3.1. Encoding z[n]-gons into side-partitions of $\mathsf{ECP}({z}[n])$

A new geometric description: convex chains between contact points, convex chains in a right triangle, and simplex product. Let us now fix $s[\kappa]\in\mathbb{N}_\kappa(n)$ . For all $z[n]\in{\mathcal{C}_\kappa}(s[\kappa])$ , consider the corresponding side lengths $c[\kappa]$ (which are thus all nonzero), and for all $j\in\{1,\ldots,\kappa\}$ , define the side-partition $u^{(j)}[N_j,c_j]=(u_1^{(j)},\ldots,u_{N_j}^{(j)})$ of the jth side length $c_j$ of $\mathsf{ECP}({z}[n])$ , which is defined in Figure 10 below and is an element of $P[c_j,N_j]$ . For any side-partition $u^{(j)}[N_j,c_j]$ thus defined, we set $u^{(j)}_0\,:\!=\,0$ , $u_{N_j+1}^{(j)}=c_j$ , so that we have $u_0^{(j)} \lt u_1^{(j)} \lt \ldots \lt u_{N_j}^{(j)} \lt u_{N_j+1}^{(j)}$ .

Figure 10.

The jth side-partition $(0=u_0^{(j)} \lt u_1^{(j)} \lt \ldots \lt u_{N_j}^{(j)} \lt u_{N_j+1}^{(j)}=c_j)$ of $c_j$ , with $s_j=2$ , $s_{{j+1}}=3$ . An alternative way of building the $u^{(j)}[N_j,c_j]$ will be given in Figure 11. Notice here that we see the contact point on $c_j$ , but we do not mark it; we treat it the same as the other points.

Main strategy of the proof. Our main strategy is to consider for all $s[\kappa]\in\mathbb{N}_\kappa(n)$ the extraction mapping, which encodes a convex z[n]-gon in terms of its $\mathsf{ECP}({z}[n])$ and its side-partitions:

(3.1) \begin{align} \begin{array}{r@{\quad}c@{\quad}c@{\quad}l} \chi_{s[\kappa]}\,:&{\mathcal{C}_\kappa}(s[\kappa])&\longrightarrow&\mathsf{NiceSet}(s[\kappa])\\[3pt] &z[n]&\longmapsto&\left(\ell[\kappa],u^{(1)}[N_1,c_1],\ldots,u^{(\kappa)}[N_\kappa,c_\kappa]\right)\end{array},\end{align}

where $\mathsf{NiceSet}(s[\kappa])\,:\!=\,\mathsf{Im}(\chi_{s[\kappa]})$ is a strict subset of $(\mathbb{R}^+)^{\kappa}\times\prod_{j=1}^\kappa (\mathbb{R}^+)^{N_j}$ that we now discuss. Recall that for all $j\in\{1,\ldots,\kappa\}$ we have set $N_j=s_j+s_{{j+1}}-1$ .

In what follows, we need to see the map $\chi_{s[\kappa]}$ as a ‘nice map’ (a piecewise linear map; see Definition 1) with a ‘nice inverse’ (i.e. with a computable Jacobian determinant), since we will later use this inverse to push forward a measure of $\mathsf{NiceSet}(s[\kappa])$ onto the Lebesgue measure on ${\mathcal{C}_\kappa}(s[\kappa])$ .

Since ${\mathcal{C}_\kappa}(s[\kappa])$ is a subset of $\mathbb{R}^{2n}$ with non-empty interior, $\mathsf{NiceSet}(s[\kappa])$ will be seen to be identifiable with a subset of a domain with the same dimension. In order to characterize $\mathsf{NiceSet}(s[\kappa])$ , it is relevant to notice that since the $s[\kappa]$ are fixed, the $u^{(j)}[N_j,c_j]$ allow us to reconstruct the vectors of the convex chains. Since these vectors have increasing slope (as we progress counterclockwise around the z[n]-gon), the $u^{(j)}[N_j,c_j]$ must satisfy a condition that we now detail.

The image set of $\chi_{s[\kappa]}$ . Set $\mathcal{L}_\kappa^*\,:\!=\,\{\ell[\kappa]\in\mathcal{L}_\kappa \text{ s.t. for all }j\in\{1,\ldots,\kappa\},c_j \gt 0 \}$ . For any $\ell[\kappa]\in\mathcal{L}_\kappa^*$ , consider the side lengths $c[\kappa]$ of the $\mathsf{ECP}$ induced by $\ell[\kappa].$ For any $\kappa$ -tuple of side-partitions $\left(u^{(1)}[N_1,c_1],\ldots,u^{(\kappa)}[N_\kappa,c_\kappa]\right)$ of $c[\kappa]$ , and all $j\in\{1,\ldots,\kappa\}$ , define the inter-point distances of the side-partition $u^{(j)}[N_j,c_j]$ by $\Delta u^{(j)}_i=u_i^{(j)}-u_{i-1}^{(j)}$ for all $i\in\{1,\ldots,N_j+1\}$ . Then define the vectors

\begin{align*}v_k^{(j)}=\begin{pmatrix} \Delta u^{{(j)}}_{s_{j}+k}\\[3pt] \Delta u^{{({j+1})}}_k\end{pmatrix},\quad \forall k\in\{1,\ldots,s_{{j+1}}\}. \end{align*}

In words, summing the vectors $v^{(j)}[s_{{j+1}}]$ allows one to join the point (0, 0) to $(c_j-u^{(j)}_{s_j},u^{({j+1})}_{s_{{j+1}}})$ . When reordered by increasing slope, these vectors form the boundary of a convex polygon whose vertices form a convex chain. This condition on the vectors must be encoded in the side-partitions when we decompose a z[n]-gon through $\chi_{s[\kappa]}$ ; this condition allows us to identify the image set $\mathsf{NiceSet}(s[\kappa]).$

We therefore define the following open subset of $\mathbb{R}^{\kappa}\times\prod_{j=1}^\kappa \mathbb{R}^{N_j}$ :

\begin{multline*} {\mathcal{S}}^{(n)}(s[\kappa])\,:\!=\,\bigg\{\left(\ell[\kappa],w^{(1)},\ldots,w^{(\kappa)}\right)\in\mathcal{L}_\kappa^*\times\prod_{j=1}^\kappa \mathbb{R}^{N_j}\\[3pt] \text{ where }w^{(j)}\,:\!=\,w^{(j)}[N_j,c_j]\in P[c_j,N_j],\\[3pt] \text{ and }\underbrace{\frac{\Delta w^{(j)}_1}{\Delta w^{({j-1})}_{s_{{j-1}}+1}} \lt \ldots \lt \frac{\Delta w^{(j)}_{s_j}}{\Delta w^{({j-1})}_{s_{{j-1}}+s_j}}}_{\text{condition on order of slopes}} \text{ for all }j\in\{1,\ldots,\kappa\}\bigg\}.\end{multline*}

Note that we set $\ell[\kappa]$ in $\mathcal{L}_\kappa^*$ so as to force the construction of any $\mathsf{ECP}$ possible (except those having a nonzero side) within $\mathfrak{C}_\kappa.$ We consider the increments for the side-partitions because they make up the vectors in each corner as described by Figure 11. Recall the family of mappings $(\varphi_{j})_{j\in\{1,\ldots,\kappa\}}$ introduced in (2.3) together with Figure 8.

