to20.5pt{\pgfpicture\makeatletter\hbox{\hskip 6.37471pt\lower-6.07469pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.375pt}{-3.07498pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$A$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{{{}}
{}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{11.38092pt}{-2.84544pt}\pgfsys@moveto{12.37686pt}{-2.84544pt}\pgfsys@curveto{12.37686pt}{-2.2954pt}{11.93097pt}{-1.8495pt}{11.38092pt}{-1.8495pt}\pgfsys@curveto{10.83087pt}{-1.8495pt}{10.38498pt}{-2.2954pt}{10.38498pt}{-2.84544pt}\pgfsys@curveto{10.38498pt}{-3.3955pt}{10.83087pt}{-3.84138pt}{11.38092pt}{-3.84138pt}\pgfsys@curveto{11.93097pt}{-3.84138pt}{12.37686pt}{-3.3955pt}{12.37686pt}{-2.84544pt}\pgfsys@closepath\pgfsys@moveto{11.38092pt}{-2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{19.12321pt}{-3.07498pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$B$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{{{}}
{}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{34.14322pt}{-2.84544pt}\pgfsys@moveto{35.13916pt}{-2.84544pt}\pgfsys@curveto{35.13916pt}{-2.2954pt}{34.69327pt}{-1.8495pt}{34.14322pt}{-1.8495pt}\pgfsys@curveto{33.59317pt}{-1.8495pt}{33.14728pt}{-2.2954pt}{33.14728pt}{-2.84544pt}\pgfsys@curveto{33.14728pt}{-3.3955pt}{33.59317pt}{-3.84138pt}{34.14322pt}{-3.84138pt}\pgfsys@curveto{34.69327pt}{-3.84138pt}{35.13916pt}{-3.3955pt}{35.13916pt}{-2.84544pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{-2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{11.38092pt}{-2.84544pt}\pgfsys@curveto{11.38092pt}{14.22638pt}{34.14322pt}{14.22638pt}{34.14322pt}{-2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{5.69046pt}\pgfsys@curveto{0.0pt}{14.22638pt}{22.76228pt}{14.22638pt}{22.76228pt}{5.69046pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,|_{st}\\\
={}&\upintop\ilimits@_{s<u<r<v<t}\upintop\ilimits@_{\Delta^{k}[s,u]\times\Delta^{k}[u,v]}|A^{\circ}|_{su}(z_{1}-w_{1})^{2H-2}\cdots(z_{k}-w_{k})^{2H-2}\mathrm{d}w_{1}\mathrm{d}z_{1}\cdots\mathrm{d}w_{k}\mathrm{d}z_{k}\\\
&\quad(v-u)^{2H-2}\mathrm{d}u\mathrm{d}v\\\
\lesssim{}&(t-s)^{2(i+k)H}\upintop\ilimits@_{s<u<r<v<t}(v-u)^{2H-2}\mathrm{d}u\mathrm{d}v\\\
\lesssim{}&(t-s)^{2(i+k)H}|(t-r)^{2H}+(r-s)^{2H}-(t-s)^{2H}|\\\
\lesssim{}&(t-s)^{2(i+k+1)H}\text{ .}\end{split}$

Finally, consider the case in which $j=0$ and $k>0$ (and $B$ may have no
single nodes):

$\begin{split}&|\\!\leavevmode\hbox to41.71pt{\vbox
to20.5pt{\pgfpicture\makeatletter\hbox{\hskip 6.37471pt\lower-6.07469pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.375pt}{-3.07498pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$A$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{{{}}
{}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{11.38092pt}{-2.84544pt}\pgfsys@moveto{12.37686pt}{-2.84544pt}\pgfsys@curveto{12.37686pt}{-2.2954pt}{11.93097pt}{-1.8495pt}{11.38092pt}{-1.8495pt}\pgfsys@curveto{10.83087pt}{-1.8495pt}{10.38498pt}{-2.2954pt}{10.38498pt}{-2.84544pt}\pgfsys@curveto{10.38498pt}{-3.3955pt}{10.83087pt}{-3.84138pt}{11.38092pt}{-3.84138pt}\pgfsys@curveto{11.93097pt}{-3.84138pt}{12.37686pt}{-3.3955pt}{12.37686pt}{-2.84544pt}\pgfsys@closepath\pgfsys@moveto{11.38092pt}{-2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{19.12321pt}{-3.07498pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$B$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{{{}}
{}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{34.14322pt}{-2.84544pt}\pgfsys@moveto{35.13916pt}{-2.84544pt}\pgfsys@curveto{35.13916pt}{-2.2954pt}{34.69327pt}{-1.8495pt}{34.14322pt}{-1.8495pt}\pgfsys@curveto{33.59317pt}{-1.8495pt}{33.14728pt}{-2.2954pt}{33.14728pt}{-2.84544pt}\pgfsys@curveto{33.14728pt}{-3.3955pt}{33.59317pt}{-3.84138pt}{34.14322pt}{-3.84138pt}\pgfsys@curveto{34.69327pt}{-3.84138pt}{35.13916pt}{-3.3955pt}{35.13916pt}{-2.84544pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{-2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{11.38092pt}{-2.84544pt}\pgfsys@curveto{11.38092pt}{14.22638pt}{34.14322pt}{14.22638pt}{34.14322pt}{-2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{5.69046pt}\pgfsys@curveto{0.0pt}{14.22638pt}{22.76228pt}{14.22638pt}{22.76228pt}{5.69046pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,|_{st}\\\
\lesssim{}&(t-s)^{2iH}\upintop\ilimits@_{s<u<v<t}\upintop\ilimits@_{[s,u]^{k}\times\Delta^{k}[u,v]}(z_{1}-w_{1})^{2H-2}\cdots(z_{k}-w_{k})^{2H-2}\mathrm{d}w_{1}\mathrm{d}z_{1}\cdots\mathrm{d}w_{k}\mathrm{d}z_{k}\\\
&\quad(v-u)^{2H-2}\mathrm{d}u\mathrm{d}v\\\
\lesssim{}&(t-s)^{2iH}\upintop\ilimits@_{s<u<z_{1}<\ldots<z_{k}<t}|(z_{1}-u)^{2H-1}-(z_{1}-s)^{2H-1}|\cdots|(z_{k}-u)^{2H-1}-(z_{k}-s)^{2H-1}|\\\
&\quad|(t-u)^{2H-1}-(z_{k}-u)^{2H-1}|\mathrm{d}u\mathrm{d}z_{1}\cdots\mathrm{d}z_{k}\\\
\lesssim{}&(t-s)^{2(i+k-1)H}\upintop\ilimits@_{s<u<z_{k}<t}|(z_{k}-u)^{2H-1}-(z_{k}-s)^{2H-1}||(t-u)^{2H-1}-(z_{k}-u)^{2H-1}|\mathrm{d}u\mathrm{d}z_{k}\text{
.}\end{split}$

Once again, the absolute values distinguish between $H\lessgtr 1/2$. Expanding
the product, we observe that three of the integrals feature products of
different terms, each to the power of $2H-1$: in these, at least one of
$z_{k}$ or $u$ only appears once, which means this variable may be integrated
out and the resulting term bounded (up to a constant) by $(t-s)^{2H}$, with
the remaining integral solved similarly. The fourth integral instead is
$\upintop\ilimits@_{s<u<z<t}(z-u)^{4H-2}\mathrm{d}u\mathrm{d}z$ which is
finite again thanks to $H>1/4$. This shows that we have
$\lesssim(t-s)^{2(i+k+1)H}$ in the above expression and concludes the proof.
$\blacksquare$

######  Remark  (Modified $|P|$).

We have stated the previous proposition under in the most natural manner; in
particular note how, in the prototypical case of fBm, the integrals $|P|_{st}$
are multiples of $P_{st}$. We will, however, additionally need a slightly
modified version of this result, in which the definition of $|P|$ is changed
as follows: maximal sequences

$\upintop\ilimits@_{\Delta^{k}[u,v]}(w_{1}-u)^{2H-1}\cdots(w_{k}-w_{k-1})^{2H-1}\mathrm{d}w_{1}\cdots\mathrm{d}w_{k}$

occurring in the middle of the expression for $|P|$, are replaced with their
bound $(v-u)^{2kH}$, and each integrand $(v-u)^{2H-2}$ is replaced with
$((v-u)\wedge 1/2)^{2H-2}$. That the statement continues despite these
modifications to hold is obvious for the first, and for the second it follows
from the facts that all integrals are still convergent (by the same proof) and
the $1/2$ can be replaced with $1/2\wedge T$ and absorbed in the constant of
proportionality.

