left(  n\log p\left(
t_{1},..,t_{m-1},t\right)  \right)  dt_{1}...dt_{m-1}dt.
\]

For each fixed $t$ in $\left[  -\pi,\pi\right]  $ we define $h\left(
t_{1},...,t_{m-1};t\right)  :=\log p\left(  t_{1},..,t_{m-1},t\right)  $. This
function has a unique maximum at $t^{\ast}=(t,...,t),$ moreover a straight
forward computation yields

\[
h\left(  t^{\ast};t\right)  =0,\partial_{q}h\left(  t^{\ast};t\right)
=\partial_{q}p\left(  t^{\ast};t\right)  =0,\text{ and }\partial_{q,r}%
\]

for $q,r=1,...,m-1$, and since

\[
\partial_{q,r}^{2}p\left(  t_{1,}...,t_{m-1};t\right)  =\left\{
\begin{array}
-2\sum_{j\neq r}^{m-1}l_{j}l_{q}\cos\left(  t_{q}-t_{j}\right)   & \text{if
2l_{q}l_{r}\cos\left(  t_{q}-t_{r}\right)   & \text{if }q\neq r
\end{array}
\right.  .
\]

we can deduce that

\[
\partial_{q,r}^{2}h\left(  t^{\ast};t\right)  =\left\{
\begin{array}
-2l_{q}\left(  1-l_{q}\right)   & \text{if }q=r\\
2l_{q}l_{r} & \text{if }q\neq r
\end{array}
\right.  .
\]

It follows that $\det\left(  \partial_{q,r}^{2}h\left(  t^{\ast};t\right)
\right)  =-2^{m-1}%
%TCIMACRO{\tprod _{j=1}^{m}}%
l_{j}$. Using Laplace's approximation we have

\[
\int_{\lbrack-\pi,\pi]^{m-1}}\exp\left(  n\log p\left(  t_{1},..,t_{m-1}%
,t\right)  \right)  dt_{1}...dt_{m-1}\sim\sqrt{\frac{\pi}{n}}^{m-1}\frac
%TCIMACRO{\tprod _{j=1}^{m}}%
\]

as $n\rightarrow\infty.$ From which we can deduce that

\[
||n!X^n||^{2}\sim2\sqrt{\pi n}\left(
%TCIMACRO{\tprod _{j=1}^{m}}%
\frac{1}{4\pi nl_{j}}\right)  ^{1/2}%
\]

as $n\rightarrow\infty.$

We see from (<ref>) that estimates on quantities of the type
$\mathbb{P}\left(  S_{n}=0\right)  $, where $S_{n}$ is a sum of $n$ i.i.d.
random variables. This observation connects the asymptotics of the signature
to the topic of local limit theorems for discrete random variable. By pursuing
this idea further it is possible to establish to full asymptotic expansion of
the signature of $\gamma$, at least in the case $m=2.$ This problem has been
studied previously in the context of Renyi entropy asymptotics, see <cit.>;
we write these in logarithmic form to make the comparison easier.

Let $\gamma$ be the curve $\gamma_{v_{1}}\ast\gamma_{v_{2}}$ where $v_{1}$ and
$v_{2}$ are orthogonal vectors with respect to the Euclidean inner-product,
such that $\left\vert v_{1}\right\vert =l=1-\left\vert v_{2}\right\vert .$ For
any $N\in%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ it holds that

\[
\log||n!X^n||^{2}=-\frac{1}{2}\log\left(  4\pi
\sigma^{2}n\right)  -\sum_{k=1}^{\left[  \frac{N-2}{2}\right]  }\frac{f_{k}%
}{n^{k}}+o\left(  n^{\left(  N-2\right)  /2}\right)  \text{ as }%
\]

where $\sigma^{2}=l\left(  1-l\right)  $ and the coefficients $f_{k}$,
$k=1,2,...$ are given explicitly in terms of the cumulants of a Bernoulli
random variable with parameter $l$.

Using Proposition <ref> we have that $\log||n!X^n||^{2}=-H\left(  \mu_{n}\right)  $ where $\mu_{n}$ denotes the
distribution of $X^n=\sum_{j=1}^{n}B_{j}$ where $B_{j}$ is a Bernoulli
random variable with parameter $l.$ The result then follows from of <cit.>,
where the explicit description of the coefficients may also be found.

The first two terms in the expansion are given by

\[
H_{2}\left(  \mu_{n}\right)  \sim\frac{1}{2}\log\left(  4\pi\sigma
^{2}n\right)  +\frac{1}{2}\frac{\kappa_{4}}{8\sigma^{4}}\frac{1}{n}+o\left(
\frac{1}{n}\right)  \text{ as }n\rightarrow\infty,
\]

where $\kappa_{4}$ is the fourth cumulant of the Bernoulli distribution with
parameter $l.$ It is known that $\kappa_{4}=\sigma^{2}\left(  1-6\sigma
^{2}\right)  $ and hence

\[
H_{2}\left(  \mu_{n}\right)  \sim\frac{1}{2}\log\left(  4\pi\sigma
^{2}n\right)  +\frac{1}{2}\frac{\left(  1-6\sigma^{2}\right)  }{8\sigma^{2}%
}\frac{1}{n}+o\left(  \frac{1}{n}\right)  \text{ as }n\rightarrow\infty.
\]

We then see that

\[
||n!S\left(  \gamma\right)  _{n}||^{2}=\exp\left(  -H_{2}\left(  \mu
_{n}\right)  \right)  =\frac{2}{\sigma\sqrt{\pi n}}\exp\left(  -\frac{\left(
1-6\sigma^{2}\right)  }{16\sigma^{2}n}+o\left(  \frac{1}{n}\right)  \right)  ,
\]

which improves on (<ref>) by yielding that

\[
||n!S\left(  \gamma\right)  _{n}||^{2}\sim\frac{1}{\sigma\sqrt{\pi}}\left(
\frac{1}{n^{1/2}}-\frac{1-6\sigma^{2}}{16\sigma^{2}n^{3/2}}\right)  +o\left(
\frac{1}{n^{3/2}}\right)
\]

as $n\rightarrow\infty.$

§ SIGNATURE NORM AND EARTH MOVER'S DISTANCES

Inspired by the second part of the previous section, we work with $C^1$ curves and propose to relate the Hilbert-Schmidt norm of the signature with discrete random variables through a development of $n!||X^n||$ in terms of earth mover's distances. We do so by revisiting a lemma  in Hambly-Lyons <cit.> that we now recall.z
For more general curves however, exploiting this bridge will require other arguments: indeed, lemma <ref> alone would lead us to work with the 2-Wasserstein distance $W_2(\tilde{\mu}_n^U, \tilde{\mu}_n^V)$ between the empirical measures

\begin{align}
\tilde{\mu}_n^U:=\frac{1}{n}\sum_{i=1}^n \delta_{\gamma'_{U_{(i)}}}, \quad \tilde{\mu}_n^V:=\frac{1}{n}\sum_{i=1}^n \delta_{\gamma'_{V_{(i)}}}.
\end{align}

This object is somewhat exotic and for the best of our knowledge, the literature does not account of an extensive study of its asymptotics.

Before proceeding with our investigation, we do a sanity check on the minimal regularity conditions that $\gamma$ must satisfy for our approach to allow us to recover known asymptotic results.
Assume $\gamma$ to a $C^1$ unit speed regular curve as considered in Lemma <ref>. Clearly, this condition alone is not sufficient to recover the Hambly-Lyons limit in its original form, which involved a third derivative in its expression (equation <ref>). However, taking the derivative on the orthogonality condition between $\gamma'$ and $\gamma''$ gives

\begin{equation}
\langle\gamma'_s,\gamma'''_s\rangle=-|\gamma''_s|^2
\end{equation}

allowing one to rewrite its right-hand side as

\begin{equation}
\lim_{n\rightarrow\infty}n!\left\vert \left\vert X^{n}\right\vert
\right\vert  = \mathbb{E}\left[\exp\left(-
\int_{0}^{l}\left(  W_{s}^{0}\right)  ^2 |\gamma_s''|^2 ds\right)  \right]  ^{1/2} \label{eq:hambly_lyons_original_but_with_second_derivative_instead_third}
\end{equation}

From this last expression, the $C^3$ assumption on $\gamma$ prescribed in Hambly-Lyons does not manifest itself explicitly. What is clear however is that $\gamma$ must have a well-defined second derivative almost everywhere. For the rest of this article, we will be working with such curves, namely a broad subclass of $C^1$ curves (as prescribed in Lemma <ref>) such that their second derivative exists, that we call $C^{1+}$.

Let $\gamma:[0,1]\rightarrow \mathbb{R}^d$ be a regular unit-speed $C^1$ curve. We say that $\gamma$ is a $\mathbf{C^{1+}}$ curve if

* its second derivative exists,

* the map $V:a\mapsto|\gamma_{a}''|^{-1}$
is a $L$-Lipschitz continuous function from $\left[  0,1\right]$ into $\mathbb{R}_{\geq0}$.

$\gamma \in C^{1+}$ is not necessarily $C^2$ as condition (i) does not restrict the second derivative $\gamma''$ to be continuous. On the other side, if $\gamma \in C^2$, then it is $C^{1+}$.

In virtue of the Lipschitz condition in Definition <ref>, let $\left(  F\left(  t\right)  \right)  _{t\in\left[  0,\tau\right]  }$
denote the unique solution to the ordinary differential equation

\begin{equation}
F^{\prime}\left(  t\right)  =|\gamma_{F\left(  t\right)  }^{\prime\prime
}|^{-1},\text{ }F\left(  0\right) = 0 \label{def:def_F_distrib_func_curvature}
\end{equation}

solved up to time $\tau:=\inf\left\{  t:F\left(  t\right)  =1\right\}  $ which
is finite by the continuity assumption on $|\gamma_{(\cdot)}^{\prime\prime}|.$
Extend $F:\mathbb{R\rightarrow}\left[  0,1\right]  $ by defining $F\left(
t\right)  =0$ for $t\leq0$ and $F\left(  t\right)  =1$ for $t\geq\tau$ so that
$F$ defines a distribution function with density $f$ given by

\begin{equation}
f\left(  t\right)  =|\gamma_{F\left(  t\right)  }^{\prime\prime}%
|^{-1}1_{[0,\tau]}\left(  t\right)  . \label{eq:def_density_function}
\end{equation}

and let $F^{-1}$ denote the left-continuous inverse of $F$. The significance of this distribution function comes from the following lemma.