Figure 11.

The map $\varphi_j$ (resp. $\varphi_{{j-1}}$ ), as introduced in Figure 8, sends the triangle $\mathsf{corner}_j$ (resp. $\mathsf{corner}_{{j-1}}$ ) to the triangle $A^{\prime}_jB^{\prime}_jC^{\prime}_j$ (resp. $A^{\prime}_{{j-1}}B^{\prime}_{{j-1}}C^{\prime}_{{j-1}}$ ). If we perform one more rotation, which is equivalent to setting $C^{\prime}_{{j-1}}=A^{\prime}_j$ and fixing $B^{\prime}_{{j-1}},A^{\prime}_j,B^{\prime}_j$ on the same line, we may interpret the side-partitions just as they appear in the right-hand panel.

A powerful diffeomorphism. It is quite easy to see that, up to a Lebesgue-null set (we want to avoid treating separately the cases in which several points of z[n] are parallel to the lines of $\mathfrak{C}_\kappa$ , or more than two $z_i$ are aligned), $\chi_{s[\kappa]}$ is a bijection between ${\mathcal{C}_\kappa}(s[\kappa])$ and ${\mathcal{S}}^{(n)}(s[\kappa])$ . The following theorem details some even more important properties of the mapping $\chi_{s[\kappa]}.$

Theorem 5. For all $s[\kappa]\in\mathbb{N}_\kappa(n)$ , the mapping

(3.2) \begin{align} \begin{array}{r@{\quad}c@{\quad}c@{\quad}l} \chi_{s[\kappa]}\,:&{\mathcal{C}_\kappa}(s[\kappa])&\longrightarrow&{\mathcal{S}}^{(n)}(s[\kappa])\\[3pt] &z[n]&\longmapsto&\chi_{s[\kappa]}(z[n])\end{array} \end{align}

is a piecewise diffeomorphism (in the sense of Definition 1) whose Jacobian determinant is constant and equals $1/\sin(\theta_\kappa)^{n-\kappa}$ (hence the Jacobian determinant does not depend on $s[\kappa]$ ).

In particular, the Lebesgue measure of the set of interest, ${\mathcal{C}_\kappa}(s[\kappa])$ , satisfies

(3.3) \begin{align} \mathsf{Leb}_{2n}\left({\mathcal{C}_\kappa}(s[\kappa])\right) = \mathsf{Leb}_{2n}\left({\mathcal{S}}^{(n)}(s[\kappa])\right)\sin(\theta_\kappa)^{n-\kappa}. \end{align}

Proof of Theorem 5.We need to detail how the inverse mapping of $\chi_{s[\kappa]}$ is defined to understand its (piecewise) linearity. Pick $\left(\ell[\kappa],u^{(1)}[N_1,c_1],\ldots,u^{(\kappa)}[N_\kappa,c_\kappa]\right)\in{\mathcal{S}}^{(n)}(s[\kappa])$ .

Linearity. Since the tuple $\ell[\kappa]$ is in $\mathcal{L}_\kappa^*$ , it defines an equiangular parallel polygon $\mathsf{ECP}$ inside $\mathfrak{C}_\kappa$ . The map which associates the $\mathsf{b}[\kappa]$ to the $\ell[\kappa]$ is piecewise linear: in the classical Cartesian coordinate system, for any $j\in\{1,\ldots,\kappa\}$ , the coordinates of $\mathsf{b}_{j}$ are linear in $\ell_j$ and $\ell_{j+1}$ , since

\begin{align*}\mathsf{b}_j=\left(r_{j-1}+\frac{\ell_j}{\tan(\theta_\kappa)}-\frac{\ell_{j+1}}{\sin(\theta_\kappa)},\ell_j\right), \end{align*}

up to a rotation.

Then the contact point $\mathsf{cp}_j$ is a translation of $\mathsf{b}_{{j-1}}$ by $u_{s_j}^{(j)}$ along the jth side of the $\mathsf{ECP}$ . This means that the constructions of the contact points are linear in the $\ell[\kappa]$ and $u_{s_j}^{(j)},j\in\{1,\ldots,\kappa\}$ . To reconstruct the rest of the points, recall the vectors

\begin{align*}v_k^{(j)}=\begin{pmatrix} \Delta u^{{(j)}}_{s_{j}+k}\\[3pt] \Delta u^{{({j+1})}}_k \end{pmatrix},\quad \forall k\in\{1,\ldots,s_{{j+1}}-1\}. \end{align*}

The convexity condition imposed on the slopes in ${\mathcal{S}}^{(n)}(s[\kappa])$ forces these vectors to appear in order of increasing slope, so that the map $\varphi_j$ sends these vectors in $\mathsf{corner}_j$ to form the boundary of a convex polygon, whose tuple of vertices is thus a convex chain. The construction of the points of this convex chain can hence be rewritten as

\begin{align*}z^{(j)}_2=\mathsf{cp}_j+A_j(\theta_\kappa)^{-1} v^{(j)}_1, \end{align*}

where $A_j$ was introduced in (2.3), and inductively for all $k\in\{2,\ldots, s_{{j+1}}-1\},$

\begin{align*}z^{(j)}_{k+1}=z^{(j)}_{k}+A_j(\theta_\kappa)^{-1} v^{(j)}_k. \end{align*}

We give an example of this construction in Figure 5. Notice that we have built only $s_{{j+1}}-1$ vectors, since the $s_{{j+1}}$ th connects the last point $z^{(j)}_{s_j}$ to $\mathsf{cp}_{{j+1}}$ and is thus determined.

Figure 12.

Vector-building.

We obtain n points $(z_1,\ldots,z_n)=(\underbrace{\mathsf{cp}_1,z^{(1)}_2,\ldots,z^{(1)}_{s_1}}_{s_1 \text{ points}},\underbrace{\mathsf{cp}_2,z^{(2)}_2,\ldots,z^{(2)}_{s_2}}_{s_2 \text{ points}},\ldots,{} \underbrace{\mathsf{cp}_\kappa,z^{(\kappa)}_2,\ldots,z^{(\kappa)}_{s_\kappa}}_{s_\kappa \text{ points}}).$ In the end, the whole construction includes only maps that are piecewise linear and piecewise differentiable (Definition 1) in the data $\left(\ell[\kappa],u^{(1)}[N_1,c_1],\ldots,u^{(\kappa)}[N_\kappa,c_\kappa]\right)$ , and thus $\chi_{s[\kappa]}$ also has these properties.