Just like in [3], we approximate $X$ piecewise linearly. Let $X^{\ell}$ be a
sequence of piecewise linear approximations of $X$ along partitions
$\pi_{\ell}$ on $[0,T]$ with step size that vanishes as $\ell\to\infty$. It
will be helpful to assume that the intervals in the mesh $\pi_{\ell}$ all have
the same length $\varrho_{\ell}$; this simplifying assumption can be made
because it is only necessary to show convergence along _a_ sequence of such
approximations, since it is known that the limit does not depend on the
particular choice of $\pi_{\ell}$ (or indeed on the type of piecewise smooth
approximation in a broad class of these) [21, Ch. 15]. For $t\in[0,T]$ we will
write $t_{\ell}^{-}$ and $t_{\ell}^{+}$ to respectively denote the endpoints
$a$ and $b$ of the interval of $\pi_{\ell}$ s.t. $t\in[a,b)$. Explicitly,
$X^{\ell}$ and its piecewise-defined derivative are given by

$\begin{split}X^{\ell}_{t}&=X_{t^{-}_{\ell}}+\varrho^{-1}_{\ell}(t-t^{-}_{\ell})X_{t^{-}_{\ell}t^{+}_{\ell}}\\\
\dot{X}^{\ell}_{t}&=\varrho^{-1}_{\ell}X_{t^{-}_{\ell}t^{+}_{\ell}}\end{split}$
(46)

where, as usual, $X_{ab}\coloneqq X_{b}-X_{a}$ denotes the increment. In order
to use Stroock’s formula (23), we will be considering Malliavin derivatives of
the signature of the piecewise-linear interpolations of $X$,

$\mathcal{S}(X^{\ell})^{\gamma_{1},\ldots,\gamma_{n}}_{st}=\upintop\ilimits@_{\Delta^{n}[s,t]}\dot{X}^{\ell;\gamma_{1}}_{u_{1}}\cdots\dot{X}^{\ell;\gamma_{n}}_{u_{n}}\mathrm{d}u_{1}\cdots\mathrm{d}u_{n}\text{
,}$

which in turn requires us to consider those of the single factors:

$\mathcal{D}_{v}\dot{X}_{u}^{\ell;\gamma}=\varrho^{-1}_{\ell}\mathbbm
1^{\gamma}_{[u^{-}_{\ell},u^{+}_{\ell})}(v)=\varrho^{-1}_{\ell}\mathbbm
1^{\gamma}_{[v^{-}_{\ell},v^{+}_{\ell})}(u)\text{ .}$ (47)

For $P\in\mathcal{P}^{n}_{m}$, we provide a discretised analogue to Section 2:

######  Definition  ($P^{\ell;\gamma_{1},\ldots,\gamma_{n}}_{st}$).

For $\gamma_{1},\ldots,\gamma_{n}\in[d]$, $0\leq s\leq t\leq T$ and
$P\in\mathcal{P}^{n}_{m}$ define

$\begin{split}P_{st}^{\ell;\gamma_{1},\ldots,\gamma_{n}}(v_{k}\mid
k\in\overline{P})&\coloneqq\upintop\ilimits@_{\Delta^{n}[s,t]}\prod_{\\{i,j\\}\in
P}\mathbb
E[\dot{X}^{\ell;\gamma_{i}}_{u_{i}}\dot{X}^{\ell;\gamma_{j}}_{u_{j}}]\mathrm{d}u_{i}\mathrm{d}u_{j}\cdot\prod_{k\in\overline{P}}\varrho_{\ell}^{-1}\mathbbm
1^{\gamma_{k}}_{[v^{-}_{k;\ell}v^{+}_{k;\ell})}(u_{k})\mathrm{d}u_{k}\text{
.}\end{split}$ (48)

as an element of $\mathcal{E}^{\otimes m}$, whose arguments are given to the
functions $\mathbbm 1^{\gamma_{k}}_{[u^{-}_{k;\ell}u^{+}_{k;\ell})}$ with
$k\in\overline{P}$.

Note how the above definition, unlike Section 2 does not distinguish between
consecutive and non-consecutive pairings: this will only become important in
the limit. Moreover, we are integrating over all $n$ variables, including the
$u_{k}$ with $k\in\overline{P}$: this is because the time arguments of the
function, $v_{k}$, are supplied separately, with the respective index
variables supplied as arguments to $\text{\textdelta}_{\gamma_{k}}$,
$k\in\overline{P}$. The functions $P^{\ell}_{st}$ are summands in the
expression of which we want to compute the limit:

######  Lemma  (Expected Malliavin derivatives of signature approximations).

$\mathbb
E\mathcal{D}^{m}\mathcal{S}(X^{\ell})_{st}^{\gamma_{1},\ldots,\gamma_{n}}=m!\sum_{P\in\mathcal{P}^{n}_{m}}P_{st}^{\ell;\gamma_{1},\ldots,\gamma_{n}}\
\in\mathcal{E}^{\odot m}$ (49)

###### Proof.

This is a consequence of (46), (47), the (iterated) Leibniz rule for the
Malliavin derivative and Wick’s formula for the mixed moments of a Gaussian
vector (as it was already used in [2, Theorem 31]). The details are a matter
of simple combinatorics; in particular note how, instead of summing over $m!$
terms corresponding to the ways of permuting the $m$ derivatives (for a fixed
$P\in\mathcal{P}^{n}_{m}$), we are only including the term corresponding to
the identity permutation and multiplying by $m!$, which identifies the same
element of $\mathcal{E}^{\otimes m}$ up to symmetry. $\blacksquare$

In order to prove convergence, it is unfortunately not possible to argue by
dominated convergence applied to Section 3: this is because the factors in the
integrand given by consecutive pairings $\mathbb
E[\dot{X}^{\ell;\gamma_{i}}_{u_{i}}\dot{X}^{\ell;\gamma_{i+1}}_{u_{i+1}}]$
converge to non-integrable functions (e.g. $(v-u)^{2H-2}$ on $\Delta^{2}[s,t]$
for fBm) and the ones corresponding to Malliavin derivatives
$\varrho_{\ell}^{-1}\mathbbm
1^{\gamma_{k}}_{[v^{-}_{k;\ell}v^{+}_{k;\ell})}(u_{k})\mathrm{d}u_{k}$ do not
converge at all (in fact they converge, as distributions, to Dirac deltas
$\text{\textdelta}_{v_{k}}$). The reason that convergence holds is that all
these quantities are integrated. To successfully exploit this, we will write
each integral $P^{\ell}_{st}$ as a nested integral, distinguishing between the
three types of integrands:

$\upintop\ilimits@(\text{non-consecutive
pairings})\upintop\ilimits@(\text{Malliavin
derivatives})\prod_{\begin{subarray}{c}\text{maximal}\\\
\text{sequences}\end{subarray}}\upintop\ilimits@(\text{consecutive
pairings})\text{ .}$ (50)

The outer integral contains the product of all terms $\mathbb
E[\dot{X}^{\ell;\gamma_{i}}_{u_{i}}\dot{X}^{\ell;\gamma_{j}}_{u_{j}}]$ with
$|j-i|>1$. These are multiplied with the second integral, which integrates all
factors coming from Malliavin derivatives. Finally, we partition the remaining
integrands $\mathbb
E[\dot{X}^{\ell;\gamma_{h}}_{u_{h}}\dot{X}^{\ell;\gamma_{h+1}}_{u_{h+1}}]$
into maximal sequences and integrate each individually: these integrals are
integrands in the second integral, alongside the Malliavin derivatives. The
operations of exchanging the order of integrals are all justified by Fubini’s
theorem, considering that all integrals are actually finite sums. We
illustrate all of this with a simple example: consider the diagram
(suppressing indices)

$P\coloneqq\leavevmode\hbox to36.82pt{\vbox
to12.92pt{\pgfpicture\makeatletter\hbox{\hskip 1.3379pt\lower-1.3379pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.13791pt}{0.0pt}\pgfsys@curveto{1.13791pt}{0.62845pt}{0.62845pt}{1.13791pt}{0.0pt}{1.13791pt}\pgfsys@curveto{-0.62845pt}{1.13791pt}{-1.13791pt}{0.62845pt}{-1.13791pt}{0.0pt}\pgfsys@curveto{-1.13791pt}{-0.62845pt}{-0.62845pt}{-1.13791pt}{0.0pt}{-1.13791pt}\pgfsys@curveto{0.62845pt}{-1.13791pt}{1.13791pt}{-0.62845pt}{1.13791pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@moveto{9.67381pt}{0.0pt}\pgfsys@curveto{9.67381pt}{0.62845pt}{9.16435pt}{1.13791pt}{8.5359pt}{1.13791pt}\pgfsys@curveto{7.90746pt}{1.13791pt}{7.398pt}{0.62845pt}{7.398pt}{0.0pt}\pgfsys@curveto{7.398pt}{-0.62845pt}{7.90746pt}{-1.13791pt}{8.5359pt}{-1.13791pt}\pgfsys@curveto{9.16435pt}{-1.13791pt}{9.67381pt}{-0.62845pt}{9.67381pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{17.07182pt}{0.0pt}\pgfsys@moveto{18.20973pt}{0.0pt}\pgfsys@curveto{18.20973pt}{0.62845pt}{17.70027pt}{1.13791pt}{17.07182pt}{1.13791pt}\pgfsys@curveto{16.44337pt}{1.13791pt}{15.93391pt}{0.62845pt}{15.93391pt}{0.0pt}\pgfsys@curveto{15.93391pt}{-0.62845pt}{16.44337pt}{-1.13791pt}{17.07182pt}{-1.13791pt}\pgfsys@curveto{17.70027pt}{-1.13791pt}{18.20973pt}{-0.62845pt}{18.20973pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{17.07182pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{25.6073pt}{0.0pt}\pgfsys@moveto{26.74521pt}{0.0pt}\pgfsys@curveto{26.74521pt}{0.62845pt}{26.23575pt}{1.13791pt}{25.6073pt}{1.13791pt}\pgfsys@curveto{24.97885pt}{1.13791pt}{24.46939pt}{0.62845pt}{24.46939pt}{0.0pt}\pgfsys@curveto{24.46939pt}{-0.62845pt}{24.97885pt}{-1.13791pt}{25.6073pt}{-1.13791pt}\pgfsys@curveto{26.23575pt}{-1.13791pt}{26.74521pt}{-0.62845pt}{26.74521pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{25.6073pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{35.28113pt}{0.0pt}\pgfsys@curveto{35.28113pt}{0.62845pt}{34.77167pt}{1.13791pt}{34.14322pt}{1.13791pt}\pgfsys@curveto{33.51477pt}{1.13791pt}{33.00531pt}{0.62845pt}{33.00531pt}{0.0pt}\pgfsys@curveto{33.00531pt}{-0.62845pt}{33.51477pt}{-1.13791pt}{34.14322pt}{-1.13791pt}\pgfsys@curveto{34.77167pt}{-1.13791pt}{35.28113pt}{-0.62845pt}{35.28113pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{11.38092pt}{34.14322pt}{11.38092pt}{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{25.6073pt}{0.0pt}\pgfsys@lineto{25.6073pt}{5.69046pt}\pgfsys@stroke\pgfsys@invoke{
} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\
\in\mathcal{P}^{6}_{2}\text{ .}$