We denote by the environment definitions the following recurring set of objects:

(i) Let $\{U_{(i)}\}_{i \in \{1,..,n\}}$ and $\{V_{{(i)}}\}_{i \in \{1,..,n\}}$ be two i.i.d. uniform order statistics collections (Definition <ref>). Furthermore, define

\begin{equation}
y_n& := \max_{j=1,...,n} |U_{(j)}-V_{(j)}|
\end{equation}

(ii) Let $\mu_{n}^U$ and $\mu_{n}^V$ be their empirical probability measures defined by

\begin{align}
\mu_n^U:= \frac{1}{n}\sum_{i=1}^n \delta_{U_{(i)}}, \quad \mu_n^V := \frac{1}{n}\sum_{i=1}^n \delta_{V_{(i)}}.
\end{align}

(iii) Let $F$ be as in equation (<ref>). Define the independent collections of i.i.d. random variables

\begin{equation}
X_{(i)} := F^{-1}(U_{(i)}), \quad Y_{(i)} := F^{-1}(V_{(i)}),\quad i \in \{1,..,n\}.
\end{equation}

We also define

\begin{equation}
X := \{X_{(i)}\}_{i \in \{1,..,n\}}, \quad Y := \{Y_{(i)}\}_{i \in \{1,..,n\}}
\end{equation}

Note that $X$ and $Y$ have $F$ as cumulative distribution function.

(iv) Let $\nu_n^X$ and $\nu_n^Y$ be the empirical probability measures of $X$ and $Y$,

\begin{equation}
\nu_{n}^X := \frac{1}{n}\sum_{i=1}^{n}\delta_{X_{(i)}},\text{ and }\nu_{n}^Y := \frac
\end{equation}

(v) Let $F_{n}^{X}$ and $F_{n}^{Y}$ denote the
corresponding distribution functions to $X$ respectively $Y$ and, $G_{n}^{X}$ and $G_{n}^{Y}$ the
respective left-continuous inverse functions.

Let $\gamma$ be a $C^{1+}$ curve and let the environment definitions hold (Definition <ref>).
Then, for $p \in \mathbb{N}^*$,

\begin{align}
D_{p,n}^-nW_{p}^{p}\left(  \nu_{n}^X,\nu_{n}^Y\right)  \leq\sum_{j=1}^{n}\left\vert
\gamma_{U_{\left(  j\right)  }}^{\prime}-\gamma_{V_{\left(  j\right)  }%
}^{\prime}\right\vert ^{p}\leq D_{p,n}^+nW_{p}^{p}\left(  \nu_{n}^X,\nu_{n}^Y\right)
\end{align}

almost surely, with

\begin{align}
D_{p,n}^{\pm}& := (1\pm M L y_n)^p \\
M& := \sup_{t \in [0,1]}|\gamma_t''|^{-1}
\end{align}

\begin{align}
Z_{j}\left(  s\right) := sF(X_{\left(  j\right)  })+\left(  1-s\right)
F(Y_{\left(  j\right)  })
\end{align}

As $U_{(j)}=F(X_{(j)})$ and $V_{(j)}=F(Y_{(j)})$ and as $\gamma \in C^3$, the mean value theorem yields

\[
\inf_{s\in\left[  0,1\right]  }\left\vert \gamma_{Z_{j}\left(  s\right)
}^{\prime\prime}\right\vert \left\vert F\left(  X_{\left(  j\right)  }\right)
-F\left(  Y_{\left(  j\right)  }\right)  \right\vert \leq\left\vert
\gamma_{U_{\left(  j\right)  }}^{\prime}-\gamma_{V_{\left(  j\right)  }%
}^{\prime}\right\vert \leq\sup_{s\in\left[  0,1\right]  }\left\vert
\gamma_{Z_{j}\left(  s\right)  }^{\prime\prime}\right\vert \left\vert F\left(
X_{\left(  j\right)  }\right)  -F\left(  Y_{\left(  j\right)  }\right)
\right\vert .
\]

Again, an application of the mean value theorem on $F$ gives for some $s_j \in [0,1]$

\begin{align}
F\left(  X_{\left(  j\right)  }\right)  -F\left(  Y_{\left(  j\right)
}\right)  =F^{\prime}\left(  Z_{j}\left(  s_{j}\right)  \right)  \left(
X_{\left(  j\right)  }-Y_{\left(  j\right)  }\right)  =|\gamma_{F\left(
Z_{j}\left(  s_{j}\right)  \right)  }^{\prime\prime}|^{-1}\left(  X_{\left(
j\right)  }-Y_{\left(  j\right)  }\right) \label{eq:rel_xi_ui_mean_val}
\end{align}

Together, the two previous lines give

\[
\left\vert X_{\left(  j\right)  }-Y_{\left(  j\right)  }\right\vert ^{p}%
\inf_{s,t\in\left[  0,1\right]  }\psi_{s,t}\left(  j\right)  \leq\left\vert
\gamma_{U_{\left(  j\right)  }}^{\prime}-\gamma_{V_{\left(  j\right)  }%
}^{\prime}\right\vert ^{p}\leq\left\vert X_{\left(  j\right)  }-Y_{\left(
j\right)  }\right\vert ^{p}\sup_{s,t\in\left[  0,1\right]  }\psi_{s,t}\left(
j\right)  ,
\]

\[
\psi_{s,t}\left(  j\right)  =\frac{\left\vert \gamma_{Z_{j}\left(  s\right)
}^{\prime\prime}\right\vert ^{p}}{\left\vert \gamma_{Z_{j}\left(  t\right)
}^{\prime\prime}\right\vert ^{p}}%
\]

By exploiting lemma <ref> and noticing $\psi_{s,t}\left(  j\right)  =\psi_{t,s}\left(  j\right)  ^{-1}$
we can deduce that

\begin{equation}
D_{p,n}^-nW_{p}^p\left(  \nu_{n}^X,\nu_{n}^Y\right)  \leq\sum_{j=1}^{n}\left\vert
\gamma_{U_{\left(  j\right)  }}^{\prime}-\gamma_{V_{\left(  j\right)  }%
}^{\prime}\right\vert ^{p}\leq D_{p,n}^+nW_{p}^p\left(  \nu_{n}^X,\nu_{n}^Y\right) \label{eq:ineq_proof_lemma_w2_sig}.
\end{equation}

\begin{align}
D_{p,n}^-  &  :=\inf\left\{  \frac{\left\vert \gamma_{Z_{j}\left(  s\right)
}^{\prime\prime}\right\vert ^{p}}{\left\vert \gamma_{Z_{j}\left(  t\right)
}^{\prime\prime}\right\vert ^{p}}:j=1,...,n\,,s,t\in\left[  0,1\right]
\right\}  ,\\
D_{p,n}^+  &  :=\sup\left\{  \frac{\left\vert \gamma_{Z_{j}\left(  s\right)
}^{\prime\prime}\right\vert ^{p}}{\left\vert \gamma_{Z_{j}\left(  t\right)
}^{\prime\prime}\right\vert ^{p}}:j=1,...,n\,,s,t\in\left[  0,1\right]
\right\}
\end{align}

Finally, it is easy to see that for all $s,t\in\left[  0,1\right]  $ we have

\begin{equation}
1-M\left\vert \gamma_{Z_{j}\left(  s\right)  }^{\prime\prime}-\gamma
_{Z_{j}\left(  t\right)  }^{\prime\prime}\right\vert \leq\frac{\left\vert
\gamma_{Z_{j}\left(  s\right)  }^{\prime\prime}\right\vert }{\left\vert
\gamma_{Z_{j}\left(  t\right)  }^{\prime\prime}\right\vert }\leq1+M\left\vert
\gamma_{Z_{j}\left(  s\right)  }^{\prime\prime}-\gamma_{Z_{j}\left(  t\right)
}^{\prime\prime}\right\vert . \label{eq:__temp_inequality_proof_lemma_speed_1}
\end{equation}

\begin{align}
\big| \gamma_{Z_j(s)}'' - \gamma_{Z_j(t)}'' \big| &\leq L\big| Z_j(s) - Z_j(t) \big| \notag\\
&= |s-t| L \; |U_{(j)}-V_{(j)}| \label{eq:__temp_inequality_proof_lemma_speed_2}
\end{align}

for some $w\in[s,t]$. Together, inequalities (<ref>) and (<ref>) yield

\begin{align}
\max_{j=1,..,n}\sup_{s,t}\frac{\left\vert \gamma_{Z_{j}\left(  s\right)
}^{\prime\prime}\right\vert }{\left\vert \gamma_{Z_{j}\left(  t\right)
}^{\prime\prime}\right\vert }\leq 1 + MLy_n  \text{
and }\min_{j=1,...,n}\inf_{s,t}\frac{\left\vert \gamma_{Z_{j}\left(  s\right)
}^{\prime\prime}\right\vert }{\left\vert \gamma_{Z_{j}\left(  t\right)
}^{\prime\prime}\right\vert }\geq 1 - MLy_n  \label{eq:D_pn_bounds_above_below}.
\end{align}

whereupon results follows straight forwardly from the equation <ref>.

This result coupled with lemma <ref> conclude this section.

Under the assumptions of Lemma <ref>,

\begin{equation}
\mathbb{E}\left[\exp
\left( -\frac{D_n^-n}{2}W_{2}^{2}\left( \nu _{n}^{X},\nu _{n}^{Y}\right) \right) %
\right]+ O(y_n^8)    \leq (n!\left\vert \left\vert X^{n}\right\vert \right\vert)^2 \leq \mathbb{E}\left[ \exp
\left( -\frac{D^+_n n}{2}W_{2}^{2}\left( \nu _{n}^{X},\mu _{n}^{Y}\right) \right) %
\right]+ O(y_n^8)    \label{eq:lower_upper_bound_sig}
\end{equation}

where $D_n^{\pm}& := (1\pm M L y_n)^2$.

On one side, equation (<ref>) gives

\begin{equation}
O\big(|X_{(j)} - Y_{(j)}|\big)=O\big(|\gamma''_{s_j}|\cdot|U_{(j)}-Y_{(j)}|\big)
\end{equation}
for a $s_j \in [U_{(j)}, V_{(j)}]$.
By the extreme value theorem, we further have

\begin{equation}
O\big(|X_{(j)} - Y_{(j)}|^k\big)=O\big(|U_{(j)}-Y_{(j)}|^k\big) = O(y_n^k).
\end{equation}

The discrete characterisation of $p$-Wasserstein distance (Lemma <ref>) yields

\begin{equation}
O(nW_p^p(\mu_N^U,\mu_n^V)) = O(y_n^p) \label{eq:nwasserstein_xy_vs_uv_asymptotics}
\end{equation}

On the other side, equation (<ref>) gives

\begin{equation}
O(D_{p,n}^{\pm}) = O(y_n^p) \label{eq:D_pn_asymptotics}
\end{equation}

The application of equations (<ref>) and (<ref>) in lemma <ref> and the alternative development of $||X^n||$ (Lemma <ref>) concludes the proof.

