3.1. Riemannian manifold

Let us recall some facts from differential geometry that will be useful. We refer the reader to [Reference Chavel8, Reference Lee26] for a more rigorous introduction to Riemannian geometry. Let $\mathcal{M}$ be a smooth d-dimensional submanifold of ${\mathbb R}^m$ .

At each point x of $\mathcal{M}$ , there is a tangent vector space $T_x\mathcal{M}$ that contains all the tangent vectors of $\mathcal{M}$ at x. The tangent bundle of $\mathcal{M}$ is denoted by $T\mathcal{M}=\sqcup_{x\in \mathcal{M}}T_x\mathcal{M}$ . For each $x \in \mathcal{M}$ , the canonical scalar product $\langle \cdot , \cdot \rangle_{\mathbb{R}^m}$ of $\mathbb{R}^m$ induces a natural scalar product on $T_x\mathcal{M}$ , denoted by $\mathbf{g}(x)$ . The application $\mathbf{g}$ , which associates each point x with a scalar product on $T_x\mathcal{M}$ , is then called the Riemannian metric on $\mathcal{M}$ induced by the ambient space $\mathbb{R}^m$ . For $\xi,\eta\in T_x\mathcal{M}$ , we use the classical notation $\langle \xi,\eta\rangle_{\mathbf{g}}$ to denote the scalar product of $\xi$ and $\eta$ with respect to the scalar product $\mathbf{g}(x)$ .

Consider a coordinate chart $\Phi=(x^1,\ldots, x^d)\,:\, U \rightarrow \mathbb{R}^d$ on a neighborhood U of x. Denote by $\left\{ \left.\frac{\partial}{\partial x^1} \right|_x, \left.\frac{\partial}{\partial x^2} \right|_x,\ldots, \left.\frac{\partial}{\partial x^d} \right|_x\right\}$ the natural basis of $T_x\mathcal{M}$ associated with the coordinates $(x^1,\ldots, x^d)$ . Then the scalar product $\mathbf{g}(x)$ is associated to a matrix $(g_{ij})_{i,j\in [ \! [ 1, d] \! ]}$ in the sense that in this coordinate chart, for $\xi$ and $\eta\in T_x\mathcal{M}$ ,

(18) \begin{equation}\langle \xi,\eta\rangle_{\mathbf{g}} = \sum_{i,j=1}^d g_{ij}(x) \xi^i\eta^j,\end{equation}

where $(\xi^i),(\eta^j) $ are the coordinates of $\xi$ and $\eta$ in the above basis of $T_x\mathcal{M}$ . Note that, for each $i,j \in [\![1,d]\!]$ ,

(19) \begin{equation} g_{ij}(x)\,:\!=\,\left\langle \left.\frac{\partial}{\partial x^i} \right|_x, \left.\frac{\partial}{\partial x^j} \right|_x \right\rangle_{\mathbf{g}},\end{equation}

and $g_{ij}\,:\,U \subset M\rightarrow \mathbb{R}$ is smooth. For a real function f on $\mathcal{M}$ , we will denote by $\widehat{f}$ its expression in the local chart: $\widehat{f}=f\circ \Phi^{-1}$ . Recall that the derivative $\frac{\partial f}{\partial x^j}$ is defined as:

\begin{equation*}\frac{\partial f}{\partial x^j}\,:\!=\,\frac{\partial \widehat{f}}{\partial x^j}\circ \Phi.\end{equation*}

Also we denote

(20) \begin{equation} \widehat{g}_{ij}\,:\!=\, g_{ij}\circ \Phi^{-1},\end{equation}

which will be called the coordinate representation of the Riemannian metric in the local chart $\Phi$ .

Charts (from $U\subset \mathcal{M}\to {\mathbb R}^d$ ) induce local parameterizations of the manifold (from ${\mathbb R}^d \to U\subset \mathcal{M}$ ). Among all possible local coordinate systems of a neighborhood of x in $\mathcal{M}$ , there are normal coordinate charts (see [Reference Lee26, pp. 131--132] or the remark below for a definition). We denote by $\mathcal{E}_x$ the Riemannian normal parameterization at x, that is, $\mathcal{E}^{-1}_x$ is the corresponding normal coordinate chart.

Remark 1 (Construction of $\mathcal{E}_x^{-1}$ ). For the sake of completeness, we briefly recall the construction of [Reference Lee26]. Let U be an open subset of $\mathcal{M}$ . There exists a local orthonormal frame $(E_i)_{i\in [ \! [ 1, d] \! ]}$ over U (see [Reference Lee26, Proposition 2.8, p. 14]). The tangent bundle TU can be identified with $U\times {\mathbb R}^d$ thanks to the smooth map

(21) \begin{equation}F \ : \ \begin{array}{c@{\quad}c@{\quad}c}U\times {\mathbb R}^d & \rightarrow & TU \\(x,(v_1,\ldots, v_d)) & \mapsto & v=\sum_{i=1}^d v_i E_i|_x.\end{array}\end{equation}

Thus, for each $x\in U$ , $F(x,\cdot)$ is an isometry between ${\mathbb R}^d$ and $T_x\mathcal{M}$ .

Recall that by [Reference Lee26, Proposition 5.19, p. 128], the exponential map $\exp(\cdot)$ of $\mathcal{M}$ can be defined on a non-empty open subset W of $T\mathcal{M}$ such that for all $x \in \mathcal{M}$ , $\overrightarrow{0}_x \in W $ , where $\overrightarrow{0}_x$ is the zero element of $T_x\mathcal{M}$ . Then the map $\exp\circ F \,:\, (x,v) \mapsto \mathcal{E}_x(v)\,:\!=\,\exp\circ\, F (x,(v_1,\ldots, v_d))$ is well defined on $F^{-1}(W \cap TU )$ and $\mathcal{E}_x^{-1}$ is a Riemannian normal coordinate chart at $x\in U$ , smooth with respect to x.

Let us state some properties of the normal coordinate charts.