According to Section 3, we have

$P^{\ell}_{st}(v_{1},v_{2})=\upintop\ilimits@_{\Delta^{6}[s,t]}\varrho_{\ell}^{-2}\mathbbm
1_{[v^{-}_{2;\ell},v^{+}_{2;\ell})}(u_{2})\mathbbm
1_{[v^{-}_{3;\ell},v^{+}_{3;\ell})}(u_{3})\mathbb
E[\dot{X}^{\ell}_{u_{1}}\dot{X}^{\ell}_{u_{6}}]\mathbb
E[\dot{X}^{\ell}_{u_{4}}\dot{X}^{\ell}_{u_{5}}]\mathrm{d}u_{1}\mathrm{d}u_{4}\mathrm{d}u_{5}\mathrm{d}u_{6}\text{
.}$

Re-organising this expression as described in (50) we obtain

$\displaystyle\upintop\ilimits@_{s<u_{1}<u_{6}<t}\mathbb
E[\dot{X}^{\ell}_{u_{1}}\dot{X}^{\ell}_{u_{6}}]$
$\displaystyle\qquad\bigg{[}\upintop\ilimits@_{u_{1}<u_{2}<u_{3}<u_{6}}\varrho_{\ell}^{-2}\mathbbm
1_{[v^{-}_{2;\ell},v^{+}_{2;\ell})}(u_{2})\mathbbm
1_{[v^{-}_{3;\ell},v^{+}_{3;\ell})}(u_{3})$
$\displaystyle\qquad\qquad\bigg{[}\upintop\ilimits@_{u_{3}<u_{4}<u_{5}<u_{6}}\mathbb
E[\dot{X}^{\ell}_{u_{4}}\dot{X}^{\ell}_{u_{5}}]\mathrm{d}u_{4}\mathrm{d}u_{5}\bigg{]}\mathrm{d}u_{2}\mathrm{d}u_{3}\bigg{]}\mathrm{d}u_{1}\mathrm{d}u_{6}\text{
.}$

Note that the domain of integration of the innermost integral can be described
in terms of variables of the two outer integrals: this extends to the case in
which there is more than one maximal sequence, by maximality, and is crucial
for the factorisation into integrals over maximal sequences to be possible.

The reason for the nested rewriting of (50) is that it will be possible to
show convergence of the integrals over maximal sequences, then by a separate
argument infer the convergence of the middle integral, and finally by
dominated convergence conclude that the outer integrals converge. We preface
the proof of convergence with a few lemmas; the first of these considers the
case of a single consecutive pairing, and will form the base case of an
induction that handles maximal sequences of arbitrary length.

######  Lemma  (One consecutive pairing).

$\lim_{\ell\to\infty}\upintop\ilimits@_{s<u<v<t}\mathbb
E[\dot{X}_{u}^{\ell}\dot{X}_{v}^{\ell}]\mathrm{d}u\mathrm{d}v=\frac{1}{2}\mathbb
E[X^{2}_{st}]=\upintop\ilimits@_{s}^{t}\big{[}\tfrac{1}{2}R^{\prime}(v)-\partial_{2}R(s,v)\big{]}\mathrm{d}v$

and the convergents are uniformly bounded by $\lesssim(t-s)^{2H}$.

###### Proof.

Considering that $\dot{X}^{\ell}$ is a piecewise-constant, and that the
integral on the right is therefore a finite sum, we can write

$\displaystyle\upintop\ilimits@_{s<u<v<t}\mathbb
E[\dot{X}^{\ell}_{u}\dot{X}^{\ell}_{v}]\mathrm{d}u\mathrm{d}v={}$
$\displaystyle\mathbb
E\upintop\ilimits@_{s<u<v<t}\dot{X}^{\ell}_{u}\dot{X}^{\ell}_{v}\mathrm{d}u\mathrm{d}v$
$\displaystyle={}$ $\displaystyle\frac{1}{2}\mathbb E[(X_{st}^{\ell})^{2}]$
$\displaystyle\xrightarrow{\ell\to\infty}{}$ $\displaystyle\frac{1}{2}\mathbb
E[X_{st}^{2}]$ $\displaystyle={}$
$\displaystyle\frac{1}{2}R(\Delta(s,t),\Delta(s,t))$ $\displaystyle={}$
$\displaystyle\frac{R(s)+R(t)}{2}-R(s,t)$ $\displaystyle={}$
$\displaystyle\upintop\ilimits@_{s}^{t}\big{[}\tfrac{1}{2}R^{\prime}(v)-\partial_{2}R(s,v)\big{]}\mathrm{d}v$

where we have used that
$(X^{\ell}_{st})^{2}\xrightarrow{\ell\to\infty}X_{st}^{2}$ in $L^{2}$. For the
second statement, we rely on the first two identities above and distinguish
between the cases $s^{-}_{\ell}=t^{-}_{\ell}$ and $s^{-}_{\ell}<t^{-}_{\ell}$:
in the former we have, using (12)

$\displaystyle|\mathbb E[(X^{\ell}_{st})^{2}]|=|\mathbb
E[\varrho_{\ell}^{-2}(t-s)^{2}X_{s^{-}_{\ell}s^{+}_{\ell}}^{2}]|\lesssim\varrho_{\ell}^{2H-2}(t-s)^{2}\leq(t-s)^{2H}$

since

$\Big{(}\frac{t-s}{\varrho_{\ell}}\Big{)}^{2-2H}\leq 1$

by $H<1$ and $t-s\leq\varrho_{\ell}$. Let now $s^{-}_{\ell}<t^{-}_{\ell}$:

$\displaystyle|\mathbb E[(X^{\ell}_{st})^{2}]|$ $\displaystyle=|\mathbb
E[(X^{\ell}_{ss^{+}_{\ell}}+X^{\ell}_{s^{+}_{\ell}t^{-}_{\ell}}+X^{\ell}_{t^{-}_{\ell}t})^{2}]|$
$\displaystyle=\Big{|}\mathbb
E\Big{[}\big{(}\varrho^{-1}(s^{+}_{\ell}-s)X^{\ell}_{ss^{+}_{\ell}}+X_{s^{+}_{\ell}t^{-}_{\ell}}+\varrho^{-1}(t-t^{-}_{\ell})X_{t^{-}_{\ell}t}\big{)}^{2}\Big{]}\Big{|}$
$\displaystyle\lesssim\varrho^{-2}(s^{+}_{\ell}-s)^{2}\mathbb
E[X_{ss^{+}_{\ell}}^{2}]+\mathbb
E[X_{s^{+}_{\ell}t^{-}_{\ell}}^{2}]+\varrho^{-2}(t-t^{-}_{\ell})^{2}\mathbb
E[X_{t^{-}_{\ell}t}^{2}]$
$\displaystyle\lesssim(s^{+}_{\ell}-s)^{2H}+(t^{-}_{\ell}-s^{+}_{\ell})^{2H}+(t-t^{-}_{\ell})^{2H}$
$\displaystyle\lesssim(t-s)^{2H}$

by $\mathscr{l}^{2}$-Jensen’s inequality, the previous case, and again (12).
$\blacksquare$

The case of several consecutive pairings is more difficult to handle, and in
Section 3 convergence of these terms will be bootstrapped from terms that only
contain shorter sequences of consecutive pairings, and the above single case,
by means of an inductive argument. It is worth remarking that the plausible
strategy of handling these integrands together with the others by integrating
only one of the variables fails:

######  Remark  (Lack of convergence of $\mathbb
E[X_{uv}^{\ell}\dot{X}_{v}^{\ell}{]}$).

One way of dealing with sequences of consecutive pairings is by rewriting them
as

$\begin{split}&\upintop\ilimits@_{s<u_{1}<v_{1}<\ldots<u_{n}<v_{n}<t}\mathbb
E[\dot{X}^{\ell}_{u_{1}}\dot{X}^{\ell}_{v_{1}}]\cdots\mathbb
E[\dot{X}^{\ell}_{u_{n}}\dot{X}^{\ell}_{v_{n}}]\mathrm{d}u_{1}\mathrm{d}v_{1}\cdots\mathrm{d}u_{n}\mathrm{d}v_{n}\\\
={}&\upintop\ilimits@_{\Delta^{n}[s,t]}\mathbb
E[X^{\ell}_{sv_{1}}\dot{X}^{\ell}_{v_{1}}]\mathbb
E[X^{\ell}_{v_{1}v_{2}}\dot{X}^{\ell}_{v_{2}}]\cdots\mathbb
E[X^{\ell}_{v_{n-1}v_{k}}\dot{X}^{\ell}_{v_{n}}]\mathrm{d}v_{1}\cdots\mathrm{d}u_{n}\text{
.}\end{split}$ (51)