This bounds will be exploited in the following section to recover the Hambly-Lyons limit in <cit.>.

New addition: approach through B-G-U; this will be integrated to the above

§ RECOVERING THE HAMBLY-LYONS LIMIT USING EARTH MOVER'S DISTANCE

In this section, we recover the Hambly-Lyons limit using a result from from del Barrio, Gin´e and Utze on the convergence of 2-Wasserstein distances <cit.> that we will now present.

Let $X$ be a real-valued random variable with probability measure $\mu$ with support on $[0,1]$ and let $f$ and $F$ be their associated probability density and cumulative distribution function. Consider also the empirical measure $\mu_n$ associated with $\mu$ and defined by

\begin{equation}
\mu_n:=\sum_{i=1}^n \delta_{X_i}
\end{equation}

where $X_i, \; i\in\{1,2,..,n\}$ is an i.i.d. sample of random variables sampled according to $\mu$. The $J_2$ functional is defined as

\begin{equation}
J_2(\mu):=\int_0^1 \frac{F(x)(1-F(x))}{f(x)}dx
\end{equation}

Moreover, define also the integral quantity $I(t)$ by

\begin{equation}
\end{equation}

add definition quantile processes here

Let $\mu$, $\mu_n$, $f$ and $F$ be as above and such that $J_2(\mu)< \infty$ and

\begin{equation}
\sup_{0<x<1}\frac{F(x)(1-F(x))}{f(x)^2}|f'(x)|<\infty
\end{equation}

\begin{equation}

\end{equation}

\begin{equation}
nW_2^2(\mu_n,\mu) \rightarrow \int_0^1 \frac{(W^0(t))^2}{I(t)^2}dt
\end{equation}

when $(n \rightarrow \infty)$ weakly in distribution in $\mathbb{R}$.

express BGU in terms of quantile processes $\mathbb{F}_s^{n,X}$ as well (keep notation consistent)

Clearly, if $I(t)=||\gamma_t''||^{-1}$, the above expression coincides exactly with the integral expression inside the Hambly-Lyons limit (<ref>).

In the following paragraphs, our goal will be to recreate the framework of the above remark for the norm of the signature $n!||X^n||$. We do so by first characterising measures $\mu$ such that the following condition is satisfied:

\begin{equation}
I(t)=||\gamma_t''||^{-1}. \label{eq:condition_rv_bgu_integral}
\end{equation}

Once done, we consider the empirical analogue $\mu_n$ of $\mu$ and relate these objects with the norm of the signature $n!||X^n||$ through the 2-Wasserstein distance $W_2(\mu_n,\mu)$.

We now characterise all the measures with support in $[0,1]$ such that condition (<ref>) is satisfied.

Let $\mu$ be a probability measure on $[0,1]$ with cumulative distribution function $F$. Then, condition $(\ref{eq:condition_rv_bgu_integral})$ is satisfied if an only if $F$ is a solution of the following ODE

\begin{equation}
F'(t)=||\gamma_{F(t)}''||^{-1}; \; F(0)=0 \label{def:ODE_cumulative}
\end{equation}

By definition, condition (<ref>) can be written as

\begin{equation}
\end{equation}

Evaluating this expression at $F(t)$ yields

\begin{equation}
\end{equation}

Finally, one concludes by exploiting the previous equation to characterise the cumulative distribution as a solution of

\begin{equation}
F'(t)=||\gamma_{F(t)}''||^{-1}; \; F(0)=0.
\end{equation}

Let $\{U_{(i)}\}_{i \in \{1,...,n\}}$ be the order statistics of a sample of i.i.d. uniform random variables. Let $F$ be a solution of the ODE (<ref>) and $\mu$ its corresponding probability measure. Define the samples

\begin{equation}
X_i:=F^{-1}(U_{(i)}), \; i \in \{1,...,n\}
\end{equation}

Then, for the empirical probability measure $\mu_n$ associated with $\mu$ and defined by $\mu_n:=\frac{1}{n}\sum_{i=1}^n \delta_{X_i}$, we have

\begin{equation}
\mathbb{E}\bigg[\exp \big( -nW_2^2(\mu_n,\mu)\big)\bigg] \longrightarrow \mathbb{E}\bigg[ \exp \bigg( \int_{0}^1 (W^0_s)^2|\gamma_s''|^2ds  \bigg)\bigg]=:c(1)^2
\end{equation}

when $(n\rightarrow \infty)$.

The right-hand side limit of the above statement is exactly the Hambly-Lyons limit (<ref>. Formally said,

\begin{equation}
\lim_{n\rightarrow \infty} (n!||X^n||)^{2}=c(1)^2=\lim_{n\rightarrow \infty} \mathbb{E}\bigg[\exp \big( -nW_2^2(\mu_n,\mu)\big)\bigg]
\end{equation}

We now propose to relate the Hilbert-Schmidt norm of the truncated signature $||X^n||$ with 2-Wasserstein distance feature in the left-hand side of the previous corollary.

To do so, we exploit the following lemma.

Let $U_{(i)}$, $V_{(i)}$, $i \in \{1,...,n\}$ be two i.i.d. order statistics sampled from a uniform probability distribution. Let $F$ be a solution of the ODE (<ref>) and define $X_i:=F^{-1}(U_{(i)})$ and $Y_i:=F^{-1}(V_{(i)})$. Let $\mu_n^X$ and $\mu_n^Y$ be their respective empirical measures. Then,

\begin{equation}
\sum_{i=1}^n    |\gamma'_{U_{(i)}}-\gamma'_{V_{(i)}}|^p=W_p^p(\mu_n^X, \mu_n^Y)
\end{equation}

First, one can write

\begin{equation}
|\gamma'_{U_{(i)}} - \gamma'_{V_{(i)}}|=|g(X_i)-g(Y_i)|
where $g:=\gamma' \circ F$.
\end{equation}

Assuming $\gamma \in C^2$, we have that $g \in C^1$. Applying the mean value theorem then gives

\begin{equation}
|\gamma'_{U_{(i)}} - \gamma'_{V_{(i)}}|=|X_i-Y_i|\cdot |g'(\xi_i)|
\end{equation}

for $\xi_i \in [X_i, Y_i]$.
By assumption, as $F$ satisfies the ODE (<ref>), we have that

\begin{equation}
|g'(s)|=|\gamma''_{F(s)}|\cdot |F'(s)|=1, \; \forall s \in [0,1]
\end{equation}

Finally, one concludes using the characterisation of Wasserstein distances in lemma <ref>.

We can now express $n!||X^n||$ in terms of $W_2^2(\mu_n^X, \mu_n^Y)$

\begin{equation}
(n!\left\vert \left\vert X^{n}\right\vert \right\vert)^2 =\mathbb{E}\left[ \exp
\left( -\frac{n}{2} W_2^2(\mu_n^X,\mu_n^Y)-\frac{n}{12}W_4^4(\mu_n^X, \mu_n^Y)+\sum_{j=1}^n O\big(|X_j-Y_j|^6\big)\right) \right]
\end{equation}

We now use B-G-U to show that we can substitute $W_2^2(\mu_n^X,\mu_n^Y)$ by the Hambly-Lyons integral. To do that we exploit a KMT embedding theorem

PLACEHOLDER FOR KMT EMBEDDING THEOREM FOR QUANTILE PROCESSES HERE

\begin{equation}
W_2^2(\mu_n^X, \mu_n^Y) \overset{D}{\longrightarrow} 2\int_{0}^1 (W_{0,s}^2)||\gamma_s''||ds
\end{equation}

Start by observe that

\begin{align}
nW_2^2(\mu_n^X, \mu_n^Y)&= \int_{0}^1 \g|F^{-1}_{X}(s)-F^{-1}_{Y}(s)\bigg|^2ds\\
&= \int_{0}^1 \bigg|(F^{-1}_{X}(s)-F^{-1}(s)) - (F^{-1}_{Y}(s)-F^{-1}(s))\bigg|^2ds \\
&=||\mathbb{F}^{n,X}||_2^2 +||\mathbb{F}^{n,Y}||_2^2 -2\int_0^1 \mathbb{F}_s^{n,X}\mathbb{F}_s^{n,Y}ds
\end{align}

We now show that $\int_{0}^1 \mathbb{F}_s^{n,X}\mathbb{F}_s^{n,Y}ds \overset{D}{\longrightarrow}0$. We do so by showing that $\int_{0}^1 \mathbb{F}_s^{n,X}\mathbb{F}_s^{n,Y}ds$ converges in mean to $0$.
By Fubini's theorem,

\begin{align}
\mathbb{E}\bigg[\int_{0}^1 \mathbb{F}_s^{n,X}\mathbb{F}_s^{n,Y}ds\bigg]&=\int_{0}^1\mathbb{E}\bigg[\mathbb{F}_s^{n,X}\mathbb{F}_s^{n,Y}\bigg]ds \label{eq:expectation__wasserstein2_bgu_lemma}
\end{align}

In virtue of the KMT embedding theorem for quantile process (QUOTE IT!, there exists two sequences of Brownian bridges $B_{t}^{n,X}$ and $B_{t}^{n,Y}$ defined on the same probability space as $v_n^X$ and $v_n^Y$ and a constant $C_Z>0$ such
that for every $k>0$ there exists $A_{Z}\left(  k\right)  =A_Z>0$ so that for every $n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ and $y>0$ we have

\begin{equation}
\mathbb{P}\left(  \sup_{t\in\left[  0,1\right]  }\left\vert \mathbb{F}_{t}%
^{n,Z}-|\gamma_{t}^{\prime\prime}|B_{t}^{n,Z}\right\vert >\epsilon_n \bigg) \leq C_Zn^{-k}
\end{equation}

for $Z \in \{X,Y\}$ and $\epsilon_n^Z:=\frac{A_Z}{\sqrt{n}}\log(n)$. For the sake of lightening the notation, we define the event $E_Z:=\bigg\{ \sup_{\omega \in \Omega} \big|\mathbb{F}_s^{n,Z}-|\gamma_t''|B_t^{n,Z} \big|>\epsilon_n\bigg\}$.