Theorem 6 (Derivatives of Riemannian metrics in normal coordinate charts) [Reference Lee26, Proposition 5.24]. For $x\in \mathcal{M}$ , let $\mathcal{E}_x^{-1} \,:\, U\subset \mathcal{M} \rightarrow \mathbb{R}^d $ be a normal coordinate chart at a point x such that $\mathcal{E}_x^{-1}(x)=0$ and let $( \widehat{g}_{ij}; 1 \le i, j \le d)$ be the coordinate representation of the Riemannian metric of $\mathcal{M}$ in the local chart $\mathcal{E}_x^{-1}$ . Then, for all i,j,

(22) \begin{equation} \widehat{g}_{ij}(0)=\delta_{ij}, \quad \widehat{g}'_{ij}(0)=0, \end{equation}

where $\delta_{ij}$ is the Kronecker delta. Additionally, for all $y \in U$ ,

(23) \begin{equation}\rho(x,y)= \| \mathcal{E}_x^{-1}(y)\|_2.\end{equation}

Notation 2. For any function $f\,:\, \mathbb{R}^{d} \rightarrow \mathbb{R}^{k}$ , we denote by $f' \,:\, \mathbb{R}^{d} \rightarrow \mathbb{R}^{k}$ the linear map that represents the first-order derivative of f. Similarly, we denote respectively by $f'' \,:\, \mathbb{R}^{d}\times \mathbb{R}^d \rightarrow \mathbb{R}^{k}$ and $f'''\,:\, \mathbb{R}^{d}\times\mathbb{R}^{d}\times \mathbb{R}^{d} \rightarrow \mathbb{R}^{k}$ the bilinear map and trilinear maps that represent the second- and third-order derivatives of f. Thus, the Taylor’s expansion of f up to third order can be written as

\begin{equation*}f(x+v)= f(x) +f'(x)(v)+\frac{1}{2}f''(x)(v,v)+\frac{1}{6}f'''(x+\varepsilon v)(v,v,v),\end{equation*}

for some $\varepsilon \in (0,1)$ .

For the normal parameterizations $\mathcal{E}_x$ , we now state some uniform controls that are key to our computations in the sequel.

3.2. Gradient operator and Laplace–Beltrami operator

Given a Riemannian manifold $(\mathcal{M},\mathbf{g})$ , the gradient operator $\nabla_{\mathcal{M}}$ and the Laplace–Beltrami operator $\Delta_{\mathcal{M}}$ are, as suggested by their names, the generalizations for differential manifolds of the gradient $\nabla_{\mathbb{R}^m}$ and the Laplacian $\Delta_{\mathbb{R}^m}$ in the Euclidean space $\mathbb{R}^m$ .

For a function f of class $\mathcal{C}^1$ on $\mathcal{M}$ , the gradient $\nabla_\mathcal{M} f$ is expressed in local coordinates as

(27) \begin{equation}\nabla_\mathcal{M} f(x)=\sum_{i,j=1}^dg^{ij}(x)\frac{\partial f}{ \partial x^i}(x)\left.\frac{\partial}{\partial x^j}\right\vert_x,\end{equation}

where ${{(g^{ij})}}_{1\leqslant i,j,\leqslant d}$ is the inverse matrix of ${{(g_{ij})}}_{1\leqslant i,j,\leqslant d}$ . Since $\sum_{j=1}^dg^{ij}g_{jk}=\delta_{ik}$ , we note that for f, h functions of class $\mathcal{C}^1$ ,

(28) \begin{equation}{{\left\langle \nabla_\mathcal{M}(f),\nabla_\mathcal{M} (h)\right \rangle }}_{\mathbf{g}}=\sum_{i,j=1}^d g^{ij}\frac{\partial f}{\partial x^i}\frac{\partial h}{\partial x^j}.\end{equation}

The Laplace–Beltrami operator is defined by (see [Reference Hsu19, Section 3.1])

(29) \begin{equation}\Delta_\mathcal{M} f\,:\!=\,\sum_{i,j=1}^d \frac{1}{\sqrt{\det(\mathbf{g})}} \frac{\partial}{ \partial x^i}{{\left(\sqrt{\det(\mathbf{g})}g^{ij}\frac{\partial f}{ \partial x^j}\right)}}.\end{equation}

When using normal coordinates, the expressions for the Laplacian and the gradient of a smooth function f at a point x match their definitions in ${\mathbb R}^d$ .

4.1. Weighted moment estimates

We begin with an auxiliary estimation. The result is related to kernel smoothing and can also be useful in density estimation on manifolds (see, for example, [Reference Berenfeld and Hoffmann4]). In particular, the following lemma does not require the compactness of the support of K, which also encompasses the Gaussian kernel function studied in [Reference Giné, Koltchinskii, Li and Zinn15], for example.

Lemma 1. Under Assumption 1, uniformly in $x \in \mathcal{M}$ , when h converges to 0, we have

(30) \begin{align} \frac{1}{h^{d+2}}\int_{\mathcal{M}} \mathbf{1}_{ \rho(x,y) \ge c_1 } K \left(\frac{\rho(x,y)}{h} \right) \mu(\mathrm{d}y)=o(h),\end{align}

(31) \begin{align} \frac{1}{h^{d+2}}\int_{\mathcal{M}} \mathbf{1}_{ \rho(x,y) \ge c_1 } K \left(\frac{\|x-y\|_2 }{h} \right) \mu(\mathrm{d}y)=o(h), \end{align}

and there is a generic constant c such that, for all points $x \in \mathcal{M}$ and positive number $h>0$ , we have

(32) \begin{align} \frac{1}{h^{d+2}}\int_{\mathcal{M}} K \left(\frac{\rho(x,y)}{h} \right) \|x -y\|_2^3 \mu(\mathrm{d}y) \le ch ,\end{align}

(33) \begin{align} \frac{1}{h^{d+2}}\int_{\mathcal{M}} K \left(\frac{\rho(x,y)}{h} \right) \|x -y\|_2^2 \mu(\mathrm{d}y) \le c, \end{align}

(34) \begin{align} \frac{1}{h^{d+2}}\int_{\mathcal{M}} K \left(\frac{\|x-y\|_2}{h} \right) \|x -y\|_2^3 \mu(\mathrm{d}y) \le ch , \end{align}

(35) \begin{align} \frac{1}{h^{d+2}}\int_{\mathcal{M}} K \left(\frac{\|x-y\|_2}{h} \right) \|x -y\|_2^2 \mu(\mathrm{d}y) \le c. \end{align}

Proof. Using Lemma 4, we have

(36) \begin{align}\int_{\mathcal{M}} \mathbf{1}_{ \rho(x,y) \ge c_1 } K \left(\frac{\rho(x,y)}{h} \right) \mu(\mathrm{d}y) & \le \mu(\mathcal{M}) \sup_{r \ge c_1} K\left( \frac{r}{h}\right) \nonumber \\& \le \mu(\mathcal{M}) \left[ H(\infty)-H\left( \frac{c_1}{h}\right) \right] \nonumber\\& = \mu(\mathcal{M}) \int_{(c_1/h,\infty)} \, \textrm{d}H(a) \nonumber \\&\le h^{d+3} \frac{ \mu(\mathcal{M})}{c_1^{d+3}} \int_{(c_1/h,\infty)} a^{d+3}\textrm{d}H(a).\end{align}