This has the benefit of expressing the convergents as integrals over $n$, and
not $2n$, variables. The problem with this strategy is that it does not hold
that $\mathbb
E[X_{uv}^{\ell}\dot{X}_{v}^{\ell}]\xrightarrow{\ell\to\infty}\frac{1}{2}R^{\prime}(v)-\partial_{2}R(u,v)$:
a simple calculation reveals

$\displaystyle\mathbb E[X_{uv}^{\ell}\dot{X}_{v}^{\ell}]$ $\displaystyle={}$
$\displaystyle\varrho^{-1}_{\ell}\Big{[}\big{(}1-\varrho^{-1}_{\ell}(v-v^{-}_{\ell})\big{)}R(v^{-}_{\ell},\Delta(v^{-}_{\ell},v^{+}_{\ell}))+\varrho^{-1}_{\ell}(v-v^{-}_{\ell})R(v^{+}_{\ell},\Delta(v^{-}_{\ell},v^{+}_{\ell}))\Big{]}$
$\displaystyle-\varrho^{-1}_{\ell}\Big{[}\big{(}1-\varrho^{-1}_{\ell}(u-u^{-}_{\ell})\big{)}R(u^{-}_{\ell},\Delta(v^{-}_{\ell},v^{+}_{\ell}))+\varrho^{-1}_{\ell}(u-u^{-}_{\ell})R(u^{+}_{\ell},\Delta(v^{-}_{\ell},v^{+}_{\ell}))\Big{]}\text{
.}$

While the second term converges to $\partial_{2}R(u,v)$ (e.g. by the
intermediate value theorem applied on the interval
$[v^{-}_{\ell},v^{+}_{\ell}]$), the first does not converge in general. To see
why, it suffices to take $X$ to be Brownian motion and $\pi_{\ell}$ to by a
diadic sequence: the first term on the right above is then equal to
$\varrho^{-1}_{\ell}(v-v^{-}_{\ell})$ which is indeterminate in view of the
fact that for $v$ in a set of full Lebesgue measure its decimal expansion
contains infinitely many $00$’s and $11$’s. The fractional case with $H<1/2$
appears even worse behaved, i.e. divergent in a possibly indeterminate
fashion.

We now move outward in (50) and prove a lemma that will guarantee convergence
of the middle integral, conditional on the convergence of the inner ones.

######  Lemma .

Let $f_{\ell}\colon[0,T]^{m}\to\mathbb R$ be a uniformly bounded sequence of
functions that are continuous and piecewise smooth on the mesh $\pi_{\ell}$.
Assume that $f_{\ell}$ converges to $f\colon[0,T]^{m}\to\mathbb R$ uniformly.
Then

$\begin{split}\upintop\ilimits@_{\Delta^{m}[s,t]}f_{\ell}(u_{1},\ldots,u_{m})\varrho^{-m}_{\ell}\prod_{k=1}^{m}\mathbbm
1_{[v^{-}_{k;\ell},v^{+}_{k;\ell})}(u_{k})\mathrm{d}u_{1},\ldots,\mathrm{d}u_{m}\xrightarrow{\ell\to\infty}\mathbbm
1_{\Delta^{m}[s,t]}(v_{1},\ldots,v_{m})f(v_{1},\ldots,v_{m})\end{split}$

where the convergence is a.e. in the variables
$(v_{1},\ldots,v_{m})\in[0,T]^{m}$. Moreover, the convergents are uniformly
bounded by $\sup_{\ell}\lVert f_{\ell}\rVert_{\infty}$.

###### Proof.

The second statement holds by uniform boundedness of $f_{\ell}$ and the fact
that

$\upintop\ilimits@_{\Delta^{m}[s,t]}\varrho^{-m}_{\ell}\prod_{k=1}^{m}\mathbbm
1_{[v^{-}_{k;\ell},v^{+}_{k;\ell})}(u_{k})\mathrm{d}u_{1},\ldots,\mathrm{d}u_{n}\leq
1\text{ .}$

We will prove pointwise convergence on the subset

$[0,T]^{m}_{*}\coloneqq\\{(v_{1},\ldots,v_{m})\in[0,T]^{m}\mid v_{i}\neq
v_{j}\text{ for }i\neq j\\}$

of $[0,T]^{m}$ of full Lebesgue measure. For
$(v_{1},\ldots,v_{m})\in[0,T]^{m}_{*}$, we may, without loss of generality,
start the sequence when $\ell$ is already large enough so that
$[v^{-}_{i;\ell},v^{+}_{i;\ell})\cap[v^{-}_{j;\ell},v^{+}_{j;\ell})=\varnothing$
for $i\neq j$, where we are including $v_{0}\coloneqq s$ and $v_{m+1}\coloneqq
t$ in this requirement. By the mean value theorem applied individually to each
$u_{k}$, there exist $w_{k;\ell}\in(v^{-}_{k;\ell},v^{+}_{k;\ell})$ s.t.

$\upintop\ilimits@_{\Delta^{m}[s,t]}f_{\ell}(u_{1},\ldots,u_{m})\varrho^{-m}_{\ell}\prod_{k=1}^{m}\mathbbm
1_{[v^{-}_{k;\ell},v^{+}_{k;\ell})}(u_{k})\mathrm{d}u_{1},\ldots,\mathrm{d}u_{m}=\mathbbm
1_{\Delta^{m}[s,t]}(v_{1},\ldots,v_{m})f_{\ell}(w_{1;\ell},\ldots,w_{m;\ell})$

and

$\displaystyle|\mathbbm
1_{\Delta^{m}[s,t]}(v_{1},\ldots,v_{m})f_{\ell}(w_{1;\ell},\ldots,w_{m;\ell})-\mathbbm
1_{\Delta^{m}[s,t]}(v_{1},\ldots,v_{m})f(v_{1},\ldots,v_{m})|$
$\displaystyle\leq{}$ $\displaystyle\mathbbm
1_{\Delta^{m}[s,t]}(v_{1},\ldots,v_{m})\big{[}|f_{\ell}(w_{1;\ell},\ldots,w_{m;\ell})-f(w_{1;\ell},\ldots,w_{m;\ell})|$
$\displaystyle+|f(w_{1;\ell},\ldots,w_{m;\ell})-f(v_{1},\ldots,v_{m})|\big{]}$
$\displaystyle\leq{}$ $\displaystyle\mathbbm
1_{\Delta^{m}[s,t]}(v_{1},\ldots,v_{m})\big{[}\lVert
f_{\ell}-f\rVert_{\infty}+\omega^{f}_{(v_{1},\ldots,v_{m})}(\varrho_{\ell})\big{]}$

where $\omega^{f}_{(v_{1},\ldots,v_{m})}$ is the modulus of continuity of $f$
at the point $(v_{1},\ldots,v_{m})$. Both summands on the right hand side
above vanish in the limit of $\ell\to\infty$, the first by uniform convergence
and the second by continuity of the uniform limit of continuous functions.
$\blacksquare$

The next two results constitute the core of our argument. They both rely on
the same induction used to reduce the length of consecutive pairings, the base
case of which is provided by Section 3. To illustrate it at level $4$, letting
$Y$ be a stochastic process (which below will be taken to be $X^{\ell}$ and
$X$) we have for $\alpha\neq\beta$

$\displaystyle 2\mathbb E\mathcal{S}(Y)^{\alpha\alpha\beta\beta}_{st}=\mathbb
E\mathcal{S}(Y)^{\alpha\alpha}_{st}\cdot\mathbb
E\mathcal{S}(Y)^{\beta\beta}_{st}-2\mathbb
E\mathcal{S}(Y)^{\alpha\beta\alpha\beta}_{st}-2\mathbb
E\mathcal{S}(Y)^{\alpha\beta\beta\alpha}_{st}$

by the shuffle property (8), using identical distribution of components to
group together $2\mathbb E\mathcal{S}(Y)^{\alpha\alpha\beta\beta}_{st}=\mathbb
E\mathcal{S}(Y)^{\alpha\alpha\beta\beta}_{st}+\mathbb
E\mathcal{S}(Y)^{\beta\beta\alpha\alpha}_{st}$ (and similar on the right hand
side), and using independence of components to write $\mathbb
E[\mathcal{S}(Y)_{st}^{\alpha\alpha}\mathcal{S}(Y)_{st}^{\beta\beta}]=\mathbb
E\mathcal{S}(Y)^{\alpha\alpha}_{st}\cdot\mathbb
E\mathcal{S}(Y)^{\beta\beta}_{st}$. While the left hand side contains a
sequence of two consecutive pairs, only sequences of consecutive pairs of
length one appear on the right.

######  Lemma  (Dominating function).

For $P\in\mathcal{P}^{n}_{m}$ it holds that the integrand of the outermost
integral of $P^{\ell}_{st}$ expressed in the nested form (50), is absolutely
bounded by an integrable function, uniformly in $\ell$ and on
$\Delta^{m}[s,t]$, so that $|P^{\ell}_{st}|\lesssim(t-s)^{2(n-m)H^{\prime}}$
for any $1/4<H^{\prime}<H$.