Using the two above sequences of Brownian bridges, we can expand the expectation inside the right-hand side integral of (<ref>)

\begin{align}
\mathbb{E}\bigg[\mathbb{F}_s^{n,X}\mathbb{F}_s^{n,Y}\bigg]=\mathbb{E}\bigg[\overbrace{\big(\mathbb{F}_s^{n,X}-|\gamma_s''|B_s^{n,X}\big)\big(\mathbb{F}_s^{n,Y}-|\gamma_s''|B_s^{n,Y}\big)}^{=:a_1^n(s)}\bigg]\notag \\
+\mathbb{E}\bigg[\overbrace{\big(\mathbb{F}_s^{n,X}-|\gamma_s''|B_s^{n,X}\big)|\gamma_s''|B_s^{n,Y}}^{=:a_2^n(s)}\bigg] \notag \\
+ \mathbb{E}\bigg[\overbrace{\big(\mathbb{F}_s^{n,Y}-|\gamma_s''|B_s^{n,Y}\big)|\gamma_s''|B_s^{n,X}}^{=:a_3^n(s)}\bigg] + \mathbb{E}\bigg[\overbrace{|\gamma_s''|^2B_s^{n,X}B_s^{n,Y}}^{=:a_4^n(s)}\bigg]
\end{align}

One can show that $\mathbb{E}[a_1^n]$ vanishes in the limit $n\rightarrow \infty$ after application of the Hungarian theorem as follows. Let $k>2$ and define the sets

\begin{align}
G_{j,n}^Z&:=\bigg\{\omega \in \Omega\;\bigg|\;\sup_{s\in[0,1]}\big|\mathbb{F}_s^{n,Z}-|\gamma_s''|B_s^{n,Z}\big| \in [j-1;j)\bigg\},\\
H_{j,n}&:=\bigg\{\omega \in \Omega\;\bigg|\;\sup_{s\in[0,1]}\big|\mathbb{F}_s^{n,X}-|\gamma_s''|B_s^{n,X}\big|\cdot \big|\mathbb{F}_s^{n,Y}-|\gamma_s''|B_s^{n,Y}\big| \in [j-1;j)\bigg\},\\
I_{j,k;n}^Z&:=H_{j,n} \cap G_{k;n}^Z, \quad Z \in \{X,Y\}.
\end{align}

These sets satisfy the following properties

\begin{align}
E_X \cup E_Y &= \dot{\bigcup}_{j=1}^\infty H_{j,n} \\
H_{j,n}&=\dot{\bigcup}_{k=1}^\infty I_{j,k;n} \\
\sum_{k=j+1}^\infty I_{j,k;n}^Z &\subset G_{1,n}^{\bar{Z}}\cap \bigcup_{k=j+1}^\infty G_{j,n}^Z
\end{align}

where $\bar{Z} \in \{X,Y\} \setminus Z$. This allows us to write

\begin{align}
H_{j,n}\subset \bigcup_{k=2}^j I_{j,k;n}^Z \cup \bigg( \bigcup_{k=j+1}^N G_{k,n}^Z \cap G_{1;n}^{\bar{Z}}\bigg) \cup \bigg( \bigcup_{k=N+1}^{\infty}G_{k,n}^Z\cap G_{1,n}^{\bar{Z}} \bigg) \label{eq:proof_2_wasserstein_mu_nu_set_decomposition}
\end{align}

Put together, for an arbitrary $M > N$, one finds

\begin{align}
\mathbb{E}[|a_1^n(s)|]&=\mathbb{E}\big[|a_1^n(s)| \cdot (\mathds{1}_{E_X \cup E_Y}+ \mathds{1}_{(E_X \cup E_Y)^c})\big] \notag \\
&\leq \sum_{j=1}^\infty j \mathbb{P}(H_{j,n})\\
&\overset{(\ref{eq:proof_2_wasserstein_mu_nu_set_decomposition})}{\leq} \sum_{j=1}^M \frac{j^2(j-1)}{2}\cdot C_Z \cdot n^{-k} + \sum_{j=1}^M \bigg((N-j-1)C_Z \cdot n^{-k}+\sum_{l=N}^M C_Z \cdot n^{-k} \bigg)j\\
&\quad +\sum_{j=M+1}^\infty\frac{j^2(j-1)}{2}\cdot C_Z \cdot n^{-k} + \sum_{j=M+1}^\infty\bigg((N-j-1)C_Z \cdot n^{-k}+\sum_{k=N}^M C_Z \cdot n^{-k} \bigg)j \\
&\quad + \epsilon_n^X \epsilon_n^Y \\

\end{align}

\begin{align}
&\overset{(n \rightarrow \infty)}{\longrightarrow 0}.
\end{align}

For $\mathbb{E}[a_2^n]$ and $\mathbb{E}[a_3^n]$, the proof exploits the same ideas as the above case. For $\mathbb{E}[a_4^n]$, one uses the following argument:
finish writing

NEXT THINGS TO SEE:
- what changes to make so that $W_2(\mu_n^X,\mu)$ can be substituted by $W_2(\mu_n^X,\mu_n^Y)$ in the expectation?
- speed of convergence to the BGU limit

§ GENERALISING THE HAMBLY-LYONS' LIMIT

Recall equation (<ref>) where we re-expressed the Hambly-Lyons limit in terms of the second derivative $\gamma''$ instead of $\gamma'''$,

\begin{equation}
\int_{0}^{l}\left(  W_{s}^{0}\right)  ^2 |\gamma_s''| ds\right)  \right]  ^{1/2}
\end{equation}

\begin{align}
r_{n,D}&:=\mathbb{E}\left[  e^{-\frac{D}{n}\sum_{j=1}%
^{n}|\gamma_{j/n}^{''}|^2\left(  B_{j/n}^{U}\right)  ^{2}}\right]^{1/2}
\end{align}

where $B^U$ and $B^V$ are two Brownian bridges vanishing at $t \in \{0,1\}$. One can interpret $r_{n,D}$ as a generalised discretised version of that limit. In particular, for $D=1$, $r_{n,D}$ converges exactly to $c(1)$ when $n\rightarrow \infty$. This section aims to recover that limit for $C^{1+}$ curves by exploiting this observation as follows.

Subsection <ref> First, we establish bounds of the following form

\begin{equation}
\alpha_n^-(y_n;\theta)\; \cdot\; r_{n,D} \; + \; \lambda_n^-(y_n;\theta) \leq \mathbb{E}\bigg[\exp\big(-n\frac{D}{2} W_2^2(\nu_n^X, \nu_n^Y)\bigg)\bigg] \leq \alpha_n^+(y_n;\theta) \; \cdot \; r_{n,D} \; + \; \lambda_n^+(y_n;\theta) \label{eq:example_bounds_w2}
\end{equation}

controlled by $y_n$ and some other control parameters $\theta$.

Subsection <ref> Then, we derive and apply local limit theorems in equation (<ref>) to obtain new bounds that only depends on $n$ and $y_n$.

Subsection <ref> Finally, we substitute the resulting bounds in Proposition <ref> to obtain

\begin{equation}
\tilde{\alpha}_n^-(y_n) \cdot r_{n,D} \; + \; \tilde{\lambda}_n^-(y_n) \leq n!||X^n||)^2 \leq \tilde{\alpha}_n^+(n,y_n) \cdot r_{n,D} + \tilde{\lambda}_n^+(y_n).
\end{equation}

and show that $n\rightarrow \infty$

\begin{align}
\hat{\alpha}_n^\pm(y_n) \rightarrow 1 \\
\hat{\lambda}_n^\pm(y_n) \rightarrow 0
\end{align}

for a suitable class of sequences $y_n$ converging to 0, yielding our desirata.

§.§ Upper and lower bounds on $W_2^2(\nu_n^X, \nu_n^Y)$

We are looking to bound $\mathbb{E}\big[\exp(-nDW_2^2(\nu_n^X, \nu_n^Y))\big]$ in such a way that $r_{n,D}$ appear as in equation (<ref>). We do so by introducing the Brownian bridges $B^U$ and $B^V$ as appearing in the definition of $r_{n,D}$ and two other pairs of stochastic processes built using the collections of samples $U_{(i)}$ and $V_{(i)}$. Some algebra on these processes allows us to bound $\mathbb{E}\big[\exp(-nDW_2^2(\nu_n^X, \nu_n^Y))\big]$ in terms of three events, each respectively defined in terms of the maximum modulus of the difference between one of the above pair.

Let the environment definitions hold (Definition <ref>) and let two independent Brownian bridges $B^{U}=\left(  B_{t}^{U}\right)
_{t\in\left[  0,1\right]  }$ and $B=\left(  B_{t}^{V}\right)  _{t\in\left[
0,1\right]  }$, which return to zero at $t=1$. Assume that $\left(  U_{(i)}\right)_{i=1}^{n}$, $\left(  V_{(i)}\right)
_{i=1}^{n}$, $B^{U}$ and $B^{V}$ are (pairwise) independent.

(i)  Define the quantile processes $\mathbb{X}$ and $\mathbb{Y}$ associated to $X$, respectively $Y$, by

\[
\mathbb{X}_{t} := \sqrt{n}\left[  G_{n}^{X}\left(  t\right) - F^{-1}(t)\right], \quad \mathbb{Y}_{t} := \sqrt{n}\left[  G_{n}^{Y}\left(  t\right)  -F^{-1}(t)\right]
\text{ for }t\text{ in }\left[  0,1\right],
\]

(ii) We also define their associated deviation processes $\left(  \Gamma_{t}^{U}\right)  _{t\in\left[
0,1\right]  }$ and $\left(  \Gamma_{t}^{V}\right)  _{t\in\left[  0,1\right]  }$

\[
\Gamma_{t}^{U}:= \frac{1}{\sqrt{n}}\left(  \mathbb{X}_{t}-\frac{|\gamma^{''}_t|}{\sqrt{2}}B_{t}^{U}\right)
,\quad \Gamma_{t}^{V}:= \frac{1}{\sqrt{n}}\left(  \mathbb{Y}_{t}-\frac{|\gamma^{''}_t|}{\sqrt{2}}B_{t}%
^{V}\right)  .\text{ }%
\]

(iii) Let $x,y,z \in \mathbb{R}_+$. We define the control events $A_n,\; B_n$ and $C_n$ associated with the processes $(X,Y)$, $(\Gamma^U,\Gamma^V)$, and $(B^U,B^V)$ respectively as

\begin{align}
A_{n}&:= A_n(My) := \left\{  \max_{j=1,..,n}\left\vert
X_{\left(  j\right)  }-Y_{\left(  j\right)  }\right\vert \leq My\right\}
B_{n}  & := B_{n}(2x) := \left\{  \max_{j=1,..,n}\left\vert \Gamma_{j/n}^{U}-\Gamma_{j/n}%
^{V}\right\vert \leq 2x\right\}, \\
C_{n}  & := C_n(\sqrt{2n}z) := \left\{  \max_{j=1,..,n}\left\vert B_{j/n}^{U}-B_{j/n}%
^{V}\right\vert |\gamma_{j/n}^{''}| \leq \sqrt{2n}z\right\},
\end{align}

where $M:=\max_{s\in[0,1]}||\gamma_s''||$.