Thanks to the boundedness of $ \int_0^{\infty}a^{d+3}\, \mathrm{d}H(a)$ , we obtain (30). Then, as a consequence of (30), by the compactness of $\mathcal{M}$ , we readily observe that uniformly in $x \in \mathcal{M}$ , when h converges to 0,

\begin{equation*}\frac{1}{h^{d+2}}\int_{\mathcal{M}} \mathbf{1}_{ \rho(x,y) \ge c_1 } K \left(\frac{\rho(x,y)}{h} \right) \|x -y\|_2^3 \mu(\mathrm{d}y)=o(h). \end{equation*}

Thus, to prove inequality (32), it remains to prove that uniformly in x, when h converges to 0,

(37) \begin{equation} I\,:\!=\,\frac{1}{h^{d+3}}\int_{\mathcal{M}} \mathbf{1}_{ \rho(x,y) < c_1 } K \left(\frac{\rho(x,y)}{h} \right) \|x -y\|_2^3 \mu(\mathrm{d}y)=O(1).\end{equation}

Recall that in Theorem 7, we showed that for each point $x \in\mathcal{M}$ , there is a local smooth parameterization $\mathcal{E}_x$ of $\mathcal{M}$ that has many nice properties, especially $\rho(x,y)= \| \mathcal{E}_x^{-1}(y)\|_2$ for all y within an appropriate neighborhood of x by (23). Thus, the term I on the left-hand side of (37) can be rewritten in its coordinate representation under the parameterization $\mathcal{E}_x$ by using the change of variables $v=\mathcal{E}_x^{-1}(y)$ :

\begin{align} I &=\frac{1}{h^{d+3}} \int_{ B_{\mathbb{R}^d}(0,c_1)}K \left(\frac{\|v\|_2}{h} \right) \| x -\mathcal{E}_x(v)\|_2^3 \sqrt{\text{det}\widehat{g}^{x}_{ij}}(v) \, \mathrm{d}v. \nonumber\end{align}

Then, using Theorems 5 and 7(ii) and (iii),

(38) \begin{align}I &\le \frac{c_2}{h^{d+3}}\int_{ B_{\mathbb{R}^d}(0,c_1)}K \left(\frac{\|v\|_2}{h} \right) \| v \|_2^3 \left(1+ c_2\|v\|_2^2\right)\,\mathrm{d}v \nonumber \\ & \le \frac{c_2}{h^{d+3}}\int_{ B_{\mathbb{R}^d}(0,c_1)}K \left(\frac{\|v\|_2}{h} \right) \| v \|_2^3 \left(1+ c_2c_1^2\right)\mathrm{d}v \nonumber \\ & \le \frac{c_2}{h^{d+3}}\int_{ \mathbb{R}^d}K \left(\frac{\|v\|_2}{h} \right) \| v \|_2^3 \left(1+ c_2c_1^2\right)\mathrm{d}v \nonumber \\ &= c_2\big(1+c_2c_1^2\big)\int_{\mathbb{R}^d} K( \|v\|_2)\|v\|_2^3 \,\mathrm{d}v. \end{align}

Using the spherical coordinate system when $d\geqslant 2$ ,

\begin{align}I & \leqslant c_2\big(1+c_2c_1^2\big) \left[ \int_{0}^{\infty} K(a) a^3 \times a^{d-1}\, \mathrm{d}a \right] \nonumber \\ & \qquad \times\left[ \int_{ [0,2\pi]\times[0,\pi]^{d-2}} \sin^{d-2}(\theta_1)\sin^{d-3}(\theta_2) \cdots \sin(\theta_{d-2})\, \mathrm{d} \theta\right]\nonumber \\ &= c_2\big(1+c_2c_1^2\big)\ S_{d-1} \left[ \int_{0}^{\infty} K(a) a^{d+2}\,\mathrm{d}a \right]. \nonumber\end{align}

For $d=1$ , we use that

\begin{equation*}\int_{\mathbb{R}^1} K(|v|)|v|^3 \mathrm{d}v=2\times \int_{0}^{\infty} K(a)a^{1+2}\,\mathrm{d}a.\end{equation*}

Hence, by Lemma 4 in the Appendix and by Fubini’s theorem, we have

(39) \begin{align} I &\le c_2\big(1+c_2c_1^2\big) S_{d-1} \int_{0}^{\infty} \left( H(\infty)-H(a) \right) a^{d+2}\,\mathrm{d}a \nonumber \\ &= c_2\big(1+c_2c_1^2\big)(d+3)^{-1}S_{d-1} \int_{[0,\infty]} b^{d+3} \, \textrm{d}H(b)<\infty .\end{align}

Therefore, inequality (32) is proved. The proof of inequality (33) is similar.

For inequalities (31), (34), and (35), we observe that they are indeed consequences of (30), (32), and (33). Consider, for example, (31), again using Lemma 4 and Theorem 5:

(40) \begin{align} &\frac{1}{h^{d+2}}\int_{ \mathcal{M}} \mathbf{1}_{\rho(x,y)\geqslant c_1}K \left(\frac{\|x-y\|_2}{h} \right) \mu(\mathrm{d}y) \nonumber \\ &\quad\le\frac{1}{h^{d+2}}\int_{ \mathcal{M}} \mathbf{1}_{\rho(x,y)\geqslant c_1}\left[ H(\infty)- H \left( \frac{\rho(x,y)}{h(1+c_3\|x-y\|_2^2)} \right) \right] \mu(\mathrm{d}y) \nonumber \\ &\quad\le\frac{1}{h^{d+2}}\int_{ \mathcal{M}} \mathbf{1}_{\rho(x,y)\geqslant c_1} \left[ H(\infty)- H \left( \frac{\rho(x,y)}{h(1+c_3\mathrm{diam}(\mathcal{M})^2)} \right) \right] \mu(\mathrm{d}y) \nonumber \\ &\quad =\frac{1}{h^{d+2}}\int_{ \mathcal{M}}\mathbf{1}_{\rho(x,y)\geqslant c_1} \tilde{K} \left(\frac{\rho(x,y)}{h} \right) \mu(\mathrm{d}y), \end{align}

for $\tilde{K} (a) \,:\!=\, H(\infty)- H\left( \frac{a}{1+c_3\mathrm{diam}(\mathcal{M})^2}\right) $ and where the second inequality uses the fact that H is a non-decreasing function. Thus, inequality (31) corresponds to inequality (30), where K is replaced with $\tilde{K}$ . Clearly, the function $\tilde{K}$ is of bounded variation and satisfies Assumption 1, which concludes the proof for (31). The arguments are similar for (34) and (35).