###### Proof.

We begin by bounding expectations corresponding to non-consecutive pairings.
As done in the proof of [3, Theorem 31], we now consider the terms

$\mathbb
E[\dot{X}_{u}^{\ell}\dot{X}^{\ell}_{v}]=\varrho^{-2}_{\ell}R(\Delta(u^{-}_{\ell},u^{+}_{\ell}),\Delta(v^{-}_{\ell},v^{+}_{\ell}))$

in three different cases: for $u^{-}_{\ell}=v^{-}_{\ell}$

$|\mathbb
E[\dot{X}^{\ell}_{u}\dot{X}^{\ell}_{v}]|\lesssim\varrho_{\ell}^{2H-2}\leq(v-u)^{2H-2}\text{
.}$

By Cauchy-Schwarz the same estimate as above holds in the case
$u^{+}_{\ell}=v^{-}_{\ell}$, with a constant in the second inequality given by
the fact that $v-u\leq 2\varrho_{\ell}$. Let $u^{+}_{\ell}<v^{-}_{\ell}$: we
have, by (9) and for any $H^{\prime}$ as in the statement

$\displaystyle|\mathbb E[\dot{X}_{u}^{\ell}\dot{X}^{\ell}_{v}]|$
$\displaystyle=\varrho^{-2}_{\ell}\bigg{|}\upintop\ilimits@_{[u^{-}_{\ell},u^{+}_{\ell}]\times[v^{-}_{\ell},v^{+}_{\ell}]}\partial_{12}R(u,v)\mathrm{d}u\mathrm{d}v\bigg{|}$
$\displaystyle\lesssim\varrho^{-2}_{\ell}\upintop\ilimits@_{[u^{-}_{\ell},u^{+}_{\ell}]\times[v^{-}_{\ell},v^{+}_{\ell}]}(v-u)^{2H-2}\mathrm{d}u\mathrm{d}v$
$\displaystyle\leq(v_{\ell}^{-}-u_{\ell}^{+})^{2H-2}$
$\displaystyle\lesssim((v^{+}_{\ell}-u^{-}_{\ell})\wedge 1/2)^{2H^{\prime}-2}$
$\displaystyle\leq((v-u)\wedge 1/2)^{2H^{\prime}-2}$

In the second-last inequality we have used that there exists some $L$ s.t. for
all $\ell\geq L$

$\vartheta^{2H-2}\leq(\vartheta+2\varrho_{\ell})^{2H^{\prime}-2}$

for all $\vartheta\in[\varrho_{\ell},1/2]$.

We now consider terms corresponding to maximal sequences of consecutive
pairings, i.e.

$\upintop\ilimits@_{s<u_{1}<v_{1}<\ldots<u_{k}<v_{k}<t}\mathbb
E[\dot{X}^{\ell}_{u_{1}}\dot{X}^{\ell}_{v_{1}}]\cdots\mathbb
E[\dot{X}^{\ell}_{u_{k}}\dot{X}^{\ell}_{v_{k}}]\mathrm{d}u_{1}\mathrm{d}v_{1}\cdots\mathrm{d}u_{k}\mathrm{d}v_{k}\text{
.}$ (52)

It is always possible (e.g. by Kolmogorov’s extension theorem) to add
independent components to $X$. With this in mind, by Wick’s theorem we may
write the above integral as $\mathbb
E\mathcal{S}(X^{\ell})_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{k}\alpha_{k}}$
with $\alpha_{i}\neq\alpha_{j}$ for all $i\neq j$. By the shuffle identity (8)
we have

$\begin{split}&\sum_{h=0}^{n}\mathcal{S}(X^{\ell})_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{h}\alpha_{h}\beta\beta\alpha_{h+1}\alpha_{h+1}\ldots\alpha_{k}\alpha_{k}}\\\
={}&\mathcal{S}(X^{\ell})_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{k}\alpha_{k}}\mathcal{S}(X^{\ell})_{st}^{\beta\beta}-\sum_{0\leq
i<j\leq
k}\mathcal{S}(X^{\ell})_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{i}\alpha_{i}\beta\alpha_{i+1}\alpha_{i+1}\ldots\alpha_{j}\alpha_{j}\beta\alpha_{j+1}\alpha_{j+1}\ldots\alpha_{k}\alpha_{k}}\\\
&-\sum_{0\leq i<j\leq
k}\mathcal{S}(X^{\ell})_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{i}\alpha_{i}\beta\alpha_{i+1}\alpha_{i+1}\ldots\alpha_{j+1}\beta\alpha_{j+1}\ldots\alpha_{k}\alpha_{k}}\\\
&-\sum_{0\leq i<j\leq
k}\mathcal{S}(X^{\ell})_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{i}\beta\alpha_{i}\ldots\alpha_{j}\alpha_{j}\beta\alpha_{j+1}\alpha_{j+1}\ldots\alpha_{k}\alpha_{k}}\\\
&-\sum_{0\leq i<j\leq
k}\mathcal{S}(X^{\ell})_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{i}\beta\alpha_{i}\ldots\alpha_{j}\beta\alpha_{j}\ldots\alpha_{k}\alpha_{k}}\\\
&-\sum_{h=0}^{k}\Big{(}\mathcal{S}(X^{\ell})_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{h}\beta\alpha_{h}\beta\alpha_{h+1}\alpha_{h+1}\ldots\alpha_{k}\alpha_{k}}+\mathcal{S}(X^{\ell})_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{h-1}\alpha_{h-1}\beta\alpha_{h}\beta\alpha_{h}\ldots\alpha_{k}\alpha_{k}}\Big{)}\text{
.}\end{split}$ (53)

When shuffling we have separated the cases in which all $\alpha_{h}\alpha_{h}$
and $\beta\beta$ occur as consecutive pairs, from those in which at least one
such pair is separated. We now take expectations: note that both independence
and equal distribution of components are used.

$(k+1)\mathbb
E\mathcal{S}(X^{\ell})_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{k}\alpha_{k}\beta\beta}=\mathbb
E\mathcal{S}(X^{\ell})_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{k}\alpha_{k}}\cdot\tfrac{1}{2}\mathbb
E[(X^{\ell})^{2}_{st}]-\sum_{Q}Q^{\ell}_{st}$ (54)

where we are summing over a finite number of diagrams $Q$ whose longest
sequence of consecutive pairings contains $k$ pairs or fewer.

We now prove the statement in the case in which $P$ has no single nodes, by
induction on $n$. For $n=0$, $P^{\ell}_{st}=\varnothing^{\ell}_{st}\equiv 1$
there is nothing to show. Let $n\geq 1$, and assume we have rewritten the
integral according to (50) (where the middle integral may be skipped, since
there are no Malliavin derivatives). If $P$ is not given by a sequence of
$n/2$ consecutive pairs, all maximal sequences of consecutive pairs in $P$
consist of fewer than $n$ pairs, and that thus the inductive hypothesis
applies to them: this means that for each such sequence $Q$ with $k$ pairs,
$|Q^{\ell}_{uv}|\lesssim(v-u)^{2kH^{\prime}}$. Using the bounds for the first
two types of integrand derived in the first part of this proof, the statement
for $P$ then follows from Section 3 applied in the modified case of Section 3
and with exponent $H^{\prime}$. Assume now $n=2(k+1)$ and let $P$ be given by
the diagram consisting of $k+1$ consecutive pairs: the only thing needed to
conclude the induction is the bound. This follows from (54) thanks to the
inductive hypothesis and the boundedness statement of Section 3.

Finally, we consider the general case in which $P$ may have single nodes. This
follows again by writing $P^{\ell}_{st}$ in nested form, bounding terms
corresponding to non-consecutive pairings as done above, and bounding the
middle integral in (50) thanks to the boundedness statement of Section 3. When
invoking this lemma, $f_{\ell}$ is going to be a product of terms of the form
(52) (with the extrema $s$ and $t$ replaced with variables $u_{i}$ and $u_{j}$
already integrated in the outer or middle integral), which as already proved
is bounded by $\lesssim(t-s)^{2H^{\prime}k}$: this yields the required bound
overall. $\blacksquare$

######  Proposition  (Convergence).

The functions $[0,T]^{m}\to\mathbb R$ of Section 3 individually converge a.e.
to those of Section 2: for $P\in\mathcal{P}^{n}_{m}$ it holds that

$P^{\ell}_{st}\xrightarrow{\ell\to\infty}P_{st}\text{ .}$ (55)

Moreover $|P_{st}|\lesssim|P|_{st}$ (the integrals of Section 3) uniformly on
$\Delta^{m}[s,t]$.

###### Proof.

The inequality is an absolute estimate of $P_{st}$ using (9) and (10). The
structure of the proof of the first statement closely mirrors that of the
previous lemma: we first consider the case in which $P$ does not have single
nodes. For $u^{-}_{\ell}<v^{-}_{\ell}$

$\displaystyle\mathbb
E[\dot{X}_{u}^{\ell}\dot{X}^{\ell}_{v}]=\varrho^{-2}_{\ell}R(\Delta(u^{-}_{\ell},u^{+}_{\ell}),\Delta(v^{-}_{\ell},v^{+}_{\ell}))=\partial_{12}R(\overline{u},\overline{v})$

for some $\overline{u}\in(u^{-}_{\ell},u^{+}_{\ell})$,
$\overline{v}\in(v^{-}_{\ell},v^{+}_{\ell})$, by the intermediate value
theorem applied twice. Pointwise convergence $\mathbb
E[\dot{X}_{u}^{\ell}\dot{X}^{\ell}_{v}]\to\partial_{12}R(u,v)$ then holds by
continuity of $\partial_{12}R$ and thanks to the fact that for any $u<v$ there
exists $L$ s.t. $u^{-}_{\ell}<v^{-}_{\ell}$ for all $\ell\geq L$. This takes
care of convergence of terms corresponding to non-consecutive pairings (of
course, the same holds for consecutive pairings, but is not useful since
$\partial_{12}R(u,v)$ may not be integrable in this case).