Let $n\in%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$, and suppose that on the same probability space we have:

* The environment definitions (Definition <ref>).

* The quantile processes $\mathbb{X}$ and $\mathbb{Y}$ associated to $X$ and $Y$ and their corresponding deviation processes $\Gamma_t^U$ and $\Gamma_t^V$ (Definition <ref>).

* The control events $A_n, \; B_n, \; C_n$ respectively associated with the processes $X,\;Y$, their deviation processes $\Gamma^U, \; \Gamma^V$, and Brownian bridges $B^U,\;B^V$ for some positive constants $x,y,z$ (Definition <ref>).

Then, for any $D \in \mathbb{R}$, it holds that

\begin{align}
\alpha^{-}_n r_{n, D} + \lambda^{-}
\; \leq \; \mathbb{E}\left[  \exp\left(-DnW_{2}^{2}\left(  \nu_{n}%
^{U},\nu_{n}^{V}\right)  \right)  \right] \; \leq \; \alpha^{+}_n r_{n, D} + \lambda^{+} \label{prop:ineq_w2_double_bounds}
\end{align}

\begin{align}
r_{n,D}&:=\mathbb{E}\left[  e^{-\frac{D}{n}\sum_{j=1}%
^{n}|\gamma_{j/n}^{''}|^2\left(  B_{j/n}^{U}\right)  ^{2}}\right]^{1/2} \\
\alpha_n^{\pm}&:=e^{\pm2Dnx\left(  My+z\right)}
\end{align}

\begin{align}
\lambda^{\pm}:&=
\left\{
\begin{array}{ll}
-e^{-2Dnx\left(My+z\right)} \cdot \mathbb{P}(G_n^c), & \text{if -}\\
e^{-DnM^2y^{2}}\mathbb{P}\left( A_{n}^{c}\right)  +\mathbb{P}( \hat{G}_{n}^{c}), & \text{if +}
\end{array}
\right.
\end{align}

\begin{align}
G_n^c := (A_{n}^c\cup B_{n}^c \cup C_{n}^c), \quad
\tilde{G}_n^c := A_{n}%
\cap\left(  B_{n}^{c}\cup C_{n}^{c}\right)
\end{align}

By definition of the inverse empirical distribution function and the quantile processes, we have that
$G_{n}^{X}\left(  i/n\right)  = X_{\left(  i\right)  }$ and $G_{n}^{Y}\left(
i/n\right)  =Y_{\left(  i\right)  }.$ It follows that

\begin{equation}
X_{\left(  i\right)  }-Y_{\left(  i\right)  }=\Gamma_{i/n}^{U}-\Gamma
_{i/n}^{V}+\frac{|\gamma^{''}_{\frac{i}{n}}|}{\sqrt{2n}}\left(  B_{i/n}^{U}-B_{i/n}^{V}\right) \label{eq:link_ui_and_events}
\end{equation}

\begin{align}
\mathbb{X}_{\frac{i}{n}}-\mathbb{Y}_{\frac{i}{n}}&=\sqrt{n}\big( X_{(i)}-Y_{(i)} \big)  \label{eq:link_ui_and_empirical}
\end{align}

From equations (<ref>) and (<ref>), we have the elementary estimate

\begin{align*}
& \left\vert \sum_{j=1}^{n}\left(  X_{\left(  j\right)  }-Y_{\left(  j\right)
}\right)  ^{2}-\frac{1}{2n}\sum_{j=1}^{n}\left(  B_{j/n}^{U}-B_{j/n}%
^{V}\right)  ^{2}|\gamma^{''}_{j/n}|^2\right\vert \\
& \leq\sum_{j=1}^{n}\left\vert \left(  \Gamma_{j/n}^{U}-\Gamma_{j/n}%
^{V}\right)  \left(   \big(X_{\left(  j\right)}-Y_{\left(  j\right)}\big) + \frac
{|\gamma_{j/n}^{''}|}{\sqrt{2n}}B_{j/n}^{U}-\frac{|\gamma_{j/n}^{''}|}{\sqrt{2n}}B_{j/n}^{V}\right)  \right\vert
\end{align*}

It is then easy to see that on the event $G_n:=A_{n}\cap B_{n}\cap C_{n}$ we

\[
\left\vert \sum_{j=1}^{n}\left(  X_{\left(  j\right)  }-Y_{\left(  j\right)
}\right)  ^{2}-\frac{1}{2n}\sum_{j=1}^{n}\left(  B_{j/n}^{U}-B_{j/n}%
^{V}\right)  ^{2} |\gamma_{j/n}^{''}|^2\right\vert \leq 2nx\left( My+z\right)  ,
\]

and therefore on this event the following inequalities hold

\begin{equation}
e^{-2Dnx\left(  My+z\right)  }\leq e^{-DnW_{2}^{2}\left(  \nu_{n}%
^{X},\nu_{n}^{Y}\right) + \frac{D}{2n}\sum_{j=1}^{n}(B_{j/n}^{U}-B_{j/n}%
^{V})^{2}\cdot |\gamma_{j/n}^{''}|^2}\leq e^{2Dnx\left(  My+z\right)  }. \notag
\end{equation}

The complementary event $G_{n}^{c}=A_{n}^{c}\cup B_{n}^{c}\cup C_{n}^{c}=A_{n}%
\cap\left(  B_{n}^{c}\cup C_{n}^{c}\right)  \cup A_{n}^{c}=:\hat{G}_{n}^{c}\dot{\cup}
A_{n}^{c}$ and on $A_{n}^{c}$ we have that

\[
\sum_{j=1}^{n}\left(  X_{\left(  j\right)  }-Y_{\left(  j\right)  }\right)
^{2}\geq nM^2 y^{2}%
\]

I do not think the above statement is right; the inequality should be taken over the max

which in turns implies

\begin{align*}
\mathbb{E}\bigg[e^{-DnW_2^2(\nu_n^X, \nu_n^Y)}\cdot 1_{G_n^c} \bigg] \leq e^{-DnM^2y^2}\cdot \mathbb{P}(A_n^c)+\mathbb{P}(\hat{G}_n^c)
\end{align*}

It follows then

\begin{align*}
& e^{-2Dnx\left(  My+z\right)  }\mathbb{E}\left[  e^{-\frac{D}{2n}\sum_{j=1}%
^{n}|\gamma_{j/n}^{''}|^2\left(  B_{j/n}^{U}-B_{h/n}^{V}\right)  ^{2}}1_{G_n}\right]  \\
&\quad \leq \mathbb{E}\left[  \exp\left(-DnW_{2}^{2}\left(  \nu_{n}%
^{X},\nu_{n}^{Y}\right)  \right)  \right] \\
&\quad \quad \leq e^{2Dnx\left(  My+z\right)  }\mathbb{E}\left[  e^{-\frac{D}{2n}%
\sum_{j=1}^{n}|\gamma_{j/n}^{''}|^2\left(  B_{j/n}^{U}-B_{j/n}^{V}\right)  ^{2}}1_{G_n}\right]
+e^{-D nM^2y^{2}} \mathbb{P}\left(  A_{n}^{c}\right)  +\mathbb{P}(
\hat{G}_{n}^{c})
\end{align*}

and then, using the equality in distribution

\[
\left\{  B_{j/n}^{U}-B_{j/n}^{V}:j=1,...,n\right\}  \overset{d}{=}\left\{
2B_{U_{\left(  j\right)  }}^{U}:j=1,...,n\right\}  ,
\]

\begin{equation}
\mathbb{E}\bigg[e^{-\frac{D}{2n}\sum_{j=1}^{n}||\gamma_{j/n}''||^2(B_{j/n}^U - B_{j/n}^V)^2}\bigg] - \mathbb{P}(G_n^c) \leq \mathbb{E}\bigg[e^{-\frac{D}{2n}\sum_{j=1}^{n}||\gamma_{j/n}''||^2(B_{j/n}^U - B_{j/n}^V)^2} 1_{G_n}\bigg],
\end{equation}

we arrive at

\begin{align*}
& e^{-2Dnx\left(  My+z\right)  }\mathbb{E}\left[  e^{-\frac{D}{2n}\sum_{j=1}%
^{n}|\gamma_{j/n}^{''}|^2\left(  B_{j/n}^{U}\right)  ^{2}}\right]
- e^{-2Dnx\left(My+z\right)  }  \mathbb{P}(G_n^c)  \\
& \quad \leq \mathbb{E}\left[  \exp\left(-DnW_{2}^{2}\left(  \nu_{n}%
^{X},\nu_{n}^{Y}\right)  \right)  \right] \leq  \\
&\quad \quad e^{2Dnx\left(  My+z\right)  }\mathbb{E}\left[  e^{-\frac{D}{2n}\sum
_{j=1}^{n}|\gamma_{j/n}^{''}|^2\left(  B_{j/n}^{U}\right)  ^{2}}\right] +e^{-DnM^2y^{2}%
}\mathbb{P}\left(  A_{n}^{c}\right)  +\mathbb{P}( \hat{G}_{n}^{c})  ,
\end{align*}

which are the claimed inequalities (<ref>).

The event $A_n$ can be controlled via the maximum distance $y_n$ between the original uniform samples $U$ and $V$. In particular, for

\begin{equation}
y_n:=\max_{j \in \{1,...,n\}}|U_{(j)}-V_{(j)}|,
\end{equation}

we have

\begin{equation}
A_n(My_n) \subset \left\{\max_{j=1,..,n}|U_{(j)}- V_{(j)}| \leq y_n\right\}.
\end{equation}

This follows from equation (<ref>) which implies

\begin{equation}
\max_{j=1,...,n}|X_{(j)}-Y_{(j)}| \leq \max_{s\in[0,1]}|\gamma_s''|\max_{j=1,...,n}|U_{(j)}-V_{(j)}|=My_n
\end{equation}

§.§ Controlled bounds on $W_2^2(\nu_n^X, \nu_n^Y)$ using local limit theorems

The additive constants $\lambda_n^{\pm}$ in Proposition <ref> explicitly depend on the probability masses $\mathbb{P}(G_n^c)$ and $\mathbb{P}({\tilde{G}_n^c})$ of the events

\begin{align}
G_n^c := (A_{n}^c\cup B_{n}^c \cup C_{n}^c), \quad
\tilde{G}_n^c := A_{n}%
\cap\left(  B_{n}^{c}\cup C_{n}^{c}\right).
\end{align}

In order to be able to take the limit $n\rightarrow \infty$, we need to understand how the probability mass of these events evolve in function of $n$ and $y_n$. To do so, we work with the control events $A_n$, $B_n$ and $C_n$ and bound their probability mass using well known local limit theorems. However, in the case of $B_n$, there are no off the shelves inequality that we can use: the closest result to our desirata is the so called Hungarian approximation <cit.>,

On a suitable probability space for independent uniform (0,1) random variables $U_1, \; U_2,\;...$,
define the empirical process

\begin{equation}
\end{equation}

then, there exists a sequence of Brownian bridges $\{B_n(t), \; 0 \leq t \leq 1\}$ such that

\begin{equation}
\mathbb{P}\bigg(\max_{0 \leq t \leq 1} \big|E_n(n) - B_n(t)\big| > \frac{1}{\sqrt{n}} (a\log(n)+x) \bigg) \leq be^{-cx}
\end{equation}

for all positive integers $n$ and all $x > 0$, where $a,\,b$ and $c$ are positive constants.