4.2. Proof of Proposition 1

We now prove Proposition 1, dealing with the approximation of the Laplace–Beltrami operator by a kernel operator. In the course of the proof, some quantities involving gradients and Laplacians will appear repetitively. The next lemma will be useful in dealing with these expressions and its proof is postponed to Appendix C.

Lemma 2 (Some auxiliary calculations). Suppose that $f,h\,:\, \mathbb{R}^m \rightarrow \mathbb{R}$ , $k\,:\, \mathbb{R}^d \rightarrow \mathbb{R}^m$ are $\mathcal{C}^2$ -continuous functions, that $k(0)=x$ and that $G\,:\, \mathbb{R}_+ \rightarrow \mathbb{R}$ is a locally bounded measurable function. Then, for all $c>0$ ,

(41) \begin{multline} \int_{B_{\mathbb{R}^d}(0,c)} G(\|v\|_2) \langle \nabla_{\mathbb{R}^m}f(x) , k'(0)(v)\rangle \langle \nabla_{\mathbb{R}^m}h(x) , k'(0)(v)\rangle \, \mathrm{d}v \\ =\langle \nabla_{\mathbb{R}^d}( f\circ k )(0), \nabla_{\mathbb{R}^d}( h\circ k )(0)\rangle \left( \frac{1}{d}\int_{B_{\mathbb{R}^d}(0,c)} G(\|v\|_2) \|v\|_2^2 \, \mathrm{d}v \right) \end{multline}

and

(42) \begin{multline} \int_{B_{\mathbb{R}^d}(0,c)} G \left(\|v\|_2 \right) \bigg[ \left\langle \nabla_{\mathbb{R}^m} f(x),k'(0)(v)+\frac{1}{2}k''(0)(v,v) \right\rangle \\ + \frac{1}{2}f''(x)(k'(0)(v),k'(0)(v)) \bigg] \,\mathrm{d}v\\ = \frac{1}{2}\Delta_{\mathbb{R}^d}(f\circ k)(0)\left( \frac{1}{d}\int_{B_{\mathbb{R}^d}(0,c)} G(\|v\|_2) \|v\|_2^2 \,\mathrm{d}v \right).\end{multline}

We are now in a position to prove Proposition 1. As $\mathcal{M}$ is compact, it is a properly embedded submanifold of $\mathbb{R}^m$ (see [Reference Lee25, p. 98]). Hence, any function of class $\mathcal{C}^3$ on $\mathcal{M}$ can be extended to a function of class $\mathcal{C}^3$ on $\mathbb{R}^m$ (see [Reference Lee25, Lemma 5.34]). Thus, without loss of generality, assume f and p are $\mathcal{C}^3$ and $\mathcal{C}^2$ functions on $\mathbb{R}^m$ with compact supports, respectively.

Recall that we want to study ${{\left| \widetilde{\mathcal{A}}_h(f)-\mathcal{A}(f) \right|}}$ , where $\mathcal{A}$ and $\widetilde{\mathcal{A}}_h$ have been respectively defined in (2) and (16). Thanks to Lemma 1, it is sufficient to prove that

\begin{equation*}\left|\mathcal{A}(f)(x)- \frac{1}{h^{d+2}}\int_{\mathcal{M}} \mathbf{1}_{\rho(y,x) < c_1}K \left(\frac{\rho(x,y)}{h} \right) ( f(y)-f(x))p(y)\mu(\mathrm{d}y) \right|=O(h),\end{equation*}

uniformly in $x\in \mathcal{M}$ , to conclude the proof of Proposition 1, where $c_1>0$ is introduced in Theorem 7.

In addition, thanks to the compactness of $\mathcal{M}$ and to the regularity of f and p, Taylor’s expansion implies that there is a constant $c_4$ such that, for all $x,y \in \mathcal{M}$ ,

(43) \begin{align}&\bigg| (f(y) -f(x))p(y) -\bigg( \langle \nabla_{\mathbb{R}^m} f(x),y-x \rangle +\frac{1}{2}f''(x)(y-x,y-x) \bigg)p(x) \nonumber\\ & \qquad\qquad\qquad\qquad\qquad - \langle \nabla_{\mathbb{R}^m} f(x),y-x \rangle \langle \nabla_{\mathbb{R}^m} p(x),y-x \rangle\bigg| \le c_4 \|x-y\|_2^3.\end{align}

Hence, by inequality (32), it is sufficient to prove that uniformly in x,

(44) \begin{align} J_1 & \,:\!=\, \frac{1}{h^{d+2}}\int_{\mathcal{M}}\mathbf{1}_{\rho(y,x) < c_1} K \left(\frac{\rho(x,y)}{h} \right) \langle \nabla_{\mathbb{R}^m} f(x),y-x \rangle \langle \nabla_{\mathbb{R}^m} p(x),y-x \rangle\mu(\mathrm{d} y) \nonumber\\ & = c_0{{\left\langle\nabla_{\mathcal{M}}(f)(x), \nabla_{\mathcal{M}}(p)(x)\right\rangle}}_{\mathbf{g}}+O(h)\end{align}

and

(45) \begin{align} J_2 & \,:\!=\, \frac{1}{h^{d+2}}\int_{\mathcal{M}} \mathbf{1}_{\rho(y,x) < c_1}K \left(\frac{\rho(x,y)}{h} \right) \bigg[ \langle \nabla_{\mathbb{R}^m} f(x),y-x \rangle +\frac{1}{2}f''(x)(y-x,y-x) \bigg] \mu(\mathrm{d}y) \nonumber\\ & = \frac{1}{2}c_0\Delta_{\mathcal{M}}(f)(x)+O(h). \end{align}