We now proceed by induction on $n$. For $n=0$ there is nothing to prove, so
let $n\geq 1$ and first consider the case in which $P$ is not given by a
sequence of $n/2$ consecutive pairs: the statement follows by dominated
convergence applied to the outer integral in (50), by the above and the
inductive hypothesis applied to sequences of consecutive nodes of length less
than $n$, in conjunction with Section 3. Let now $n=2(k+1)$ and let $P$ be
given by the diagram consisting of $k+1$ consecutive pairs: recalling the
argument (and indexing notation) of the previous proof, we have
$P_{st}^{\ell}=\mathbb
E\mathcal{S}(X^{\ell})_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{k}\alpha_{k}\beta\beta}$,
which is convergent since $\mathcal{S}(X^{\ell})_{st}\to\mathcal{S}(X)_{st}$
in $L^{2}$. By the same calculation of (53) applied to $X$ instead of to
$X^{\ell}$, and taking expectations

$\begin{split}&(k+1)\lim_{\ell\to\infty}P_{st}^{\ell}\\\ ={}&(k+1)\mathbb
E\mathcal{S}(X)_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{k}\alpha_{k}\beta\beta}\\\
={}&\mathbb
E\mathcal{S}(X)_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{k}\alpha_{k}}\cdot\tfrac{1}{2}\mathbb
E[(X)^{2}_{st}]-\sum_{0\leq i<j\leq k}\mathbb
E\mathcal{S}(X)_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{i}\alpha_{i}\beta\alpha_{i+1}\alpha_{i+1}\ldots\alpha_{j}\alpha_{j}\beta\alpha_{j+1}\alpha_{j+1}\ldots\alpha_{k}\alpha_{k}}\\\
&-\sum_{0\leq i<j\leq k}\mathbb
E\mathcal{S}(X)_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{i}\alpha_{i}\beta\alpha_{i+1}\alpha_{i+1}\ldots\alpha_{j+1}\beta\alpha_{j+1}\ldots\alpha_{k}\alpha_{k}}\\\
&-\sum_{0\leq i<j\leq k}\mathbb
E\mathcal{S}(X)_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{i}\beta\alpha_{i}\ldots\alpha_{j}\alpha_{j}\beta\alpha_{j+1}\alpha_{j+1}\ldots\alpha_{k}\alpha_{k}}\\\
&-\sum_{0\leq i<j\leq k}\mathbb
E\mathcal{S}(X)_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{i}\beta\alpha_{i}\ldots\alpha_{j}\beta\alpha_{j}\ldots\alpha_{k}\alpha_{k}}\\\
&-\sum_{h=0}^{k}\Big{(}\mathbb
E\mathcal{S}(X)_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{h}\beta\alpha_{h}\beta\alpha_{h+1}\alpha_{h+1}\ldots\alpha_{k}\alpha_{k}}+\mathbb
E\mathcal{S}(X)_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{h-1}\alpha_{h-1}\beta\alpha_{h}\beta\alpha_{h}\ldots\alpha_{k}\alpha_{k}}\Big{)}\text{
.}\end{split}$ (56)

We now expand the product: by the inductive hypothesis and Section 3, and
using Fubini’s theorem we have (setting $u_{0}=s=w_{0}$)

$\displaystyle\mathbb
E\mathcal{S}(X)_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{k}\alpha_{k}}\cdot\mathbb
E[X^{2}_{st}]$ $\displaystyle={}$
$\displaystyle\upintop\ilimits@_{s<u_{1}<\ldots<u_{k}<t}\big{[}\tfrac{1}{2}R^{\prime}(u_{1})-\partial_{2}R(u_{0},u_{1})\big{]}\cdots\big{[}\tfrac{1}{2}R^{\prime}(u_{k})-\partial_{2}R(u_{k-1},u_{k})\big{]}\mathrm{d}u_{1}\cdots\mathrm{d}u_{k}$
$\displaystyle\quad\cdot\upintop\ilimits@_{s}^{t}\big{[}\tfrac{1}{2}R^{\prime}(v)-\partial_{2}R(s,v)\big{]}\mathrm{d}v$
$\displaystyle={}$
$\displaystyle\upintop\ilimits@_{\begin{subarray}{c}s<u_{1}<\ldots<u_{k}<t\\\
s<v<t\end{subarray}}\big{[}\tfrac{1}{2}R^{\prime}(u_{1})-\partial_{2}R(u_{0},u_{1})\big{]}\cdots\big{[}\tfrac{1}{2}R^{\prime}(u_{k})-\partial_{2}R(u_{k-1},u_{k})\big{]}$
$\displaystyle\quad\cdot\big{[}\tfrac{1}{2}R^{\prime}(v)-\partial_{2}R(s,v)\big{]}\mathrm{d}u_{1}\cdots\mathrm{d}u_{k}\mathrm{d}v$
$\displaystyle={}$
$\displaystyle\sum_{j=0}^{k}\upintop\ilimits@_{s<u_{1}<\ldots<u_{j}<v<u_{j+1}<\ldots<u_{k}<t}\big{[}\tfrac{1}{2}R^{\prime}(u_{1})-\partial_{2}R(u_{0},u_{1})\big{]}\cdots\big{[}\tfrac{1}{2}R^{\prime}(u_{k})-\partial_{2}R(u_{k-1},u_{k})\big{]}$
$\displaystyle\qquad\cdot\big{[}\tfrac{1}{2}R^{\prime}(v)-\partial_{2}R(s,v)\big{]}\mathrm{d}u_{1}\cdots\mathrm{d}u_{k}\mathrm{d}v\text{
.}$

Note that the use of Fubini’s theorem is justified in view of (10) applied to
absolutely bound each integral above, and Section 3. Writing

$\partial_{2}R(\Delta(x,y),z)\coloneqq\partial_{2}R(y,z)-\partial_{2}R(x,z)=\upintop\ilimits@_{x}^{y}\partial_{12}R(w,z)\mathrm{d}w\text{
,}$

we expand each summand:

$\displaystyle\upintop\ilimits@_{s<u_{1}<\ldots<u_{j}<v<u_{j+1}<\ldots<u_{k}<t}\big{[}\tfrac{1}{2}R^{\prime}(u_{1})-\partial_{2}R(u_{0},u_{1})\big{]}\cdots\big{[}\tfrac{1}{2}R^{\prime}(u_{k})-\partial_{2}R(u_{k-1},u_{k})\big{]}$
$\displaystyle\quad\cdot\big{[}\tfrac{1}{2}R^{\prime}(v)-\partial_{2}R(s,v)\big{]}\mathrm{d}u_{1}\cdots\mathrm{d}u_{k}\mathrm{d}v$
$\displaystyle={}$
$\displaystyle\upintop\ilimits@_{s<u_{1}<\ldots<u_{j}<v<u_{j+1}<\ldots<u_{k}<t}\big{[}\tfrac{1}{2}R^{\prime}(u_{1})-\partial_{2}R(u_{0},u_{1})\big{]}\cdots\big{[}\tfrac{1}{2}R^{\prime}(u_{j})-\partial_{2}R(u_{j-1},u_{j})\big{]}$
$\displaystyle\quad\cdot\big{[}\tfrac{1}{2}R^{\prime}(v)-\partial_{2}R(u_{j},v)+\textstyle\sum_{i=0}^{j-1}\partial_{2}R(\Delta(u_{i},u_{i+1}),v)\big{]}$
$\displaystyle\quad\cdot\big{[}\tfrac{1}{2}R^{\prime}(u_{j+1})-\partial_{2}R(v,u_{j+1})+\partial_{2}R(\Delta(u_{j},v),u_{j+1})\big{]}$
$\displaystyle\quad\cdot\big{[}\tfrac{1}{2}R^{\prime}(u_{j+2})-\partial_{2}R(u_{j+1},u_{j+2})\big{]}\cdots\big{[}\tfrac{1}{2}R^{\prime}(u_{k})-\partial_{2}R(u_{k-1},u_{k})\big{]}\mathrm{d}u_{1}\cdots\mathrm{d}u_{k}\mathrm{d}v$
$\displaystyle={}$
$\displaystyle\upintop\ilimits@_{s<u_{1}<\ldots<u_{j}<v<u_{j+1}<\ldots<u_{k}<t}\big{[}\tfrac{1}{2}R^{\prime}(u_{1})-\partial_{2}R(u_{0},u_{1})\big{]}\cdots\big{[}\tfrac{1}{2}R^{\prime}(v)-\partial_{2}R(u_{j},v)\big{]}$
$\displaystyle\quad\cdot\big{[}\tfrac{1}{2}R^{\prime}(u_{j+1})-\partial_{2}R(u_{j+1},v)\big{]}\cdots\big{[}\tfrac{1}{2}R^{\prime}(u_{k})-\partial_{2}R(u_{k-1},u_{k})\big{]}\mathrm{d}u_{1}\cdots\mathrm{d}u_{k}\mathrm{d}v$
$\displaystyle+\upintop\ilimits@_{s<u_{1}<\ldots<u_{j}<r<v<u_{j+1}<\ldots<u_{k}<t}\big{[}\tfrac{1}{2}R^{\prime}(u_{1})-\partial_{2}R(u_{0},u_{1})\big{]}\cdots\big{[}\tfrac{1}{2}R^{\prime}(u_{j})-\partial_{2}R(u_{j-1},u_{j})\big{]}$
$\displaystyle\quad\cdot\big{[}\tfrac{1}{2}R^{\prime}(v)-\partial_{2}R(r,v)+\partial_{2}R(\Delta(u_{j},r),v)\big{]}\partial_{12}R(r,u_{j+1})\mathrm{d}u_{1}\cdots\mathrm{d}u_{k}\mathrm{d}r\mathrm{d}v$
$\displaystyle+\sum_{i=0}^{j-1}\upintop\ilimits@_{s<u_{1}<\ldots<u_{i}<q<u_{i+1}<\ldots<u_{j}<v<u_{j+1}<\ldots<u_{k}<t}\big{[}\tfrac{1}{2}R^{\prime}(u_{1})-\partial_{2}R(u_{0},u_{1})\big{]}$
$\displaystyle\qquad\cdots\partial_{12}R(q,v)\big{[}\tfrac{1}{2}R^{\prime}(u_{i+1})-\partial_{2}R(q,u_{i+1})+\partial_{2}R(\Delta(u_{i},q),u_{i+1})\big{]}\cdots\big{[}\tfrac{1}{2}R^{\prime}(u_{j+1})-\partial_{2}R(v,u_{j+1})\big{]}$
$\displaystyle\qquad\cdots\big{[}\tfrac{1}{2}R^{\prime}(u_{k})-\partial_{2}R(u_{k-1},u_{k})\big{]}\mathrm{d}u_{1}\cdots\mathrm{d}u_{k}\mathrm{d}q\mathrm{d}v$
$\displaystyle+\sum_{i=0}^{j-1}\upintop\ilimits@_{s<u_{1}<\ldots<u_{i}<q<u_{i+1}<\ldots<u_{j}<r<v<u_{j+1}<\ldots<u_{k}<t}\big{[}\tfrac{1}{2}R^{\prime}(u_{1})-\partial_{2}R(u_{0},u_{1})\big{]}$
$\displaystyle\qquad\cdots\partial_{12}R(q,v)\big{[}\tfrac{1}{2}R^{\prime}(u_{i+1})-\partial_{2}R(q,u_{i+1})+\partial_{2}R(\Delta(u_{i},q),u_{i+1})\big{]}\cdots\partial_{12}R(r,u_{j+1})$
$\displaystyle\qquad\cdots\big{[}\tfrac{1}{2}R^{\prime}(u_{k})-\partial_{2}R(u_{k-1},u_{k})\big{]}\mathrm{d}u_{1}\cdots\mathrm{d}u_{k}\mathrm{d}q\mathrm{d}r\mathrm{d}v$
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$\displaystyle={}$ $\displaystyle(k+1)P_{st}+\mathbb
E\mathcal{S}(X)_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{j}\alpha_{j}\beta\alpha_{j+1}\alpha_{j+1}\beta\alpha_{j+2}\alpha_{j+2}\ldots\alpha_{k}\alpha_{k}}+\mathbb
E\mathcal{S}(X)^{\alpha_{1}\alpha_{1}\ldots\alpha_{j}\alpha_{j}\beta\alpha_{j+1}\beta\alpha_{j+1}\ldots\alpha_{k}\alpha_{k}}_{st}$
$\displaystyle+\sum_{i=0}^{j-1}\mathcal{S}(X)_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{i}\alpha_{i}\beta\alpha_{i+1}\alpha_{i+1}\ldots\alpha_{j}\alpha_{j}\beta\alpha_{j+1}\alpha_{j+1}\ldots\alpha_{k}\alpha_{k}}+\sum_{i=0}^{j-1}\mathcal{S}(X)_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{i+1}\beta\alpha_{i+1}\ldots\alpha_{j}\alpha_{j}\beta\alpha_{j+1}\alpha_{j+1}\ldots\alpha_{k}\alpha_{k}}$
$\displaystyle+\sum_{i=0}^{j-1}\mathcal{S}(X)_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{i}\alpha_{i}\beta\alpha_{i+1}\alpha_{i+1}\ldots\alpha_{j+1}\beta\alpha_{j+1}\ldots\alpha_{k}\alpha_{k}}+\sum_{i=0}^{j-1}\mathcal{S}(X)_{st}^{\alpha_{1}\alpha_{1}\ldots\alpha_{i+1}\beta\alpha_{i+1}\ldots\alpha_{j+1}\beta\alpha_{j+1}\ldots\alpha_{k}\alpha_{k}}\text{
.}$

It now follows by substitution into the sum $\sum_{j=0}^{k}$ and simplifying
in (56) that $\lim_{\ell}P^{\ell}_{st}=P_{st}$.

Finally, we consider diagrams that contain single nodes. In order to invoke
Section 3 we must argue that $f_{\ell}\to f$ uniformly (uniform boundedness
holds by the previous lemma). This again follows from the fact that $f_{\ell}$
can be written as a product of expected signatures of $X^{\ell}$, each of
which converges uniformly in $\ell$ as a function of its extrema: recalling
the notations for truncation and projection introduced in Section 1 and the
definition of inhomogeneous $p$-variation distance [21, §8.1.2], we have

$\displaystyle\sup_{u<v}|\mathbb E\mathcal{S}(X)_{uv}^{(n)}-\mathbb
E\mathcal{S}(X^{\ell})_{uv}^{(n)}|$ $\displaystyle\leq\mathbb
E\sup_{u<v}|\mathcal{S}(X)_{uv}^{(n)}-\mathcal{S}(X^{\ell})_{uv}^{(n)}|$
$\displaystyle\leq\lVert\rho_{p\mathchar
45\relax\mathrm{var}}(\mathcal{S}^{\lfloor
p\rfloor}(X^{\ell}),\mathcal{S}^{\lfloor p\rfloor}(X))\rVert_{L^{1}}$
$\displaystyle\xrightarrow{\ell\to\infty}0$

for $p>(1/H)\vee n$, where we have used [18, Theorem 1]. The statement now
follows once again by dominated convergence and Fubini’s theorem.
$\blacksquare$

We are ready to put it all together:

###### Proof of Section 2.

$\displaystyle\mathscr{w}^{m}\mathcal{S}(X)_{st}^{\gamma_{1},\ldots,\gamma_{n}}$
$\displaystyle=\frac{1}{m!}\sko\hskip 0.90005pt^{m}\big{(}\mathbb
E\mathcal{D}^{m}\mathcal{S}(X)_{st}^{\gamma_{1},\ldots,\gamma_{n}}\big{)}$
(57) $\displaystyle=\frac{1}{m!}\sko\hskip
0.90005pt^{m}\lim_{\ell\to\infty}\big{(}\mathbb
E\mathcal{D}^{m}\mathcal{S}(X^{\ell})_{st}^{\gamma_{1},\ldots,\gamma_{n}}\big{)}$
(58) $\displaystyle=\sko\hskip
0.90005pt^{m}\sum_{P\in\mathcal{P}^{n}_{m}}\lim_{\ell\to\infty}P_{st}^{\ell;\gamma_{1},\ldots,\gamma_{n}}$
(59) $\displaystyle=\sum_{P\in\mathcal{P}^{n}_{m}}\sko\hskip
0.90005pt^{m}P^{\gamma_{1},\ldots,\gamma_{n}}_{st}$ (60)

In (57) we have used Stroock’s formula (23), which is possible since
$\mathcal{S}(X)_{st}^{\gamma_{1},\ldots,\gamma_{n}}\in\mathbb D^{\infty,2}$:
this is because $\mathcal{S}(X^{\ell})_{st}\to\mathcal{S}(X)_{st}$ in
$\bigoplus_{k\leq n}\mathscr{W}^{k}$ which is closed in $L^{2}\Omega$. In (58)
we have used that convergence of $\mathcal{S}(X^{\ell})_{st}$ actually holds
in $\mathbb D^{\infty,2}$, since the norm of $\mathbb D^{\infty,2}$ is
dominated by the $L^{2}$ norm in bounded Wiener chaos [35, Proposition 1.2.2].
(59) uses Section 3 and (60) is just the statement (required by our definition
of membership of a function to $\mathcal{H}^{\otimes m}$ Section 1) that
$P_{st}^{\ell;\gamma_{1},\ldots,\gamma_{n}}$ converges a.e. boundedly to
$P^{\ell}_{st}$, which holds by Section 3 and Section 3. As argued in the
previous two proofs, $P_{st}^{\gamma_{1},\ldots,\gamma_{n}}$ can always be
expressed as the expected signature evaluated on a word, up to augmenting $X$
with independent copies of itself: this can be used to infer that each
$P_{st}^{\gamma_{1},\ldots,\gamma_{n}}$ — not just their sum — belongs to
$\mathbb D^{\infty,2}(\mathcal{H}^{\otimes m})$. This concludes the proof of
the main result. $\blacksquare$

## Conclusions and further directions

Conclusions and further directions

By providing a single formula for the expected signature of fractional
Brownian motion that holds for any Hurst parameter $H\in(1/4,1)$, this article
closes a gap in the literature left open by [3]. Along the way, we have had
opportunity to consider numerous other aspects of our computation, such as
similar formulae for higher levels of the Wiener chaos expansion of the
signature, and other examples of Gaussian processes.

We believe this work recommends a variety of applications and further
investigations. First and foremost, it would be interesting to write
stochastic Taylor expansions as suggested by Section 2, under precise
conditions on the vector fields, and by providing bounds on the mean square
error. Making this calculation rigorous and providing precise asymptotic
estimates such as those in [37] would be an interesting result, which could be
applied to approximation problems for Gaussian RDEs on manifolds such as those
considered in [1] for SDEs (although for this precise problem, the joint
signature $\mathcal{S}(X,t)$ would have to be considered). A further step
would involve proving conditional versions of the results in this paper, which
would make it possible to estimate the error generated by multiple steps in an
Euler scheme.