Jacobian.  Let us compute the Jacobian determinant of the inverse mapping $(\chi_{s[\kappa]})^{-1}$ . This requires first the Jacobian determinant of the construction of the contact points $\mathsf{cp}[\kappa]$ . To build a contact point, we build the vertices $\mathsf{b}[\kappa]$ : we fix the y-coordinate of $\mathsf{b}_\kappa$ and $\mathsf{b}_1$ as $\ell_1$ . Now, rotate the figure by $\pi/2-\theta_\kappa$ : in this new system of coordinates, the y-coordinate of $\mathsf{b}_1$ and $\mathsf{b}_2$ is $\ell_2$ . This determines the coordinates of $\mathsf{b}_1$ , and from one rotation to the other, those of $\mathsf{b}_j$ for all $j\in\{1,\ldots,\kappa\}$ . The Jacobian determinant of the whole construction of the $\mathsf{b}[\kappa]$ is the determinant of a product of rotation matrices, and is thus 1.

Then, as said before, the contact point $\mathsf{cp}_j$ is built as a translation of $u_{s_j}^{(j)}$ from $\mathsf{b}_{{j-1}}$ on the jth side of $\mathsf{ECP}$ . This operation has Jacobian determinant 1 as well.

For $j\in\{1,\ldots,\kappa\}$ , the building of $z_k^{(j)}$ , $k\in\{2,\ldots,s_j\}$ , is a translation from $z^{(j)}_{k-1}$ with the product of the matrix $A_j(\theta_\kappa)^{-1}$ with the vector $v^{(j)}_{k-1}$ , for all $j\in\{1,\ldots,\kappa\}$ . So we have

(3.4) \begin{align}\nonumber \mathrm{Jac}\left((\chi_{s[\kappa]})^{-1}\right)&=\left|\prod_{j=1}^{\kappa}\det\left(A_j(\theta_\kappa)^{-1}\right)^{s_j-1}\right|\\[3pt] &=\sin(\theta_\kappa)^{n-\kappa}. \end{align}

3.2. Working at fixed $\ell[\kappa]$

Above, we performed a first ‘conditioning’ based on the size vector $s[\kappa]$ of the vectors forming the boundary of any z[n]-gon. From this point, the map $\chi_{s[\kappa]}$ encodes z[n] in two parts: the ‘coordinates’ $\ell[\kappa]$ of the $\mathsf{ECP}({z}[n])$ (in the sense that their data is equivalent) and the side-partitions $\left(u^{(1)}[N_1,c_1],\ldots,u^{(\kappa)}[N_\kappa,c_\kappa]\right)$ . We may now perform a second conditioning on the coordinates $\ell[\kappa]$ , by introducing the set

(3.5) \begin{align} {\mathcal{S}}^{(n)}(\ell[\kappa],s[\kappa])=\bigg\{\left(w^{(1)},\ldots,w^{(\kappa)}\right) \text{ such that }\left(\ell[\kappa],w^{(1)},\ldots,w^{(\kappa)}\right)\in{\mathcal{S}}^{(n)}(s[\kappa])\bigg\}.\end{align}

This conditioning actually reveals the mass of z[n]-gons contained in an $\mathsf{ECP}$ of coordinates $\ell[\kappa]$ with a repartition $s[\kappa]$ . Indeed, we have the following lemma.

Lemma 5. For all $\ell[\kappa]\in\mathcal{L}_\kappa$ , $s[\kappa]\in\mathbb{N}_\kappa(n)$ ,

(3.6) \begin{align} \mathsf{Leb}_{2n}({\mathcal{C}_\kappa}(s[\kappa]))&=\sin(\theta_\kappa)^{n-\kappa}\int_{\mathbb{R}^\kappa}\mathbf{1}_{\left\{{\ell[\kappa]\in\mathcal{L}_\kappa}\right\}}\mathsf{Leb}_{2n-\kappa}\left({\mathcal{S}}^{(n)}(\ell[\kappa],s[\kappa])\right)\text{d} \ell[\kappa]. \end{align}

Proof. We have

(3.7) \begin{multline} \mathsf{Leb}_{2n}(\mathcal{S}^{(n)}(s[\kappa]))=\\[3pt] \int_{\mathbb{R}^\kappa}\mathbf{1}_{\left\{{\ell[\kappa]\in\mathcal{L}_\kappa}\right\}}\underbrace{\int_{\mathbb{R}^{2n-\kappa}} \mathbf{1}_{\left\{{\left(u^{(1)}[N_1,c_1],\ldots,u^{(\kappa)}[N_\kappa,c_\kappa]\right)\in{\mathcal{S}}^{(n)}(\ell[\kappa],s[\kappa])}\right\}} \text{d} u[2n-\kappa]}_{\mathsf{Leb}_{2n-\kappa}\left({\mathcal{S}}^{(n)}(\ell[\kappa],s[\kappa])\right)}\text{d} \ell[\kappa], \end{multline}

where we let $\text{d} u[2n-\kappa]=\prod_{j=1}^\kappa \text{d} u^{(j)}[N_j,c_j]$ to lighten the notation. Hence, (3.3) allows us to conclude.

This lemma encodes an n-tuple z[n] in convex position in terms of a new geometric description embodied in the coordinates $\left(\ell[\kappa],u^{(1)}[N_1,c_1],\ldots,u^{(\kappa)}[N_\kappa,c_\kappa]\right)$ . This change of variables comes at the price of the Jacobian computed in Theorem 5. The next step, as suggested by Lemma 5, is to compute, for fixed $(\ell[\kappa],s[\kappa])$ , the Lebesgue measure of the set ${\mathcal{S}}^{(n)}(\ell[\kappa],s[\kappa]).$

The Lebesgue measure of ${\mathcal{S}}^{(n)}(\ell[\kappa],s[\kappa])$ . Pick $\ell[\kappa]\in\mathcal{L}_\kappa$ and $\left(u^{(1)},\ldots,u^{(\kappa)}\right)\in{\mathcal{S}}^{(n)}(\ell[\kappa],s[\kappa])$ . This tuple of side-partitions $\left(u^{(1)},\ldots,u^{(\kappa)}\right)$ can be seen as an element of the set $\prod_{j=1}^\kappa P[c_j,N_j]$ . Indeed, a side-partition $u^{(j)}\,:\!=\,u^{(j)}[N_j,c_j]$ marks $N_j$ points on the segment $[0,c_j]$ . Nonetheless, just as we did after (2.5) and (2.6), we may instead consider the tuples of distances between points, and reorder each $u^{(j)}$ into increasing increments so as to form $(\Delta\widetilde{u}^{(1)}[N_1+1],\ldots,\Delta\widetilde{u}^{(\kappa)}[N_\kappa+1])$ , which is thus an element of $\prod_{j=1}^\kappa I[c_j,N_j+1]$ . Considering the elements of $\prod_{j=1}^\kappa I[c_j,N_j+1]$ rather than those of $\prod_{j=1}^\kappa P[c_j,N_j]$ prevents us from forming the same convex chain twice. Next we define