The proof is similar to the study of I given by (37) in the proof of Lemma 1. We rewrite the integrals considered in coordinate representations. Using the change of variables $v=\mathcal{E}_x^{-1}(y)$ , we have

\begin{multline*}J_1=\frac{1}{h^{d+2}}\int_{B_{\mathbb{R}^d}(0,c_1)} K \left(\frac{\|v\|_2}{h} \right) \langle \nabla_{\mathbb{R}^m} f(x),\mathcal{E}_x(v)-x \rangle\langle \nabla_{\mathbb{R}^m} p(x),\mathcal{E}_x(v)-x \rangle \sqrt{ \mathrm{det}\widehat{g}^x_{ij}(v)} \,\mathrm{d}v.\end{multline*}

By properties (ii) and (iii) in Theorem 7 we have

\begin{align} \bigg|J_1 - &\int_{B_{\mathbb{R}^d}(0,c_1)} K \left(\frac{\|v\|_2}{h} \right) \langle \nabla_{\mathbb{R}^m} f(x),\mathcal{E}_x(v)-x \rangle \langle \nabla_{\mathbb{R}^m} p(x),\mathcal{E}_x(v)-x \rangle \, \mathrm{d}v\bigg| \nonumber \\ \le &\frac{c_2}{h^{d+2}} \|\nabla_{\mathbb{R}^m} f(x)\|_2 \| \nabla_{\mathbb{R}^m} p(x)\|_2 \int_{B_{\mathbb{R}^d}(0,c_1)} K \left(\frac{\|v\|_2}{h} \right) \|v\|_2^2 \| \mathcal{E}_x(v)-x\|_2^2 \,\mathrm{d}v \nonumber \\ \le &\frac{c_2^3}{h^{d+2}} \|\nabla_{\mathbb{R}^m} f(x)\|_2 \| \nabla_{\mathbb{R}^m} p(x)\|_2 \int_{B_{\mathbb{R}^d}(0,c_1)} K \left(\frac{\|v\|_2}{h} \right) \|v\|_2^4 \, \mathrm{d}v \nonumber \\ \le & \frac{c_2^3c_1}{h^{d+2}} \|\nabla_{\mathbb{R}^m} f(x)\|_2 \| \nabla_{\mathbb{R}^m} p(x)\|_2 \int_{B_{\mathbb{R}^d}(0,c_1)} K \left(\frac{\|v\|_2}{h} \right) \|v\|_2^3 \, \mathrm{d}v. \nonumber\end{align}

As in the proof of Lemma 1, we deduce that the latter is bounded by O(h).

In addition, again using property (iii) in Theorem 7, we have that, uniformly in x,

(46) \begin{align}&\Big|\frac{1}{h^{d+2}}\int_{B_{\mathbb{R}^d}(0,c_1)} K \left(\frac{\|v\|_2}{h} \right) \langle \nabla_{\mathbb{R}^m} f(x),\mathcal{E}_x(v)-x \rangle\langle \nabla_{\mathbb{R}^m} p(x),\mathcal{E}_x(v)-x \rangle \, \mathrm{d}v-J_{11}\Big| \nonumber\\&\quad =O(h),\end{align}

with

\begin{align*}J_{11}\,:\!=\, &\frac{1}{h^{d+2}}\int_{B_{\mathbb{R}^d}(0,c_1)} K \left(\frac{\|v\|_2}{h} \right) \langle \nabla_{\mathbb{R}^m} f(x),\mathcal{E}_x'(0)(v) \rangle \langle \nabla_{\mathbb{R}^m} p(x),\mathcal{E}_x'(0)(v) \rangle \, \mathrm{d}v.\end{align*}

Let us now compare $J_{11}$ with the first term of the generator $\mathcal{A}$ . Using equation (41) of Lemma 2, with $G(||v||_2)=K{{\left(\frac{||v||_2}{h}\right)}}$ , $k=\mathcal{E}_x$ , and Proposition 4, we have

\begin{align*}J_{11} & = \frac{1}{h^{d+2}}\left( \frac{1}{d}\int_{B_{\mathbb{R}^d}(0,c_1)} K\left(\frac{\|v\|_2}{h}\right) \|v\|_2^2 \, \mathrm{d}v \right) \langle \nabla_{\mathbb{R}^d}( f\circ \mathcal{E}_x )(0), \nabla_{\mathbb{R}^d}( p \circ \mathcal{E}_x )(0)\rangle\\&= \left( \frac{1}{d}\int_{B_{\mathbb{R}^d}(0,c_1/h)} K\left(\|v\|_2\right) \|v\|_2^2 \mathrm{d}v \right) \langle \nabla_{\mathcal{M}} f(x), \nabla_{\mathcal{M}} p(x) \rangle_{\mathbf{g}}\\&= \left( \frac{1}{d}\int_{\mathbb{R}^d} K\left(\|v\|_2\right) \|v\|_2^2 \mathrm{d}v \right) \langle \nabla_{\mathcal{M}} f(x), \nabla_{\mathcal{M}} p(x) \rangle_{\mathbf{g}}+o(h),\end{align*}

where the last estimation is uniform in $x \in \mathcal{M}$ and comes from the second estimation in (B2) in Lemma 4. Thus, we have proved Equation (44) for $J_1$ .

The proof for $J_2$ , given by (45), is similar to what we have done for $J_1$ . To identify the Laplace–Beltrami operator in the last step of the proof, we use (42) of Lemma 2 and the second point of Proposition 4. Therefore, we have proved Proposition 1.

4.3. Proof of Proposition 2

Let us now prove Proposition 2. This proposition deals with the difference between the geodesic distance on $\mathcal{M}$ and the Euclidean norm of ${\mathbb R}^m$ .

By inequalities (30) and (31) of Lemma 1, we know that uniformly in x, when h converges to 0,

\begin{equation*}\int_{ \mathcal{M}} {{\left(K\left( \frac{\| x-y\|_2}{h}\right)+K\left( \frac{\rho( x-y)}{h}\right)\right)}} \mathbf{1}_{ \rho(x,y) \ge c_1} \mu( \mathrm{d}y) =o(h^{d+3}).\end{equation*}

Thus, by regularity of f, boundedness of p and compactness of $\mathcal{M}$ , uniformly in x, when h converges to 0,

\begin{equation*}\int_{ \mathcal{M}} {{\left(K\left( \frac{\| x-y\|_2}{h}\right)+K\left( \frac{\rho( x-y)}{h}\right)\right)}} |f(x)-f(y)| {p(y)}\mathbf{1}_{ \rho(x,y) \ge c_1} \mu( \mathrm{d}y)=o(h^{d+3}).\end{equation*}