Indeed, after rewriting the event $B_n^c$ as

\begin{align}
B_n^c:= \bigg\{ \max_{j=1,...,n} \bigg|\bigg(\mathbb{X}_{j/n}-\frac{|\gamma_{j/t}''|}{\sqrt{2}} B^U(t)\bigg)+\bigg(\mathbb{Y}_{j/n}-\frac{|\gamma_{j/t}''|}{\sqrt{2}} B^V(t)\bigg)  \bigg|>\frac{2x}{\sqrt{n}} \bigg\},
\end{align}

it becomes clear that the main difference lies in the fact that the Hungarian approximation features the empirical process $E_n(n)$ while our desirata would involve a quantile process such as $\mathbb{X}$ or $\mathbb{Y}$. This subsection will therefore consist in the derivation of a quantile process version of the Hungarian approximation that we then use alongside other local limits theorems in Proposition <ref> to obtain explicit bounds in terms of $n$ and $y_n$.

Let $F, \;X, \;F^X$ and $G_n^X$ be as defined in the environment definitions (Definition <ref>). To lighten the notation, we omit the subscript and write $F_n := F_n^X$ and $G_n := G_n^X$.

The uniform quantile process $\mathbb{U}_t^n$ associated with $X$ is given by

\[
\mathbb{U}_{t} := \mathbb{U}_{t}^{n} := \sqrt{n}\left(  G_{n}^X\left(  t\right)
-t\right)  .
\]

We also recall that the quantile process $\mathbb{F}_n^t$ associated with $X$ is defined by

\[
\mathbb{F}_{t}^{n}:=\sqrt{n}\left(  F_{n}^{-1}\left(  t\right)  -F^{-1}\left(
t\right)  \right)  .
\]

Using the mean value theorem, there exists $\theta_{t,n}\in\lbrack t\wedge
G_{n}^{-1}\left(  t\right)  ,t\vee G_{n}^{-1}\left(  t\right)  ]$ such that

\begin{equation}
\mathbb{F}_{t}^{n}=\left(  F^{-1}\right)  ^{\prime}\left(  \theta
_{t,n}\right)  \mathbb{U}_{t}^{n}=|\gamma_{\theta_{t,n}}^{\prime\prime
\end{equation}

which gives the ingredients we need to prove the following result.

Let $\gamma:[0,1] \rightarrow \mathbb{R}^d$ be a $C^{1+}$ curve (Definition <ref>). Then there
exists a sequence of Brownian bridges $\left(  B_{t}^{n}\right)  _{t\in\left[
0,1\right]  }$ with the following property: there exists a constant $B>0$ such
that for every $k>0$ there exists $A\left(  k\right)  =A>0$ so that for every
%TCIMACRO{\U{2115} }%
\mathbb{N}

\begin{equation}
\mathbb{P}\left(  \sup_{t\in\left[  0,1\right]  }\left\vert \mathbb{F}_{t}%
^{n}-|\gamma_{t}^{\prime\prime}|B_{t}^{n}\right\vert >\frac{A}{\sqrt{n}}\log
n\right)  \leq Bn^{-k}.\label{KMT modified}%
\end{equation}

Using the standard Hungarian theorem for the uniform quantile process <cit.> we know that there
exist universal constants $a,b>0$ and a sequence of Brownian bridges such

\begin{equation}
\mathbb{P}\left(  \sup_{t\in\left[  0,1\right]  }\left\vert \mathbb{U}_{t}%
^{n}-B_{t}^{n}\right\vert >\frac{a}{\sqrt{n}}\log n\right)  \leq
bn^{-k},\text{ }n=1,2...\label{KMT quant}%
\end{equation}

Using (<ref>) we have that

\begin{align*}
\left\vert \mathbb{F}_{t}^{n}-|\gamma_{t}^{\prime\prime}|B_{t}^{n}\right\vert
& =\left\vert |\gamma_{\theta_{t,n}}^{\prime\prime}|\mathbb{U}_{t}^{n}%
-|\gamma_{t}^{\prime\prime}|B_{t}^{n}\right\vert \\
& \leq\left\vert \left(  |\gamma_{\theta_{t,n}}^{\prime\prime}|-|\gamma
_{t}^{\prime\prime}|\right)  \mathbb{U}_{t}^{n}\right\vert +\left\vert
|\gamma_{t}^{\prime\prime}|\left(  \mathbb{U}_{t}^{n}-B_{t}^{n}\right)
\right\vert \\
& \leq L\left\vert \theta_{t,n}-t\right\vert |\mathbb{U}_{t}^{n}|+L\left\vert
\left(  \mathbb{U}_{t}^{n}-B_{t}^{n}\right)  \right\vert ,
\end{align*}

where $L:=\max\{|\gamma^{\prime\prime}|_{\text{Lip}},|\gamma^{\prime\prime
}|_{\infty}\}.$ It holds that $\left\vert \theta_{t,n}-t\right\vert $
$\leq\frac{1}{\sqrt{n}}|\mathbb{U}_{t}^{n}|,$ and therefore

\[
\left\vert \mathbb{F}_{t}^{n}-|\gamma_{t}^{\prime\prime}|B_{t}^{n}\right\vert
\leq\frac{L}{\sqrt{n}}\left(  \mathbb{U}_{t}^{n}\right)  ^{2}+L\left\vert
\left(  \mathbb{U}_{t}^{n}-B_{t}^{n}\right)  \right\vert .
\]

A simple estimate then shows that for any $c>0$ we have

\begin{align}
& \mathbb{P}\left(  \sup_{t\in\left[  0,1\right]  }\left\vert \mathbb{F}%
_{t}^{n}-|\gamma_{t}^{\prime\prime}|B_{t}^{n}\right\vert >\frac{c}{\sqrt{n}%
}\log n\right)  \nonumber\\
& \leq\mathbb{P}\left(  \sup_{t\in\left[  0,1\right]  }\left\vert
\mathbb{U}_{t}^{n}\right\vert >\left(  \frac{c}{2L}\log n\right)
^{1/2}\right)  +\mathbb{P}\left(  \sup_{t\in\left[  0,1\right]  }\left\vert
\mathbb{U}_{t}^{n}-B_{t}^{n}\right\vert >\frac{c}{2L\sqrt{n}}\log n\right)
\end{align}

Using the Dvoeretsky-Kiefer-Wolowitz theorem <cit.> for the uniform quantile process,
the first term is bounded by

\[
\mathbb{P}\left(  \sup_{t\in\left[  0,1\right]  }\left\vert \mathbb{U}_{t}%
^{n}\right\vert >\frac{c}{2L\sqrt{n}}\log n\right)  \leq d\exp\left(
-\frac{c}{2L}\log n\right)  ,
\]

and this in turn is bounded by $dn^{-k}$ as soon as $c\geq2Lk.$ The second
term in (<ref>) can be handled using (<ref>) in that provided
$c\geq2La,$ where $a$ is absolute constant in (<ref>), it holds

\[
\mathbb{P}\left(  \sup_{t\in\left[  0,1\right]  }\left\vert \mathbb{U}_{t}%
^{n}-B_{t}^{n}\right\vert >\frac{c}{2L\sqrt{n}}\log n\right)  \leq bn^{-k}.
\]

By taking $A:=2L\max\left(  k,a\right)  $ we see that the conclusion
(<ref>) holds with $B:=b+d.$

We are now in position to rework the additive constant $\lambda_{n}^{\pm}$ in proposition <ref> and obtain bounds for $W_2(\nu_n^X, \nu_n^Y)$ expressed in terms of $y_n$ and $n$ only.

Under the assumptions of Proposition <ref>,
there exists a constant $B>0$ such
that for every $k>0$ there exists $A\left(  k\right)  =A>0$ so that for every $n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ and $y>0$ we have

\begin{align}
& \beta_n^-(y)\cdot r_{n,D/2}  - \kappa_{n,D}^-(y)
\; \leq \; \mathbb{E}\bigg[  \exp\left(  -DnW_{2}^{2}\left(  \nu_{n}%
^{X},\nu_{n}^{Y}\right)  \right)  \bigg] \; \leq \; \beta_n^+(y)\cdot r_{n,D/2} + \kappa_{n,D}^+(y) \notag
\end{align}

\begin{align}
\beta_{n,D}^{\pm}(y)&:= n^{\pm2ADMy} \\
\kappa_{n,D}^{\pm}(y)&:=
\left\{
\begin{array}{ll}
n^{-2ADMy}\cdot p_n, & \text{if -}\\
4be^{-DnM^2y^{2}}n^{-k} + p_n, & \text{if +}
\end{array}
\right.
\end{align}

where $p_n:=bn^{-k}+4e^{-\frac{1}{2}nM^2y^2} + 4e^{-2ny^2}$

We start by using the version of the Hungarian approximation for general quantile process we presented in proposition <ref>. Hence, there exist a constant $B>0$ such that for every $k>0$, there exists $A(k)=A>0$ and two
independent Brownian bridges, $B_{n}^{U}$ and $B_{n}^{V}$, defined on the
same probability space as the random variables $\left\{
U_{i,}V_{i}:i=1,...,n\right\},$ such that for every $n\in\mathbb{N}$

\begin{equation}
\mathbb{P}\left(  \sup_{t\in\left[  0,1\right]  }\left\vert \mathbb{U}_{t}%
^{n}-|\gamma_{t}^{\prime\prime}|B_{t}^{n}\right\vert >\frac{A}{\sqrt{n}}\log
n\right)  \leq Bn^{-k}.%
\end{equation}