\begin{align*} \begin{array}{r@{\quad}c@{\quad}c@{\quad}l} \mathsf{Order}_{\ell[\kappa],s[\kappa]}:&{\mathcal{S}}^{(n)}(\ell[\kappa],s[\kappa])&\longrightarrow&\prod_{j=1}^\kappa I[c_j,N_j+1]\\[3pt] &\left(u^{(1)},\ldots,u^{(\kappa)}\right)&\longmapsto&(\Delta\widetilde{u}^{(1)}[N_1+1],\ldots,\Delta\widetilde{u}^{(\kappa)}[N_\kappa+1])\end{array},\end{align*}

a piecewise linear mapping. Given $(\Delta\widetilde{u}^{(1)}[N_1+1],\ldots,\Delta\widetilde{u}^{(\kappa)}[N_\kappa+1])\in\prod_{j=1}^\kappa I[c_j,N_j+1]$ , how many distinct n-tuples $\left(u^{(1)},\ldots,u^{(\kappa)}\right)\in{\mathcal{S}}^{(n)}(\ell[\kappa],s[\kappa])$ can we build out of this object? We answer this question in the following lemma.

Lemma 6. Let $s[\kappa]\in\mathbb{N}_\kappa(n)$ , $\ell[\kappa]\in\mathcal{L}_\kappa$ , with the corresponding $c[\kappa]$ . Consider a tuple $(\Delta\widetilde{u}^{(1)}[N_1+1],\ldots,\Delta\widetilde{u}^{(\kappa)}[N_\kappa+1])\in\prod_{j=1}^\kappa I[c_j,N_j+1]$ . Then

(3.8) \begin{align} \#\mathsf{Order}_{\ell[\kappa],s[\kappa]}^{-1}\left(\Delta\widetilde{u}^{(1)}[N_1+1],\ldots,\Delta\widetilde{u}^{(\kappa)}[N_\kappa+1]\right)=\prod_{j=1}^{\kappa} \left(\begin{smallmatrix}{s_j+s_{{j+1}}} \\[3pt] {s_j}\end{smallmatrix}\right)s_j!. \end{align}

Figure 13.

In the first drawing, given a partition in $I[c_j,s_{{j-1}}+s_j]$ and a partition in $I[c_{{j+1}},s_j+s_{{j+1}}]$ , we randomly pair $s_j$ pieces of $c_j$ with $s_j$ pieces of $c_{{j+1}}$ to form the vectors in the jth corner. Note that an affine transformation is hiding in the construction of these vectors. In the second drawing, vectors have been reordered by increasing slope. The points $\mathsf{cp}_j,\mathsf{cp}_{{j+1}}$ naturally appear as the edges of the convex chain formed by those vectors. In these particular drawings, we took $s_j=3$ , $s_{{j-1}}=2$ , $s_{{j+1}}=2$ .

This allows us to compute the Lebesgue measure of ${\mathcal{S}}^{(n)}(\ell[\kappa],s[\kappa])$ . Indeed, the map $\mathsf{Order}_{\ell[\kappa],s[\kappa]}$ carries the Lebesgue measure of ${\mathcal{S}}^{(n)}(\ell[\kappa],s[\kappa])$ onto that of $\prod_{j=1}^\kappa I[c_j,N_j+1]$ .

Corollary 1. For $s[\kappa]\in\mathbb{N}_\kappa(n)$ , and $\ell[\kappa]\in\mathcal{L}_\kappa$ fixed, we have

(3.9) \begin{align} \mathsf{Leb}_{2n-\kappa}\left({\mathcal{S}}^{(n)}(\ell[\kappa],s[\kappa])\right)=\prod_{j=1}^{\kappa}\frac{c_j^{s_j+s_{{j+1}}-1}}{s_j!(s_j+s_{{j+1}}-1)!}. \end{align}

Proof. Indeed, by the previous lemmas, we obtain

(3.10) \begin{align} \mathsf{Leb}_{2n-\kappa}\left({\mathcal{S}}^{(n)}(\ell[\kappa],s[\kappa])\right)&=\prod_{j=1}^{\kappa}\left(\begin{smallmatrix}{s_j+s_{{j+1}}} \\[3pt] {s_j}\end{smallmatrix}\right)s_j! \cdot\mathsf{Leb}_{N_j}\left(I[c_j,N_j+1]\right), \end{align}

and we conclude by (2.7).

3.3. The joint distribution of the pair $(\boldsymbol{\ell}^{(n)}[\kappa],\mathbf{s}^{(n)}[\kappa])$

Theorem 5 concretizes our understanding of this new equivalent geometric description of the set ${\mathcal{C}_\kappa}(s[\kappa])$ in terms of the $\mathsf{ECP}$ . Let $\mathbf{z}[n]$ have distribution ${\mathbb{D}}^{(n)}_{\kappa}$ , and set $\boldsymbol{\ell}^{(n)}[\kappa]=\ell[\kappa](\mathbf{z}[n])$ , $\mathbf{s}^{(n)}[\kappa]=s[\kappa](\mathbf{z}[n])$ . By computing the Lebesgue measure of the set ${\mathcal{S}}^{(n)}(\ell[\kappa],s[\kappa])$ , we managed to understand the weight of all z[n]-gons contained in any $(\ell[\kappa],s[\kappa])$ -fibration, which is the key to the computation of the joint distribution of the pair $(\boldsymbol{\ell}^{(n)}[\kappa],\mathbf{s}^{(n)}[\kappa])$ .

Theorem 6. Let $\mathbf{z}[n]$ have distribution ${\mathbb{D}}^{(n)}_{\kappa}$ , and consider the random variables $\boldsymbol{\ell}^{(n)}[\kappa]=\ell[\kappa](\mathbf{z}[n])$ , $\mathbf{s}^{(n)}[\kappa]=s[\kappa](\mathbf{z}[n])$ . Then for a given $s[\kappa]\in\mathbb{N}^\kappa$ , the pair $(\boldsymbol{\ell}^{(n)}[\kappa],\mathbf{s}^{(n)}[\kappa])$ has the joint distribution

(3.11) \begin{align} \mathbb{P}\left(\boldsymbol{\ell}^{(n)}[\kappa]\in\text{d} \ell[\kappa],\mathbf{s}^{(n)}[\kappa]=s[\kappa]\right)=f^{(n)}_{\kappa}\left(\ell[\kappa],s[\kappa]\right)\text{d} \ell[\kappa], \end{align}

where

(3.12) \begin{align} f^{(n)}_{\kappa}\left(\ell[\kappa],s[\kappa]\right)=\frac{n!\sin(\theta_\kappa)^{n-\kappa}}{\widetilde{\mathbb{P}}_\kappa(n)} \mathbf{1}_{\left\{{s[\kappa]\in\mathbb{N}_\kappa(n)}\right\}} \mathbf{1}_{\left\{{\ell[\kappa]\in\mathcal{L}_\kappa}\right\}}\prod_{j=1}^{\kappa}\frac{c_j^{s_{{j-1}}+s_j-1}}{s_j!(s_{{j-1}}+s_j-1)!}. \end{align}