Thus, we only have to prove that, uniformly in x,

\begin{equation*}\int_{ \mathcal{M}} \left| K\left( \frac{\rho(x,y)}{h}\right)-K\left( \frac{\| x-y\|_2}{h}\right) \right| |f(x) -f(y)| {p(y)} \mathbf{1}_{ \rho(x,y) < c_1} \mu( \mathrm{d}y)=O(h^{d+3}),\end{equation*}

or equivalently, using the change of variables $v=\mathcal{E}_x^{-1}(y)$ and $\rho(x,y)=\|\mathcal{E}_x^{-1}(y)\|_2$ by (23),

\begin{align*} &\int_{B_{\mathbb{R}^d(0,c_1)}} \left| K \left(\frac{\|v\|_2}{h} \right)- K \left(\frac{\|\mathcal{E}_x(v)-x\|_2}{h} \right)\right| \left| f\circ \mathcal{E}_x(v)- f(x)\right|{p\circ \mathcal{E}_x(v)}\sqrt{ \text{det} \widehat{g}^{x}_{ij}(v)} \,\mathrm{d}v \\ &\quad = O(h^{d+3}).\end{align*}

In addition, by regularity of f, boundedness of p, and compactness of $\mathcal{M}$ , there is a constant c such that $|f(x)-f(y)| \le c\|x-y\|_2 $ . Moreover, by property (ii) of Theorem 7, the function $v\mapsto\text{det} \widehat{g}^{x}_{ij}(v)$ is bounded on $B_{\mathbb{R}^d(0,c_1)}$ . Hence, it is sufficient to show that, uniformly in x,

\begin{equation*}I\,:\!=\,\int_{B_{\mathbb{R}^d(0,c_1)}} \left| K \left(\frac{\|v\|_2}{h} \right)- K \left(\frac{\|\mathcal{E}_x(v)- x\|_2}{h} \right)\right| \| \mathcal{E}_x(v)-x\|_2 \,\mathrm{d}v=O(h^{d+3}).\end{equation*}

Recall that $\|\mathcal{E}_x(v)- x\|_2\leqslant\|v\|_2$ (by Theorem 7). By (B1) in Lemma 4, we have

\begin{align*} I\leqslant &\int_{B_{\mathbb{R}^d(0,c_1)}} \left( \int_{ \left( \frac{\|\mathcal{E}_x(v)- x\|_2}{h}, \frac{\|v\|_2}{h}\right]} \mathrm{d}H(a) \right) \| v\|_2 \mathrm{d}v \\ = &\int_{B_{\mathbb{R}^d(0,c_1)}} \left( \int_{ \mathbb{R}_+} \mathbf{1}_{ \|\mathcal{E}_x(v)- x\|_2 < ah \le \|v\|_2} \mathrm{d}H(a) \right)\| v\|_2 \mathrm{d}v.\end{align*}

Also by Theorem 5, there exists a constant $c_3$ such that, for all $x,y\in\mathcal{M}$ ,

\begin{equation*}\rho(x,y)\leqslant c_3{{\left\| x-y\right\|}}_2^3+{{\left\| x-y\right\|}}.\end{equation*}

In addition, the polynomial function $z\mapsto z+c_3z^3$ is an increasing bijective function and we denote by $\varphi$ its inverse. Thus, for all $x,y \in \mathcal{M}$ , $ \varphi( \rho(x,y))\le \|x-y\|_2$ . Then, using that $\varphi(\|v\|_2)= \varphi(\rho(x,\mathcal{E}_x(v)))$ , we deduce that $\varphi(\| v\|_2) \le \|\mathcal{E}_x(v)-x \|_2.$ Therefore,

\begin{align*}I \le & \int_{B_{\mathbb{R}^d(0,c_1)}} \left( \int_{ \mathbb{R}_+} \mathbf{1}_{ \varphi(\|v\|_2) < ah \le \|v\|_2} \mathrm{d}H(a) \right)\| v\|_2 \mathrm{d}v \\ = & \int_{\mathbb{R}^+} \left( \int_{B_{\mathbb{R}^d(0,c_1)}} \| v\|_2.\mathbf{1}_{ ah \le \|v \|_2 < ah+c_3(ah)^3} \mathrm{d}v \right)\mathrm{d}H(a),\end{align*}

by Fubini’s theorem. Finally, using the spherical coordinate system as in the proof of Lemma 1, we see that

\begin{align*}I & \leqslant S_{d-1}\int_{\mathbb{R}^+} \left( \int_{0}^{c_1} r^{d}\mathbf{1}_{ ah \le r < ah+c_3(ah)^3} \mathrm{d}r \right)\,\mathrm{d}H(a) \\ & \le S_{d-1}\int_{\mathbb{R}^+} \left( \mathbf{1}_{ ah \le c_1} \times \int_{ah}^{ah+c_3(ah)^3} r^{d} \,\mathrm{d}r \right)\,\mathrm{d}H(a) \\ & \le S_{d-1}\int_{\mathbb{R}^+} \big( \mathbf{1}_{ ah \le c_1} \times c_3(ah)^3\big[ah+c_3(ah)^3\big]^d \big)\,\mathrm{d}H(a) \\ & \le S_{d-1} \int_{\mathbb{R}_+} c_3(ah)^{d+3}\big(1+c_3c_1^2\big)^{d} \,\mathrm{d}H(a) \\ & = S_{d-1} c_3\big(1+c_3c_1^2\big)^{d} h^{d+3}\int_{\mathbb{R}^+} a^{d+3} \,\mathrm{d}H(a).\end{align*}

This concludes the proof of Proposition 2.

5.1. On Vapnik–Chervonenkis theory

Before starting the proof, we recall the main definitions and an important result from Vapnik–Chervonenkis theory for the Borelian space $({\mathbb R}^m,\mathcal{B}({\mathbb R}^m))$ we will need. Other useful results are given in Appendix A. For more details on the Vapnik–Chervonenkis theory, we refer the reader to [Reference Devroye, Györfi and Lugosi11, Reference Giné and Nickl16, Reference Nolan and Pollard29]. In this section, we will recall upper bounds that exist for

\begin{equation*} \sup_{f\in \mathcal{F}}\mathrm{E}{{\left[ \left| \sum_{i=1}^n {{\left( f(X_i)- \mathrm{E}{{[f(X_i)]}}\right)}} \right|\right]}} \end{equation*}

when the functions f range over certain Vapnik–Chervonenkis classes of functions that are defined below.