In the notation of the
previous proposition, this gives that

\begin{equation*}
\mathbb{P}\left( B_{n}^{c}\right) =\mathbb{P}\left(
\max_{j=1,..,n}\left\vert \Gamma _{U_{\left( j\right) }}^{U}-\Gamma
_{V_{\left( j\right) }}^{V}\right\vert >x\right) \leq 2\mathbb{P}\left(
\sup_{t\in \left[ 0,1\right] }\left\vert \Gamma _{t}^{U}\right\vert >\frac{x%
}{2}\right) \leq b n^{-k},
\end{equation*}

when $x$ is taken to equal $(2n)^{-1}\left( A\log n\right) .$
Using the
Dvoretzky–Kiefer–Wolfowitz inequality we also have that

\begin{equation*}
\mathbb{P}\left( A_{n}^{c}\right) =\mathbb{P}\left(
\max_{j=1,..,n} \left\vert U_{\left( j\right) }-V_{\left( j\right)}
\right\vert >y\right) \leq \mathbb{P}\left( \sup_{t\in \left[ 0,1\right]
}\left\vert F_{n}^{U}\left( t\right) -F_{n}^{V}\left( t\right) \right\vert
>y\right) \leq 4e^{-ny^{2}/2}.
\end{equation*}

It is known that $\mathbb{P}\left( \sup_{t\in \left[ 0,1\right]
}B_{t}^{U}>z\right) =e^{-2z^{2}}$ when $z>0,$ which allows us to obtain the

\begin{eqnarray*}
\mathbb{P}\left( C_{n}^{c}\right)  &=&\mathbb{P}\left(
\max_{j=1,..,n}\left\vert B_{U_{\left( j\right) }}^{U}-B_{V_{\left( j\right)
}}^{V}\right\vert |\gamma_{j/n}^{''}| >\sqrt{n}z\right)  \\
&\leq &2\mathbb{P}\left( \sup_{t\in \left[ 0,1\right] }\left\vert
B_{t}^{U}\right\vert > \frac{\sqrt{n}z}{M}\right) \leq 4\mathbb{P}\left( \sup_{t\in %
\left[ 0,1\right] }B_{t}^{U}>\frac{\sqrt{n}z}{M}\right) =4e^{-2nz^{2}/M^2},
\end{eqnarray*}

where the second line uses the symmetry of the Brownian bridge, $B^{U}%
\overset{d}{=}-B^{U}$ and where $M:=\sup_{t\in[0,1]} |\gamma^{''}(t)|$. By putting these estimates together we obtain

\begin{eqnarray*}
\mathbb{P}(G_n^c) &=&\mathbb{P}\left( A^{c}\cup B^{c}\cup C^{c}\right)  \\
&\leq &bn^{-k}+4e^{-nM^2y^{2}/2}+4e^{-2nz^{2}/M^2}.
\end{eqnarray*}

Using $\hat{G}_n^c \subset G_n^c$, taking $z=My$, and applying Proposition <ref> for $x=(2n)^{-1}\left( A\log n\right)$ yield

\begin{align}
& n^{-2ADMy}r_{n,D}  - n^{-2ADMy} (bn^{-k}+4e^{-\frac{1}{2}nM^2y^2} + 4e^{-2ny^2})  \notag \\
& \quad \leq \mathbb{E}\left[  \exp\left(  -DnW_{2}^{2}\left(  \nu_{X}%
^{U},\nu_{n}^{Y}\right)  \right)  \right] \leq n^{2ADMy}r_{n,D} \\
&\quad \quad + 4be^{-DnM^2y^{2}}n^{-k} + bn^{-k}
+ 4e^{-\frac{1}{2}nM^2y^{2}}+4e^{-2ny^{2}} \notag
\end{align}

We conclude this section by applying the previous corollary in proposition
<ref>, yielding lower and upper bounds on $n!||X^n||$ in terms of $y_n$ and $n$ only.

Under the assumptions of Corollary <ref>, for $y_n:=\max_{i = 1,..,n}|U_{(i)}-V_{(i)}|$,

\begin{align}
& \beta_n^-(y_n) \cdot r_{n,\frac{1}{2}D_n^-}  - \kappa_{n, D_n^-}^-(y_n) + O(y_n^8)
\; \leq \; (n!||X^n||)^2 \; \leq \; \beta_n^+ (y_n) \cdot r_{n,\frac{1}{2}D_n^+} + \kappa_{n, D_n^+}^+(y_n) + O(y_n^8) \notag
\end{align}

\begin{align}
D_n^{\pm}:=(1\pm LMy_n)^2
\end{align}

Apply corollary <ref> on proposition <ref> for $D\in\{\frac{1}{2} D_n^\pm \}$ and $y=y_n$.

§.§ Recovering Hambly Lyons limit

Thanks to proposition (<ref>), we now can investigate the behaviour of $n!||X^n||$ for large $n$. A natural question to ask is whether there exist a sequence of $y_n$ such that $\beta_n^\pm \rightarrow 1$ and $\kappa_n^\pm \rightarrow 0$ for $n\rightarrow \infty$. If this is the case, one would consequently recovers a generalisation of the Hambly-Lyons limit <cit.> presented in equation (<ref>) in the introduction of this article for $C^{1+}$ curves.

Let $\gamma:[0,1] \rightarrow \mathbb{R}$ be a $\mathcal{C}^{1+}$ curve (Definition <ref>). Then,

\begin{equation}
\lim_{n\rightarrow \infty} n!||X^n||^2=\mathbb{E}\bigg[  \exp\bigg(-\int_{0}^1 (W_{s}^{0})^{2}\;|\gamma_s''|^2ds\bigg) \bigg]^{1/2}
\end{equation}

where $\left(  W_{s}^{0}\right)  _{s\in\lbrack0,l]}$ is a Brownian bridge
which returns to zero at time $1$.

Start from the bounds in proposition <ref> for a $k\geq1$. Let $y_n$ be a sequence in $\mathbb{R}$ such that $y_n\rightarrow 0$ when $n\rightarrow \infty$. We are looking to determine constraints on its order of convergence for $\kappa_n^\pm \rightarrow 0$ $(n\rightarrow \infty)$ to hold. Expressing $\kappa_n^\pm$ as a power of $n$ using elementary logarithmic identities and constraining the exponents to converge to a strictly negative value ($-\infty$ included) yield the following constraints

\begin{align}
\begin{cases}
\lim_{n \rightarrow \infty} \frac{ny_n^2}{\log(n)} \in \mathbb{R}_+ \cup \{\infty\} \\
\lim_{n\rightarrow \infty}ny_n^2 \in \mathbb{R}_+ \cup \{\infty\}
\end{cases} \label{eq:condition_yn_bounds_converge}
\end{align}

which are satisfied when

\begin{align}
\liminf_{n\rightarrow \infty} \frac{y_n}{g(n)}>0,
\end{align}

for $g(n):=\sqrt{\frac{\log(n)}{n}$.
\par Moreover, if $y_n$ satisfies condition (\ref{eq:condition_yn_bounds_converge}), $D_n^{\pm}\rightarrow 1,\; (n\rightarrow \infty)$. Finally,
\begin{equation}
\lim_{n\rightarrow \infty} n!||X^n|| = \lim_{n \rightarrow \infty} r_{n, D^{\pm}_n/2} = \mathbb{E}\bigg[  \exp\bigg(-\int_{0}^1 (W_{s}^{0})^{2}\;|\gamma_s''|^2 ds \bigg) \bigg]^{1/2} \label{eq:limit_asymptotics_sig}
\end{equation}
\end{proof}

\par We appreciate this generalisation by explicitly computing the limit for a $C^{1+}$ curve that is neither $C^3$ or $C^2$.

\begin{example}[The piecewise circular curve]
\par For a $k\geq2$, let $\gamma_t:[0,1]\rightarrow \mathbb{R}^2$ be a curve defined by $\gamma_t:=\tilde{\gamma^1} * ... * \tilde{\gamma^k}$ where
\begin{equation}
\tilde{\gamma}^i_t:=C_k^{-1}\big(\cos(A_i \cdot C_k t + B_i), \sin(A_i \cdot C_k t + B_i)\big)
\end{equation}
are planar circles of curvature $C_k$ satisfying $\sum_{i=1}^{k}|supp(\tilde{\gamma}^i)|=1$ with $A_i \in \{\pm 1\}$, $B_i \in \{0, \pi \}$ and $C_k\in \mathbb{R}_{+}$ such that $\gamma$ is unit speed and $C^1$. Corollary (\ref{thm:charac_limit_c2_generalisation}) and theorem 3.2 of Revuz and Yor \cite{revuz2013continuous} gives
\begin{equation}
\lim_{n \rightarrow \infty}n!||X^n||_{HS}=\bigg(\frac{\sqrt{2C_k}}{\sinh(\sqrt{2C_k})}\bigg)^{1/4}=:c_k(1)
\end{equation}
Morever, as $C_k \rightarrow \infty$ when $k \rightarrow \infty$ (consequence of the fact that $\gamma$ is unit speed), one finds
\begin{equation}
\lim_{k \rightarrow \infty} c_k(1)=0.
\end{equation}
\end{example}

\begin{figure}[H]
\includegraphics[width=\linewidth]{graphs/circular_curves_approx_sin.png}
\caption{Unit-speed piecewise circular paths (blue) with constant curvature norm approximating a sinus (in orange). Top and bottom curves respectively satisfy $|\gamma_{\text{top}}(t)''|=\frac{1}{5}|\gamma_{\text{bot}}(t)''|=c\in\mathbb{R}_+$ for all $t$.}
\end{figure}

\section{Full signature asymptotics expansion}\label{sec:full_asymptotic_expansion}

\par We conclude this article by presenting a way to fully recover all high-order terms in the asymptotic development of $n!||X^n||$. We illustrate our approach by explicitly computing high-order terms up to $O(n^{-3})$. This already improves the asymptotic equivalence (equation \ref{eq:original_asymptotics_hilbert_schmidt}) we presented in the introduction, explicitly
\begin{equation}
r_{n,2D_n^{\pm}}\sim 1+\frac{1}{n}\log c\left(  1\right)  +o\left(  \frac
{1}{n}\right)  \text{ as }n\rightarrow\infty \notag
\end{equation}
for a $C^3$ unit-speed curve of length one.