Proof. Write

(3.13) \begin{align} \text{d} {\mathbb{D}}^{(n)}_{\kappa}({z}[n])=\frac{n!}{\widetilde{\mathbb{P}}_\kappa(n)}\mathbf{1}_{\left\{{z[n]\in\mathcal{C}_\kappa(n)}\right\}}\mathbf{1}_{\left\{{s[\kappa](z[n])\in\mathbb{N}_\kappa(n)}\right\}}\text{d} {z}[n] \\[-32pt] \nonumber \end{align}

(3.14) \begin{align} =\frac{n!}{\widetilde{\mathbb{P}}_\kappa(n)}\sum_{s[\kappa]\in\mathbb{N}_\kappa(n)}\mathbf{1}_{\left\{{{z}[n]\in{\mathcal{C}_\kappa}(s[\kappa])}\right\}}\text{d} {z}[n]. \end{align}

For any continuous bounded test function $\eta\,:\,\mathbb{R}^{\kappa}\times\mathbb{N}^{\kappa}\to\mathbb{R}$ , we have

\begin{align}\nonumber \mathbb{E}\bigg[\eta\left(\boldsymbol{\ell}^{(n)}[\kappa],\mathbf{s}^{(n)}[\kappa]\right)\bigg] =\int_{(\mathbb{R}^2)^n}\eta(\left(\ell[\kappa]({z}[n]),s[\kappa]({z}[n])\right)\text{d} {\mathbb{D}}^{(n)}_{\kappa}({z}[n]), \end{align}

which, after the change of variables $\chi_{s[\kappa]}({z}[n])=\left(\ell[\kappa],u^{(1)}[N_1,c_1],\ldots,u^{(\kappa)}[N_\kappa,c_\kappa]\right)$ , performed at fixed $(\ell[\kappa],s[\kappa])$ , gives

(3.15) \begin{multline} \mathbb{E}\bigg[\eta\left(\boldsymbol{\ell}^{(n)}[\kappa],\mathbf{s}^{(n)}[\kappa]\right)\bigg]=\frac{n!}{\widetilde{\mathbb{P}}_\kappa(n)}\sum_{s[\kappa]\in\mathbb{N}_\kappa(n)}\mathrm{Jac}\left((\chi_{s[\kappa]})^{-1}\right)\int_{\mathbb{R}^{\kappa}}\eta(\ell[\kappa],s[\kappa])\\[3pt] \times\left[\int_{\mathbb{R}^{2n-\kappa}}\mathbf{1}_{\left\{{\left(\ell[\kappa],u^{(1)}[N_1,c_1],\ldots,u^{(\kappa)}[N_\kappa,c_\kappa]\right)\in{\mathcal{S}}^{(n)}(s[\kappa])}\right\}}\text{d} u[2n-\kappa]\right]\text{d} \ell[\kappa]. \end{multline}

Now, $\mathrm{Jac}\left((\chi_{s[\kappa]})^{-1}\right)=\sin(\theta_\kappa)^{n-\kappa}$ for all $s[\kappa]\in\mathbb{N}_\kappa(n)$ by (3.4), and the last bracket in (3.15) is nothing but the Lebesgue measure of ${\mathcal{S}}^{(n)}(\ell[\kappa],s[\kappa])$ , which we computed in Corollary 1! Hence substituting (3.9) in (3.15) gives Theorem 6.

In the next section, we are going to exploit the asymptotic stochastic behavior of the pair $(\boldsymbol{\ell}^{(n)}[\kappa],\mathbf{s}^{(n)}[\kappa])$ to deduce an equivalent of $\widetilde{\mathbb{P}}_\kappa(n)$ . However, in the particular cases $\kappa\in\{3,4\}$ , we have ${\mathbb{D}}^{(n)}_{\kappa}=\mathbb{Q}^{(n)}_{\kappa}$ , and the set $\mathcal{L}_\kappa$ is easily computable. Hence we can immediately compute the exact value of $\mathbb{P}_\kappa(n)$ from $\mathbb{Q}^{(n)}_{\kappa}$ . In Appendix B, we take a look at these computations to recover Valtr’s famous results for the triangle and the parallelogram.

5.1. Basic brick fluctuations

Notation. We say that a point z is the $\alpha$ -barycenter of (a, b) if $z= \alpha a+(1-\alpha) b$ ; in this case, $\alpha$ is called the barycenter parameter.

Basic building bricks of a z [ n ]-gon. For any $\mathbf{z}[n]$ with distribution ${\mathbb{D}}^{(n)}_{\kappa}$ , we have provided an alternative geometric description of the $\mathbf{z}[n]$ -gon in terms of its $\mathsf{ECP}(\mathbf{z}[n])$ , and more specifically in terms of the following:

• • the boundary distances $\boldsymbol{\ell}^{(n)}[\kappa]$ of this $\mathsf{ECP},$

• • the tuple $\mathbf{s}^{(n)}[\kappa]$ counting the vectors in the corners of the $\mathsf{ECP}$ , and

• • the fragmentation of the sides $\mathbf{c}^{(n)}[\kappa]$ into side-partitions $\mathbf{u}^{(1)}[\mathbf{N}^{(n)}_1],\ldots,\mathbf{u}^{(\kappa)}[\mathbf{N}^{(n)}_\kappa]$ , where $\mathbf{N}^{(n)}_j=\mathbf{s}^{(n)}_{j}+\mathbf{s}^{(n)}_{{j+1}}-1$ for all $j\in\{1,\ldots,\kappa\}.$

Notice that the contact points ${\boldsymbol{\mathsf{cp}}}^{(n)}[\kappa]$ and the vertices ${\boldsymbol{\mathsf{b}}}^{(n)}[\kappa]$ of the $\mathsf{ECP}(\mathbf{z}[n])$ can be recovered using these three data. Indeed, $\boldsymbol{\ell}^{(n)}[\kappa]$ determines the $\mathsf{ECP}$ and thus its vertices ${\boldsymbol{\mathsf{b}}}^{(n)}[\kappa]$ .