Let (T, d) be a pseudo-metric space. Let $\varepsilon>0$ and $N\in\mathbb{N}\cup \{+\infty\}$ . A set of points $\{ x_1,\ldots, x_N\}$ in T is an $\varepsilon$ -cover of T if, for any $x \in T$ , there exists $i \in [1,N]$ such that $ d( x, x_i) \le \varepsilon$ . Then the $\varepsilon$ -covering number of T is defined as

\begin{align*}N(\varepsilon, T,d) \,:\!=\, &\inf\{ N \in \mathbb{N}\cup\{ +\infty\} \,:\, \text{there are }N \text{ points in } T \\ &\hskip 1cm \text{such that they form an }\varepsilon\text{-cover of } (T, d) \}. \end{align*}

For a collection of real-valued measurable functions $\mathcal{F}$ on $\mathbb{R}^m$ , a real measurable function F defined on $\mathbb{R}^m$ is called an envelope of $\mathcal{F}$ if, for any $x \in \mathbb{R}$ ,

\begin{equation*} \sup_{f \in \mathcal{F}} |f(x)|\le F(x) . \end{equation*}

This allows us to define Vapnik–Chervonenkis classes of functions (see Definition 3.6.10 in [Reference Giné and Nickl16]). Recall that, for a probability measure Q on the measurable space $(\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m))$ , the $L^2(Q)$ -distance given by

\begin{equation*}\displaystyle{(f,g)\mapsto{{\left( \int{{\left| f(x)-g(x) \right|}}^2Q(\mathrm{d} x)\right)}}^{1/2}}\end{equation*}

defines a pseudo-metric on the collection of all bounded real measurable functions on $\mathbb{R}^m$ .

Definition 1 (Vapnik-Chervonenkis class of functions). A class of measurable functions $\mathcal{F}$ is of Vapnik–Chervonenkis type with respect to a measurable envelope F of $\mathcal{F}$ if there exist finite constants A, v such that, for all probability measures Q and $\varepsilon\in(0,1)$ ,

\begin{equation*}N( \varepsilon \|F\|_{L^2},\mathcal{F}, L^2(Q)) \le (A/\varepsilon)^v.\end{equation*}

We will write

\begin{equation*}N(\varepsilon, \mathcal{F})\,:\!=\,\sup\nolimits_{Q} N(\varepsilon, \mathcal{F}, L^2(Q)).\end{equation*}

We now present a version of the useful inequality (2.5) of Giné and Guillou in [Reference Giné and Guillou14] that gives a bound for the expected concentration rate. For a class of function $\mathcal{F}$ , let us define for any real valued function $\varphi\,:\, \mathcal{F} \rightarrow \mathbb{R}$ ,

\begin{equation*}\| \varphi(f) \|_{ \mathcal{F}}=\sup\nolimits_{f \in \mathcal{F}}| \varphi(f)|.\end{equation*}

Theorem 8 (See [Reference Giné and Guillou14, Proposition 2.1 and Inequality (2.5)]). Consider n i.i.d. random variables $X_1,\ldots,X_n$ with values in $(\mathbb{R}^m, \mathcal{B}(\mathbb{R}^m))$ . Let $\mathcal{F}$ be a measurable uniformly bounded Vapnik–Chervonenkis-type class of functions on $(\mathbb{R}^m, \mathcal{B}(\mathbb{R}^m))$ . We introduce two positive real numbers, $\sigma^2$ and U, such that

\begin{equation*} \sigma^2 \ge \sup\nolimits_{f \in \mathcal{F}} \mathrm{Var}(f(X_1)), \quad U \ge \sup\nolimits_{f \in \mathcal{F}}\|f \|_{\infty},\quad \text{and}\quad 0 < \sigma \le 2U. \end{equation*}

Then there exists a constant R, depending only on the Vapnik–Chervonenkis parameters A, v of $\mathcal{F}$ and on U, such that

\begin{equation*} \mathrm{E}{{\left[ \left\| \sum_{i=1}^n {{\left( f(X_i)- \mathrm{E}{{[f(X_i)]}}\right)}} \right\|_{ \mathcal{F}}\right]}} \le R {{\big( \sqrt{ n} \sigma \sqrt{ {{\left| \log\sigma \right|}}}+ {{\left| \log\sigma \right|}}\big)}}. \end{equation*}

Note that there also exists a formulation of the previous result in terms of deviation probability (see, for example, [Reference Massart28, Theorem 3]) that would lead to results similar to the ones established in [Reference Calder and Garca Trillos7].

5.2. Step I: Decomposition of $Z_n$

We first upper-bound the quantity $Z_n$ with a sum of simpler terms.

Lemma 3. There is a constant $c>0$ such that, for all $n\geqslant 1$ ,

(50) \begin{equation} nh_n^{d+2}Z_n \le \sum_{\alpha =1}^{m} Y^{\alpha}_n + \sum_{\alpha ,\beta=1}^{m} Y^{\alpha,\beta}_n + Y^{(3)}_n +2 nch_n^{d+3},\end{equation}

where

\begin{align*} Y^{\alpha}_n &\,:\!=\, \sup_{x \in \mathcal{M}}\left| \sum_{i=1}^n \bigg[ K\left( \frac{ \|X_i-x\|_2}{h_n}\right) \big(X_{i}^{\alpha}-x_{i}^{\alpha}\big) -\mathrm{E}\left(K\left( \frac{ \|X-x\|_2}{h_n}\right) (X^{\alpha}-x^{\alpha}) \right)\bigg]\right|, \\ Y^{\alpha,\beta}_n &\,:\!=\, \sup_{x \in \mathcal{M}}\left| \sum_{i=1}^n \bigg[K\left( \frac{ \|X_i-x\|_2}{h_n}\right) \big(X_{i}^{\alpha}-x^{\alpha}\big)\big(X_{i}^{\beta}-x^{\beta}\big)\right. \\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.-\mathrm{E}\left(K\left( \frac{ \|X-x\|_2}{h_n}\right) (X^{\alpha}-x^{\alpha})(X^{\beta}-x^{\beta}) \right) \bigg]\right|, \\ Y^{(3)}_n &\,:\!=\, \sup_{x \in \mathcal{M}} \left| \sum_{i=1}^n K\left( \frac{ \|X_i-x\|_2}{h_n}\right)\|X_i-x\|_2^3 - \mathrm{E}\left[K\left( \frac{ \|X-x\|_2}{h_n}\right)\|X-x\|_2^3\right] \right|.\end{align*}