The fact that (e.g. for fBm) the integral $\mathbb
E\mathcal{S}(X)^{\alpha_{1}\alpha_{1}\cdots\alpha_{k}\alpha_{k}}_{st}$ with
$\alpha_{i}\neq\alpha_{j}$ is actually convergent for any $H>0$ raises the
question of whether something can be said about the sequence
$\mathcal{S}(X^{\ell})^{\alpha_{1}\alpha_{1}\cdots\alpha_{k}\alpha_{k}}_{st}$,
i.e. by considering the particular word on which $\mathcal{S}(X^{\ell})$,
which is not convergent in probability for $H\leq 1/4$, is evaluated.

It would be interesting to express the expected signature of a Gaussian
process as the exponential of a formal series of tensors, thus computing its
signature cumulants [8]: this is how the expected signature of Brownian motion
(2) is usually presented (with the series a finite sum), but the analogous
formulation for Gaussian processes that are not martingales appears more
difficult to write down.

A more computational goal, though not one that appears trivial, is to
explicitly compute Section 2 for certain semimartingales, such as the Brownian
bridge, for which the integrals are all analytically solvable. An interesting
question is how the relationship between Brownian motion and Brownian bridge
is reflected by their expected signatures. It would also be helpful to see
whether similar formulae to ours can be made available for non-centred
Gaussian processes, e.g. general Ornstein-Uhlenbeck processes. Finally, it
would be interesting to try to apply the main theorem to the Riemann-Liouville
process Section 2.

##  Appendix A Equivalence with [3] for the expected signature of fBm at level
$4$

In [3, Theorem 34] the authors check that the explicit integral expressions
for $\mathbb E\mathcal{S}(X)^{(n)}_{01}$ with $n=2,4$, previously derived for
$X$ a fractional Brownian motion of Hurst parameter $H\in(1/2,1)$, continue to
be valid for $H\in(1/4,1)$. This is done by performing transformations on
$\mathbb E\mathcal{S}(X^{\ell})_{01}$ before passing to the limit. This
calculation is specific to levels $2$ and $4$ and to fBm, and for this reason
the expression for the expected signature is not immediately comparable to
that obtained as a special case of Section 2. We devote this appendix to
checking that the two agree.

Level $2$ is easy to check, since (31) reduces to
$\text{\textdelta}^{\alpha\beta}/2$. Referring to Section 2, we consider level
$4$ using (39): starting with the first integral above, we have

$\displaystyle\upintop\ilimits@_{0<u<v<1}\big{[}\tfrac{1}{2}R(\mathrm{d}u)-R(s,\mathrm{d}u)\big{]}\big{[}\tfrac{1}{2}R(\mathrm{d}v)-R(u,\mathrm{d}v)\big{]}\mathrm{d}u\mathrm{d}v$
$\displaystyle={}$ $\displaystyle
H^{2}\upintop\ilimits@_{0<u<v<1}u^{2H-1}(v-u)^{2H-1}\mathrm{d}u\mathrm{d}v$
$\displaystyle={}$ $\displaystyle
H^{2}\upintop\ilimits@_{0}^{1}u^{2H-1}\bigg{[}\frac{(v-u)^{2H-1}}{2H}\bigg{]}_{u=0}^{1}\mathrm{d}u$
$\displaystyle={}$
$\displaystyle\frac{H^{2}}{2}\upintop\ilimits@_{0}^{1}u^{2H-1}\bigg{[}\frac{(v-u)^{2H}}{2H}\bigg{]}_{v=u}^{1}\mathrm{d}u$
$\displaystyle={}$
$\displaystyle\frac{H}{2}\upintop\ilimits@_{0}^{1}u^{2H-1}(1-u)^{2H}\mathrm{d}u$
$\displaystyle={}$
$\displaystyle\frac{H}{4}\upintop\ilimits@_{0}^{1}u^{2H-1}(1-u)^{2H-1}\mathrm{d}u$

where the last identity can be verified by showing that the difference of the
two integrands is odd about the point $u=1/2$, which in turn is seen by
observing that

$\frac{H}{4}u^{2H-1}(1-u)^{2H-1}-\frac{H}{2}u^{2H-1}(1-u)^{2H}+\frac{H}{4}(1-u)^{2H-1}u^{2H-1}-\frac{H}{2}(1-u)^{2H-1}u^{2H}$

has zero derivative and vanishes at $u=1/2$. This shows equality with [3,
coefficient of the first term of $\Gamma^{2}_{H}$ in Corollary 33]. We proceed
with the second integral in Section 2:

$\displaystyle\upintop\ilimits@_{s<u<v<w<t}R(\mathrm{d}u,\mathrm{d}w)\big{[}\tfrac{1}{2}R(\mathrm{d}v)-R(u,\mathrm{d}v)\big{]}\mathrm{d}u\mathrm{d}v\mathrm{d}w$
$\displaystyle={}$ $\displaystyle
H^{2}(2H-1)\upintop\ilimits@_{0<u<v<w<1}(w-u)^{2H-2}(v-u)^{2H-1}\mathrm{d}u\mathrm{d}v\mathrm{d}w$
$\displaystyle={}$ $\displaystyle
H^{2}\upintop\ilimits@_{0<u<v<1}\Big{[}(1-u)^{2H-1}(v-u)^{2H-1}-(v-u)^{4H-2}\Big{]}\mathrm{d}u\mathrm{d}v$
$\displaystyle={}$
$\displaystyle\bigg{(}\frac{H}{2}-\frac{H^{2}}{4H-1}\bigg{)}\upintop\ilimits@_{0}^{1}(1-u)^{4H-1}\mathrm{d}u$
$\displaystyle={}$ $\displaystyle\frac{2H-1}{8(4H-1)}\text{ .}$

For the third integral we have

$\displaystyle\upintop\ilimits@_{0<u<v<w<z<1}R(\mathrm{d}u,\mathrm{d}w)R(\mathrm{d}v,\mathrm{d}z)$
$\displaystyle={}$ $\displaystyle
H^{2}(2H-1)^{2}\upintop\ilimits@_{0<u<v<w<z<1}(w-u)^{2H-2}(z-v)^{2H-2}\mathrm{d}u\mathrm{d}v\mathrm{d}w\mathrm{d}z$
$\displaystyle={}$ $\displaystyle
H^{2}(2H-1)\upintop\ilimits@_{0<u<v<z<1}\Big{[}(z-u)^{2H-1}(z-v)^{2H-2}-(v-u)^{2H-1}(z-v)^{2H-2}\Big{]}\mathrm{d}u\mathrm{d}v\mathrm{d}z$
$\displaystyle={}$
$\displaystyle\frac{H(2H-1)}{2}\upintop\ilimits@_{0<v<z<1}\Big{[}z^{2H}(z-v)^{2H-2}-(z-v)^{4H-2}-v^{2H}(z-v)^{2H-2}\Big{]}\mathrm{d}v\mathrm{d}z$
$\displaystyle={}$
$\displaystyle\frac{H(2H-1)}{2}\upintop\ilimits@_{0<v<z<1}(z^{2H}-v^{2H})(z-v)^{2H-2}\mathrm{d}v\mathrm{d}z-\frac{H(2H-1)}{4H-1}\upintop\ilimits@_{0}^{1}(1-v)^{4H-1}\mathrm{d}v$
$\displaystyle={}$
$\displaystyle\frac{H(2H-1)}{2}\upintop\ilimits@_{0<v<z<1}(z^{2H}-v^{2H})(z-v)^{2H-2}\mathrm{d}v\mathrm{d}z-\frac{2H-1}{4(4H-1)}$
$\displaystyle={}$
$\displaystyle\frac{H}{2}\upintop\ilimits@_{0}^{1}(1-v^{2H})(1-v)^{2H-1}\mathrm{d}v-H^{2}\upintop\ilimits@_{0<v<z<1}z^{2H-1}(z-v)^{2H-1}-\frac{2H-1}{4(4H-1)}$
$\displaystyle={}$
$\displaystyle\frac{H}{4(4H-1)}-\frac{H}{4}\upintop\ilimits@_{0}^{1}v^{2H-1}(1-v)^{2H-1}\mathrm{d}v\text{
.}$

In the integration by parts we have used that $\lim_{z\to
v^{+}}(z^{2H}-v^{2H})(z-v)^{2H-1}=0$ which can be shown by using that for
$1/4<H<1/2$

$0\leq(z^{2H}-v^{2H})(z-v)^{2H-1}\leq(z-v)^{4H-1}\xrightarrow{z\to v^{+}}0$

since $z^{2H}-v^{2H}<(z-v)^{2H}$ for $0<v<z$ and $H<1/2$. In the last identity
we have used a similar symmetry argument as the one used in the first
calculation, solved trivial integrals and rearranged terms. Note how this
calculation would have been simpler if $H\geq 1/2$ since it would not have
been necessary to integrate by parts to avoid integrating $(z-v)^{2H-2}$ (cf.
[3, Lemma 32]).

### Data availability statement

The research presented in this manuscript is theoretical in nature and does
not refer to empirical data. All necessary information is contained in the
cited references.

### Author contribution statement

Thomas Cass and Emilio Ferrucci are both authors of this paper. Emilio
Ferrucci is the corresponding author.

### Conflict of interest statement

The authors declare no conflicts of interest.

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