In (3.15), we see that, conditional on the jth side length $\mathbf{c}^{(n)}_j=c_j$ and on the size vector $\mathbf{s}^{(n)}[\kappa]$ , the side-partition $\left(u_1^{(j)} \lt \ldots \lt u_{\mathbf{N}_j}^{(j)}\right)$ has the law of a reordered $\mathbf{N}_j$ -tuple of i.i.d. uniform random variables drawn in $[0,c_j].$ The contact point ${\mathsf{cp}}^{(n)}_j$ is placed on the jth side of the $\mathsf{ECP}$ (on the segment $[{\boldsymbol{\mathsf{b}}}^{(n)}_{{j-1}},{\boldsymbol{\mathsf{b}}}^{(n)}_j]$ ), at the coordinate $\mathbf{u}^{(j)}_{\mathbf{s}^{(n)}_{j}}$ for all $j\in\{1,\ldots,\kappa\}$ . This means that conditional on $(\boldsymbol{\ell}^{(n)}[\kappa],\mathbf{s}^{(n)}[\kappa])$ , the tuple ${\mathsf{cp}}^{(n)}[\kappa]$ has independent entries, and ${\mathsf{cp}}^{(n)}_j$ is a ${\boldsymbol\beta}_j^{(n)}$ -barycenter of $\left({\boldsymbol{\mathsf{b}}}^{(n)}_{{j-1}},{\boldsymbol{\mathsf{b}}}^{(n)}_j\right)$ , where ${\boldsymbol\beta}_j^{(n)}$ is $\beta$ -distributed with parameters $\left(\mathbf{s}^{(n)}_{j},\mathbf{s}^{(n)}_{{j+1}}\right)$ .

For $j\in\{1,\ldots,\kappa\}$ , the random variable

(5.1) \begin{align} \boldsymbol{\delta}^{(n)}_j\,:\!=\,\sqrt{\frac{n}{\kappa}}\left({\boldsymbol\beta}_j^{(n)}-1/2\right),\text{ for all }j\in\{1,\ldots,\kappa\},\end{align}

provides the fluctuations of the barycenter parameter and thus encodes the fluctuations of the contact point ${\mathsf{cp}}_{j}^{(n)}$ on the jth side $\mathbf{c}^{(n)}_{j}$ of $\mathsf{ECP}(\mathbf{z}[n]).$

Fluctuations of basic bricks. We have proved that the boundary distances $\boldsymbol{\ell}^{(n)}[\kappa]$ behaved as $\frac{1}{n}$ times an exponential distribution, and we will prove in the sequel that the contact points ${\mathsf{cp}}^{(n)}[\kappa]$ are typically at distance $\frac{1}{\sqrt{n}}$ around their limit (this is visible in (5.1)). So in order to describe the fluctuations of the $\mathbf{z}[n]$ -gon around its limit, we will state a theorem describing the joint distribution of all basic bricks taken together, rather than providing the fluctuations in $\frac{1}{\sqrt{n}}$ of a complicated object for the whole process, which would crush the behavior of some of the bricks. A comprehensive picture of the fluctuations would rely on the concatenation of all the corners’ fluctuations, adjusted to take into account the contact points; we believe that presenting such a picture would not bring any new insight, so we leave it to the interested reader as an exercise.

The ◿-convex chains. Recall Lemma 1 and its notation. Consider for all $j\in\{1,\ldots,\kappa\}$ the convex chain $\mathsf{CC}^{(n)}_j=\mathsf{CC}_j^{(n)}(\mathbf{z}[n])$ lying in the jth corner of $\mathsf{ECP}(\mathbf{z}[n])$ , defined as an element of $\mathsf{Chain}_{\mathbf{s}^{(n)}_{j}}(\mathsf{corner}_j(\mathbf{z}[n]))$ (this accounts for the decomposition of $\mathbf{z}[n]$ into convex chains). In order to understand the fluctuations of the jth corner, we will use the normalized version of each of these convex chains:

(5.2) \begin{align} \unicode{x25FF}\mathsf{CC}^{(n)}_j=\mathsf{Aff}_j(\mathsf{CC}^{(n)}_j) \text{ where }\mathsf{Aff}_j=\mathsf{Aff}_{\mathsf{corner}_j(\mathbf{z}[n])},\end{align}

where, for a given non-flat triangle ABC, the mapping $\mathsf{Aff}_{ABC}$ is the unique map that sends A, B, C to (0, 0),(1, 0),(1, 1), as introduced in Lemma 2. The law of $\unicode{x25FF}\mathsf{CC}^{(n)}_j$ depends only on $\mathbf{s}^{(n)}_{j}$ , but the affine mapping $\mathsf{Aff}_j$ depends on the coordinates of $\mathsf{ECP}(\mathbf{z}[n])$ . However, determining the fluctuations of $\mathsf{CC}^{(n)}_j$ amounts to looking at those of $\unicode{x25FF}\mathsf{CC}^{(n)}_j$ .

Recall that a generic ◿-normalized convex chain $\unicode{x25FF}\mathsf{CC}_m$ of size m is a random variable whose law is that of a convex chain taken uniformly in $ \mathsf{Chain}_{m}$ (◿). Therefore, a consequence of Lemma 1 and Lemma 2 is the following lemma.

Thanks to this lemma, it should be clear that we can work on each convex chain separately when $\mathbf{s}^{(n)}[\kappa]$ is fixed. We introduce some processes in order to describe these fluctuations.

5.2. A parametrization of the normalized convex chains

Let $m\geq0$ , and let $\unicode{x25FF}\mathsf{CC}_m=((0,0),\mathbf{z}_1,\ldots,\mathbf{z}_{m-1},(1,1))$ be a generic ◿-normalized convex chain of size m. Rather than considering the tuple of points $((0,0),\mathbf{z}_1,\ldots,\mathbf{z}_{m-1},(1,1))$ , we consider the m vectors composing the convex chain. Recall that these vectors are obtained by forming the tuples $\mathbf{u}[m],\mathbf{v}[m]$ of increments of two elements taken uniformly (and independently) in the simplex $P[1,m-1]$ . Then the vectors $(\mathbf{u}_{i},\mathbf{v}_{i})$ , $i\in\{1,\ldots,m\}$ , are reordered by increasing slope to form this chain.

In order to use the toolbox of stochastic processes, it is convenient for us to see $\unicode{x25FF}\mathsf{CC}_m$ as a linear process. For this we will need a suitable parametrization (several choices are possible). For technical reasons, we choose the local slope of $\unicode{x25FF}\mathsf{CC}_m$ as the time parameter, and apply the $\arctan$ function in order to remain in a compact set.

Consider the point $\left(x_{m}(u),y_{m}(u)\right)$ for $u\in[0, 1]$ , corresponding to the contributions of the previous vectors whose slope are smaller than $\tan\left(\frac{\pi}{2}u\right)$ , and the process $\mathsf{C}_m$ defined as $\mathsf{C}_m(u)\,:\!=\,\left(x_{m}(u),y_{m}(u)\right)$ for $u\in[0, 1]$ . Hence, we have

(5.3) \begin{align}x_m(u)=\sum_{i=1}^m \mathbf{u}_i\mathbf{1}_{\left\{{\frac{\mathbf{v}_i}{\mathbf{u}_i}\leq \tan\left(\frac{\pi}{2}u\right)}\right\}}\quad\text{ and }\quad y_m(u)=\sum_{i=1}^m \mathbf{v}_i\mathbf{1}_{\left\{{\frac{\mathbf{v}_i}{\mathbf{u}_i}\leq \tan\left(\frac{\pi}{2}u\right)}\right\}},\end{align}

where we set $\tan\left(\frac{\pi}{2}\cdot1\right)=+\infty$ so that $x_m(1)=y_m(1)=1.$ Note that the tuple $\unicode{x25FF}\mathsf{CC}_m$ (seen as a set) coincides with the (finite) set $\{\mathsf{C}_m(u),u\in[0, 1]\}$ .