% \begin{proposition}[Asymptotics in $||\cdot||_{\text{HS}}$]\label{prop:asymptotic_dev_augmented}
% \par Let $\gamma:[0,1] \rightarrow \mathbb{R}$ be a unit-speed $\mathcal{C}^{1+}$ curve (Definition \ref{def:c1+_curves}). Then, as $n \rightarrow \infty$,
% \begin{align}
%     (n!||X^n||_{\text{HS}})^{\frac{1}{n}} \sim  1 + \frac{1}{n}\log(c(1)) -\frac{2}{n^4}\sum_{i=1}^{n-1} \sum_{s=1}^{i} s \Gamma_s + \frac{1}{1n^6}\sum_{i=1}^{n-1}\xi_2^i + o(n^{-3}) \label{eq:final_asymptotics}
% \end{align}
% where
% \begin{align}
%     c(1)&:=\mathbb{E}\bigg[  \exp\bigg(-\int_{0}^1 (W_{s}^{0})^{2}\;|\gamma_s''|^2 ds \bigg) \bigg]^{1/2} \\
%     \xi_i^j &:= \sum_{1\leq s_1 < ... < s_i \leq j} \bigg( \sum_{s=1}^{s_1}s\Gamma_s \bigg)\cdot \Gamma_{s_2}\Gamma_{s_3} \ldots \Gamma_{s_i}, \quad i,j\in\{2,...,n\} \\
%     \Gamma_i&:=||\gamma_{i/n}''||^2
% \end{align}
% \end{proposition}

\textcolor{blue}{update result with handwritten notes}
\begin{theorem}[Asymptotics in $||\cdot||_{\text{HS}}$]\label{thm:asymptotic_dev_augmented}
\par Let $\gamma:[0,1] \rightarrow \mathbb{R}$ be a unit-speed $\mathcal{C}^{1+}$ curve (Definition \ref{def:c1+_curves}). Then, as $n \rightarrow \infty$,
\begin{align}
(n!||X^n||_{\text{HS}})^{\frac{1}{n}} \sim  1 + \frac{1}{n}\log(c(1)) -\frac{\lambda}{n^4}\sum_{i=1}^{n-1} \sum_{s=1}^{i} s \Gamma_s + \frac{\lambda}{2n^6}\sum_{i=1}^{n-1}\xi_2^i + o(n^{-3})
\end{align}
\begin{align}
\lambda&:=\frac{c(1)^4}{8} \\
c(1)&:=\mathbb{E}\bigg[  \exp\bigg(-\int_{0}^1 (W_{s}^{0})^{2}\;|\gamma_s''|^2 ds \bigg) \bigg]^{1/2} \\
\xi_i^j &:= \sum_{1\leq s_1 < ... < s_i \leq j} \bigg( \sum_{s=1}^{s_1}s\Gamma_s \bigg)\cdot \Gamma_{s_2}\Gamma_{s_3} \ldots \Gamma_{s_i}, \quad i,j\in\{2,...,n\} \\
\Gamma_i&:=||\gamma_{i/n}''||^2
\end{align}
\end{theorem}

\begin{proof}
\par First, start from equation (\ref{eq:limit_asymptotics_sig}) and let $c_n$ be a converging sequence in $\mathbb{R}_+$ converging to a strictly positive number. Let $r_{n, c_n}$ be defined by
\begin{equation}
r_{n, c_n}:=E\bigg[\exp\big(-\frac{c_n}{2n}\sum_{i=1}^{n}B_{i/n}^2\cdot ||\gamma_{i/n}''||^2\big)\bigg]^{1/2}. \label{def:r_n_equation}
\end{equation}
\par We first reexpress $r_{n,c_n}$ as an expectation over Bessel bridges before exploiting theorem 3.2 in \cite{revuz2013continuous} which relates the latter object with a distributional differential equation (similarly to the proof of lemma \ref{lem:yor_revuz_integral_into_dde}).

\par First, let
\begin{equation}
\mu_n:=\frac{c_n}{2n}\sum_{i=1}^{n} ||\gamma_{i/n}''||^2\cdot  \delta_{\frac{i}{n}}.
\end{equation}
Using theorem 3.2 in \cite{revuz2013continuous}, we have
\begin{equation}
r_{n,c_n}=\phi(1)^{-1/4} \label{eq:relation_rn_to_phi1}
\end{equation} where
\begin{equation}
\phi'' = \phi \cdot \mu_n; \; \phi(0)=0, \; \phi'(0)=1
\end{equation}
in the distributional sense.
\par We solve this using the following piecewise linear Ansatz
\begin{equation}
\phi(s)=t\mathds{1}_{[0;1/n)} + \sum_{i=1}^{n-2} (a_i+b_i t) \mathds{1}_{[i/n;\;(i+1)/n)} + (a_{n-1}+b_{n-1}t)\mathds{1}_{[(n-1)/n;\;1]} \label{eq:dde_rn}
\end{equation}
where we impose the following continuity conditions
\begin{align}
\begin{cases}
1/n=a_1+b_1/n \\
a_i+b_i \frac{i+1}{n}=a_{i+1}+b_{i+1} \frac{i+1}{n}, \quad i \in \{1,...,n-2\}
\end{cases}
\end{align}
which can be reworked into
\begin{align}
\begin{cases}
a_1=-\frac{b_1}{n} + \frac{1}{n} \\
a_i = \frac{1}{n} \big(\sum_{s=1}^{i-1}b_s- ib_i + 1\big), \quad i \in \{2,...,n-1\}
\end{cases} \label{eq:continuity_cond_ai_bi}
\end{align}

\par Moreover, the Ansatz $\phi$ is a solution to the distributional equation (\ref{eq:dde_rn}) if the following constraints on $a_i$ and $b_i$ are satisfied
\begin{align}
\begin{cases}
b_1 = \frac{c_n}{2n^2}\Gamma_1+1 \\
\end{cases}\label{eq:dde_constraints_ai_bi}
\end{align}
for $i \in \{1,..,n-2\}$ where $\Gamma_i:=||\gamma_{i/n}''||^2$.

\par To see this, define the bump function
\begin{align}
\begin{cases}
\exp(-\frac{1}{1-k(x-x_0)^2}), &x\in(x_0-\frac{1}{k}; x_0+\frac{1}{k}) \\
0, & \text{else}
\end{cases}
\end{align}
centered on $x_0$ with width $\frac{2}{k}$. Let $k \in \mathbb{N}$ and $k>n$. Then, considering $f_{\frac{i}{n}, k}$ as a test function in (\ref{eq:dde_rn}) and taking the limit $k\rightarrow \infty$ yields the result.

\par Equivalently, the two above systems can be written as
\begin{align}
\begin{cases}
a_1=-\frac{b_1}{n}+\frac{1}{n} \\
a_i = \frac{1}{n} \big(\sum_{s=1}^{i-1}b_s- ib_i + 1\big), \quad i \in \{2,...,n-1\} \\
b_1=\frac{c_n}{2n^2}\Gamma_1 + 1 \\
b_{i+1} = \frac{c_n}{2n^2}\Gamma_{i+1} + \frac{c_n}{2n^2}\Gamma_{i+1} \sum_{s=1}^{i}b_s+b_i, \quad i \in \{1,...,n-2\} \\
\end{cases} \label{eq:augmented_system_dde_continuity}
\end{align}
\par A lengthy induction using the two bottom equations in system (\ref{eq:augmented_system_dde_continuity}) gives \begin{equation}
b_i=1+\kappa \sum_{s=1}^{i}s\Gamma_s + \kappa^2 \xi_2^i+ \kappa^3 \sum_{s=3}^i \Gamma_s \bigg(\sum_{i=2}^{s-1} \xi_2^i \bigg) + ... + \kappa^i \prod_{s=1}^{i}\Gamma_s
\label{eq:exact_expression_bi}
\end{equation}
\begin{align}
\kappa&:= \frac{c_n}{2n^2} \\
\xi_i^j &:= \sum_{1\leq s_1 < ... < s_i \leq j} \bigg( \sum_{s=1}^{s_1}s\Gamma_s \bigg)\cdot \Gamma_{s_2}\Gamma_{s_3} \ldots \Gamma_{s_i}, \quad i,j\in\{2,...,n\}
\end{align}

\par This expression allows to explicitly express $\phi(1):=a_{n-1}+b_{n-1}$ by noting that the second equation of system (\ref{eq:augmented_system_dde_continuity}) for $i=n-1$ can be written as
\begin{align}
a_{n-1}+b_{n-1}=\frac{1}{n}\bigg(\sum_{i=1}^{n-1}b_i\bigg) + \frac{1}{n} \label{eq:substitution_an_b_n}
\end{align}

\par In virtue of equation (\ref{eq:relation_rn_to_phi1}), we have
\begin{align}
r_{n,c_n}^{-4} &\sim 1 - \frac{c_n}{2n^3}\sum_{i=1}^{n-1}\sum_{s=1}^{i} s\Gamma_s + \frac{c_n^2}{4n^5} \sum_{i=1}^{n-1}\xi_2^i
+ \frac{c_n^3}{8n^7} \sum_{i=1}^{n-1} \sum_{s=3}^{i}\Gamma_s \bigg(\sum_{i=2}^{s-1}\xi_2^i \bigg) + \mathcal{O}(n^{-3})
\end{align}

\par Note that the second, third and fourth terms respectively scale as $\mathcal{O}(1)$, $\mathcal{O}(n^{-1})$ and $\mathcal{O}(n^{-2})$ because of the summations.

\textcolor{blue}{MISSING $C_N$ BELOW, ADD IT}

\par Using a Taylor expansion of the logarithm around $x_0=c(1)^{-4}$ and the fact that $\exp(\log(x)) = x$, we find
\begin{align}
(r_{n,c_n/2})^{\frac{1}{n}} \sim  1 + \frac{1}{n}\log(c(1)) -\frac{c_n}{n^4}\sum_{i=1}^{n-1} \sum_{s=1}^{i} s \Gamma_s + \frac{c_n^2}{2n^6}\sum_{i=1}^{n-1}\xi_2^i + o(n^{-3})
\end{align}
as $n \rightarrow \infty$, where
\begin{align}
\lambda&:=\frac{c(1)^4}{8} \\
c(1)&:=\mathbb{E}\bigg[  \exp\bigg(-\int_{0}^1 (W_{s}^{0})^{2}\;|\gamma_s''|^2 ds \bigg) \bigg]^{1/2}.
\end{align}

In particular,
\begin{equation}

\end{equation}

\end{proof}

\textcolor{blue}{add first expansion of $r_{n,c}$ alone, than add it in the bounds and show how $(n!||X^n||)^{2}$ grows (using the two bounds). Once done, }

\begin{remark}
Note that because of the summations, the third and fourth term in the right hand side of equation (\ref{eq:final_asymptotics}) respectively scale as $O(n^{-1})$ and $O(n^{-2})$.
\end{remark}

\begin{remark}
\par We note the appearance of a particular structure in the high-order coefficients that seems to have a lot to say, namely the summations $\xi_i^j$. These terms can somehow be seen as a generalisation of weighted moving-averages which are extensively used in time series analysis, where the index $i$ denote the number of higher order correlations that is captured and $j$ the terminal time.
\end{remark}

\par The same approach can be adopted to compute higher order terms: to do so, one needs to retrieve higher order coefficients $\lambda_s$ by running the iteration prescribed by the system of equations (\ref{eq:augmented_system_dde_continuity}). Once done, one simply considers the same Taylor expansions used in the step preceding equation (\ref{eq:right_after_taylor}) with a higher truncation level.

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