Define further the curve $\mathsf{C}_\infty(u)\,:\!=\,\left(x(u),y(u)\right)$ for all $u\in[0, 1]$ , where

(5.4) \begin{align}x(u)\,:\!=\,1-\frac{1}{\left(1+\tan\left(\frac{\pi}{2}u\right)\right)^2}\quad\text{ and }\quad y(u)\,:\!=\,\frac{\tan\left(\frac{\pi}{2}u\right)^2}{\left(1+\tan\left(\frac{\pi}{2}u\right)\right)^2}.\end{align}

We have once more $x(1)=y(1)=1$ . This curve is actually the parametrization of the parabolic arc lying in ◿, tangent at (0, 0) and (1, 1). It is not surprising to encounter $\mathsf{C}_\infty$ here, since the convergence in distribution of $\mathsf{C}_m$ to $\mathsf{C}_{\infty}$ that will be stated in Theorem 9 can be seen as a consequence of Bárány’s work on affine perimeters (not a direct consequence, however).

For all $j\in\{1,\ldots,\kappa\}$ , we let $\mathsf{C}^{(n)}_{j}$ be the parametrization in terms of the slope of the jth ◿-normalized convex chain $\unicode{x25FF}\mathsf{CC}_j^{(n)}$ . So now by Lemma 9, we have

\begin{align*}\mathcal{L}\left(\mathsf{C}_j^{(n)}, 1\leq j \leq \kappa \,| \,\mathbf{s}^{(n)}[\kappa]\right)=\mathcal{L}\left(\mathsf{C}_{\mathbf{s}^{(n)}_{j}}, 1\leq j \leq \kappa\right), \end{align*}

and conditional on $\mathbf{s}^{(n)}[\kappa]$ , the $\mathsf{C}^{(n)}_{j}$ , $j\in\{1,\ldots,\kappa\}$ , are independent.

Theorem 9 will also include the convergence in distribution of the fluctuation process $\mathsf{Y}_m\,:\!=\,\sqrt{m}(\mathsf{C}_m-\mathsf{C}_{\infty})$ to a Gaussian process. This convergence is actually the key that will help us understand the fluctuations of the convex chain $\unicode{x25FF}\mathsf{CC}_j^{(n)}$ around its limit $\unicode{x25FF}\mathsf{CC}_\infty$ . Indeed, conditional on $(\boldsymbol{\ell}^{(n)}[\kappa],\mathbf{s}^{(n)}[\kappa])$ , the processes

(5.5) \begin{align}\mathsf{Y}_j^{(n)}\,:\!=\,\sqrt{\frac{n}{\kappa}}\left(\mathsf{C}^{(n)}_{j}-\mathsf{C}_\infty\right),\,\, 1\leq j \leq \kappa,\end{align}

describe the successive fluctuations of the ◿-convex chain in their corners. We are now able to state the most important result of this section (in which we borrow the notation of Theorem 7).

Proof. We start with the proof of Part 2. Recall (5.1). Conditional on $(\boldsymbol{\ell}^{(n)}[\kappa],\mathbf{s}^{(n)}[\kappa])$ , the ${\mathsf{cp}}^{(n)}[\kappa]$ are independent, which provides the independence of the $\boldsymbol{\delta}^{(n)}[\kappa]$ (given $(\boldsymbol{\ell}^{(n)}[\kappa],\mathbf{s}^{(n)}[\kappa])$ ). It suffices then, to prove the convergence of the marginals; by symmetry, we will prove only the convergence of $\boldsymbol{\delta}_j^{(n)}= \sqrt{\frac{n}{\kappa}}\left({\boldsymbol\beta}_j(n)-\frac{1}{2}\right)$ .

According to the Skorokhod representation theorem, up to a change of probability space, we may assume that

$$\frac{\mathbf{s}^{(n)}_{j}-\frac{n}{\kappa}}{\sqrt{n/\kappa}}\xrightarrow[n]{(a.s.)} \mathbf{x}_j$$

for all $j\in\{1,\ldots,\kappa\}$ . We may then assume that $\mathbf{s}^{(n)}_{j}=\frac{n}{\kappa}+\mathbf{x}_j\sqrt{\frac{n}{\kappa}}+o(\sqrt{n})$ . In the sequel we drop the $o(\sqrt{n})$ since it provides only negligible contributions.

Since $\beta_j^{(n)}$ is $\beta$ -distributed with parameters $(\mathbf{s}^{(n)}_{j},\mathbf{s}^{(n)}_{{j+1}})$ , we have $\beta_j^{(n)}\mathrel{\mathop{\kern 0pt=}\limits^{(d)}}\frac{\mathbf{T}_j}{\mathbf{T}_j+\mathbf{T}_{{j+1}}}$ where $\mathbf{T}_j$ (resp. $\mathbf{T}_{{j+1}}$ ) is a random variable that is gamma-distributed with parameter $\mathbf{s}^{(n)}_{j}$ (resp. $\mathbf{s}^{(n)}_{{j+1}}$ ), these variables being independent.

By the central limit theorem, we have

\begin{align*} \left(\frac{\mathbf{T}_j-\frac{n}{\kappa}}{\sqrt{n/\kappa}},\frac{\mathbf{T}_{{j+1}}-\frac{n}{\kappa}}{\sqrt{n/\kappa}}\right)\xrightarrow[n]{(d)}\left(\mathbf{x}_j+\mathbf{q}_1,\mathbf{x}_{{j+1}}+\mathbf{q}_2\right),\end{align*}

for $\mathbf{q}_1,\mathbf{q}_2$ two i.i.d. standard Gaussian random variables $\mathcal{N}(0,1)$ . Hence

(5.6) \begin{align} \sqrt{\frac{n}{\kappa}}\left({\boldsymbol\beta}_j^{(n)}-\frac{1}{2}\right)&\mathrel{\mathop{\kern 0pt=}\limits^{(d)}}\sqrt{\frac{n}{\kappa}}\left(\frac{\mathbf{T}_j}{\mathbf{T}_j+\mathbf{T}_{{j+1}}}-\frac{1}{2}\right) \\[-32pt] \nonumber \end{align}

(5.7) \begin{align} \mathrel{\mathop{\kern 0pt=}\limits^{(d)}}\frac{n/\kappa}{2(\mathbf{T}_j+\mathbf{T}_{{j+1}})}\left(\frac{\mathbf{T}_j-\frac{n}{\kappa}}{\sqrt{n/\kappa}}-\frac{\mathbf{T}_{{j+1}}-\frac{n}{\kappa}}{\sqrt{n/\kappa}}\right) \\[-32pt] \nonumber \end{align}

(5.8) \begin{align} \xrightarrow[n]{(d)} \frac{1}{4}\left(\mathbf{x}_j-\mathbf{x}_{{j+1}}+\mathbf{q}_1-\mathbf{q}_2\right).\end{align}

This gives the expected result.

Notice now that Part 3 implies Part 0. Indeed, the weak convergence of the processes $\mathsf{C}^{(n)}_1,\ldots,\mathsf{C}^{(n)}_\kappa$ to $\mathsf{C}_\infty$ is a consequence of that of $\mathsf{Y}^{(n)}_1,\ldots,\mathsf{Y}^{(n)}_\kappa$ to $\mathsf{Y}_\infty$ . This latter convergence will be the main object of the rest of this section. This is by far the most complicated proof, and we will need to introduce quite a lot of tools to achieve it. We will come back to this proof later.

5.3. Convergence of the normalized convex chain

In order to prove the convergence of $\mathsf{Y}^{(n)}[\kappa]$ , a new parametrization is required.

Notation. Define the maps $g\,:\,t\in[0, 1]\mapsto\tan\left(\frac{\pi}{2}t\right)$ and $\displaystyle h\,:\,t\in[0, 1]\mapsto\frac1{1+g(t)}$ . Let us introduce the following mappings for all $(s,t)\in[0, 1]^2$ :

(5.9) \begin{align}f(s,t)&=h(s)-h(t), \\[-32pt] \nonumber \end{align}

(5.10) \begin{align}e_1(s,t)&=h(s)^2-h(t)^2, \\[-32pt] \nonumber \end{align}

(5.11) \begin{align} e_2(s,t)&=\left(g(t)h(t)\right)^2-\left(g(s)h(s)\right)^2, \\[-32pt] \nonumber \end{align}

(5.12) \begin{align} v_1(s,t)&=2\left(h(s)^3-h(t)^3\right)-e_1(s,t)^2, \\[-32pt] \nonumber \end{align}

(5.13) \begin{align} v_2(s,t)&=2\left(\left(g(t)h(t)\right)^3-\left(g(s)h(s)\right)^3\right)-e_2(s,t)^2.\end{align}

In the following, we will denote by $\left(\mathsf{Z}^{(1)},\mathsf{Z}^{(2)}\right)$ the coordinates of any process $\mathsf{Z}$ taking values in $\mathbb{R}^2$ .

Recall the definition of the process $\mathsf{Y}_m$ given in (5.5).

Theorem 9. The convergence in distribution

(5.14) \begin{align} \mathsf{C}_m\xrightarrow[n]{(d)} \mathsf{C}_\infty\end{align}

holds in $D([0, 1])^2$ .

The sequence of processes $(\mathsf{Y}_m)$ converges in distribution to a centered Gaussian process $\mathsf{Y}_\infty$ in $D([0, 1])^2$ , whose coordinates can be represented as

(5.15) \begin{align} \mathsf{Y}_\infty^{(1)}(t)&=\overline{\mathsf{Y}}^{(1)}(t)+\mathbf{g}_1\cdot x(t)+(\mathbf{g}_1-\mathbf{g}_2)\cdot \frac2{\pi}\frac{\tan\left(\frac{\pi}{2}t\right)}{1+\tan\left(\frac{\pi}{2}t\right)^2}\cdot x^{\prime}(t), \\[-32pt] \nonumber \end{align}

(5.16) \begin{align} \mathsf{Y}_\infty^{(2)}(t)&=\overline{\mathsf{Y}}^{(2)}(t)+\mathbf{g}_1\cdot y(t)+(\mathbf{g}_1-\mathbf{g}_2)\cdot \frac2{\pi}\frac{\tan\left(\frac{\pi}{2}t\right)}{1+\tan\left(\frac{\pi}{2}t\right)^2}\cdot y^{\prime}(t),\end{align}

and where the following hold:

1. (i) The mappings x’,y’ are the derivatives of $t\mapsto x(t)=\mathsf{C}_\infty^{(1)}(t),t\mapsto y(t)=\mathsf{C}_\infty^{(2)}(t)$ (they are deterministic processes).

2. (ii) The random variables $\mathbf{g}_1,\mathbf{g}_2$ are independent standard Gaussian variables $\mathcal{N}(0,1)$ .

3. (iii) The process $\overline{\mathsf{Y}}$ is a centered Gaussian process with variance function

   (5.17) \begin{align} \mathbb{V}\left[\overline{\mathsf{Y}}^{(p)}(t)\right]=f(0,t)v_p(0,t)+f(0,t)(1-f(0,t))e_p(0,t)^2\quad \forall p\in\{1,2\}\end{align}
   and with covariance function determined, for $(s \lt t)\in[0, 1]^2$ , by
   (5.18) \begin{align} \mathrm{cov}\left(\overline{\mathsf{Y}}^{(p)}(s),\overline{\mathsf{Y}}^{(q)}(t)-\overline{\mathsf{Y}}^{(q)}(s)\right)=-f(0,s)f(s,t)e_p(0,s)e_q(s,t)\quad \forall (p,q)\in\{1,2\}^2.\end{align}

4. (iv) The processes $\overline{\mathsf{Y}}^{(1)},\overline{\mathsf{Y}}^{(2)}$ are independent from $\mathbf{g}_1$ and $\mathbf{g}_2$ .

The reason why the functions $f,e_p,v_p$ appear will be revealed in Proposition 3. Notice that

$$\lim_{t\to1}\frac2{\pi}\frac{\tan\left(\frac{\pi}{2}t\right)}{1+\tan\left(\frac{\pi}{2}t\right)^2}\cdot x^{\prime}(t) \quad \text{and} \quad \lim_{t\to1}\frac2{\pi}\frac{\tan\left(\frac{\pi}{2}t\right)}{1+\tan\left(\frac{\pi}{2}t\right)^2}\cdot y^{\prime}(t)$$

are finite, so the process $\mathsf{Y}_\infty$ is also well-defined at $t=1$ .

Once more, the first assertion (5.14) is a consequence of the second; we will prove only the second one.

Remark 4. (Back to Theorem 8.) The proof of Part 4 of Theorem 8 is an immediate consequence of Theorem 9! Indeed, for all $j\in\{1,\ldots,\kappa\}$ we have

(5.19) \begin{align} \mathsf{Y}_j^{(n)}=\sqrt{\frac{n/\kappa}{\mathbf{s}^{(n)}_{j}}}\sqrt{\mathbf{s}^{(n)}_{j}}\left(\mathsf{C}_{\mathbf{s}^{(n)}_{j}}-\mathsf{C}_\infty\right).\end{align}

By Theorem 9, the term $\sqrt{\mathbf{s}^{(n)}_{j}}\left(\mathsf{C}_{\mathbf{s}^{(n)}_{j}}-\mathsf{C}_\infty\right)$ converges in distribution to $\mathsf{Y}_\infty$ , and we know furthermore that $\frac{\mathbf{s}^{(n)}_{j}}{n/\kappa}\xrightarrow[n]{(d)}1$ . Slutsky’s lemma allows us to conclude.