Proof. Since, for any $f\in\mathcal{F}$ , the differentials up to third order have operator norms bounded by 1, we have, by the Taylor expansion theorem, for any $(x,y) \in (\mathbb{R}^m)^2$ ,

\begin{align*}f(y)-f(x)&=f'(x)(y-x)+ \frac{1}{2}f''(x)(y-x,y-x)+\tau_f(y;x),\end{align*}

where $\tau_f$ is some function satisfying

(51) \begin{equation}\sup\nolimits_{f \in \mathcal{F}} |\tau_f(y,x)| \le \| f'''\|_{\infty} \|x-y\|_2^3= \|x-y\|_2^3.\end{equation}

Thus, using the notation of the lemma, we deduce

\begin{equation*} nh_n^{d+2}Z_n \le \sum_{\alpha =1}^{m} Y^{\alpha}_n + \sum_{\alpha =1,\beta=1}^{m} Y^{\alpha,\beta}_n +Y^{r}_n,\end{equation*}

with

\begin{align*} Y^{r}_n &\,:\!=\, \sup\nolimits_{\underset{x\in\mathcal{M}}{f \in \mathcal{F}}} \left| \sum_{i=1}^n \left(K\left( \frac{ \|X_i-x\|_2}{h_n}\right)\tau_f( X_i,x) - \mathrm{E}\left[K\left( \frac{ \|X-x\|_2}{h_n}\right)\tau_f(X,x) \right]\right) \right|.\end{align*}

Using (51), we now control $Y^r_n$ by $Y^{(3)}_n$ as follows:

\begin{align*} Y^r_n & {\le} \sup_{x \in \mathcal{M}} \left| \sum_{i=1}^n K\left( \frac{ \|X_i-x\|_2}{h_n}\right)\|X_i-x\|_2^3 \right| + n\sup_{x \in \mathcal{M}} \mathrm{E}\left[K\left( \frac{ \|X-x\|_2}{h_n}\right)\|X-x\|_2^3\right] \\ &\le Y^{(3)}_n+2n\sup_{x \in \mathcal{M}} \mathrm{E}\left[K\left( \frac{ \|X-x\|_2}{h_n}\right)\|X-x\|_2^3\right].\end{align*}

Since the function p is bounded on the compact $\mathcal{M}$ , using (34) of Lemma 1, we deduce that $Y^r_n\leqslant Y^{(3)}_n+2nch_n^{d+3}$ , which concludes the proof.

5.3. Step II: Application of the Vapnik–Chervonenkis theory

5.3.1. Control the first-order term $\mathrm{E}[Y^{\alpha}_n ]$

Let $\alpha\in [\![1,m]\!]$ be fixed. Given the kernel K, to bound the first-order term $Y^{\alpha}_n$ , we introduce three families of real functions on $\mathcal{M}$ :

\begin{align*}\mathcal{G}&\,:\!=\,\{ \varphi_{h,y,z} \,:\, y,z \in \mathcal{M} , h>0 \},\\ \mathcal{G}_1&\,:\!=\,\{ \psi_{h,y} \,:\, y\in \mathcal{M},h >0 \} \\ \mathcal{G}_2&\,:\!=\,\{ \zeta_y(x) \,:\, y \in \mathcal{M} \},\end{align*}

with

\begin{equation*} \begin{array}{llcl} \varphi_{h,y,z} \,: \, \,&x& \,\,\, \longmapsto \,\,\, &K\left( \frac{\| x-y\|_2}{h} \right) (x^{\alpha}-z^{\alpha}),\\[2pt] \psi_{h,y} \,:\,&x& \,\,\, \longmapsto \,\,\, & K\left( \frac{\| x-y\|_2}{h} \right),\\[2pt] \zeta_y \,:\,&x& \,\,\, \longmapsto \,\,\, & x^{\alpha}-y^{\alpha}. \end{array}\end{equation*}

Since K is of bounded variation, by [Reference Nolan and Pollard29, Lemma 22], $\mathcal{G}_1$ is of Vapnik–Chervonenkis type with respect to a constant envelope. Since $\mathcal{M}$ is a compact manifold, by Lemma 3, $\mathcal{G}_2$ is also of Vapnik–Chervonenkis type with respect to a constant envelope. Thus, using Lemma 2, we deduce that $\mathcal{G}$ is a Vapnik–Chervonenkis-type class of functions because $\mathcal{G}=\mathcal{G}_1 \cdot \mathcal{G}_2$ . Thus, by Definition 1, there exist real values $A \ge 6, v \ge 1$ depending only on the Vapnik–Chervonenkis characteristics of $\mathcal{G}_1$ and $\mathcal{G}_2$ such that, for all $ \varepsilon \in(0,1)$ ,

\begin{equation*}N( \varepsilon, \mathcal{G}) \le \left( \frac{A}{2\varepsilon}\right)^v.\end{equation*}

Now let us consider the following sequence of families of real functions on $\mathcal{M}$ :

\begin{equation*}\mathcal{H}_n=\left\{ \varphi_{n,y} \,:\, y \in \mathcal{M} \right\},\quad \text{with }\varphi_{n,y} \,:\, x \longmapsto K\left(\frac{\| x-y\|_2}{h_n} \right) (x^{\alpha}-y^{\alpha}) .\end{equation*}

Proposition 5. Let $(X_i)_{i\geqslant 1}$ be a sample of i.i.d. random variables with distribution $p(x)\mu(\mathrm{d} x)$ on the compact manifold $\mathcal{M}$ and X a random variable with the same distribution. We assume that p is bounded on $\mathcal{M}$ .

Then, if the kernel K satisfies Assumption 1 and the sequence $(h_n)_{n\geqslant 0}$ satisfies (5), then we have

(52) \begin{align} \frac{1}{nh_n^{d+2}} \mathrm{E}{{\left[ \left\| \sum_{i=1}^n {{\left(f(X_i)-\mathrm{E}[f(X)]\right)}} \right\|_{\mathcal{H}_n}\right]}} = O\left( \sqrt{ \frac{ \log h_n^{-1}}{nh_n^{d+2}}}\right).\end{align}

Proof. Since $\mathcal{H}_n\subset \mathcal{G}$ , by Lemma 1, for all n, we have $N(\varepsilon, \mathcal{H}_n)\le \left(\frac{A}{\varepsilon}\right)^v$ . Hence, by Theorem 8, there exists a constant R, depending only on A, v, and U, such that

\begin{align*} \mathrm{E}{{\left[ \left\| \sum_{i=1}^n f(X_i)-\mathrm{E}(f(X)) \right\|_{\mathcal{H}_n}\right]}} \le R\left( \sqrt{n}\sigma\sqrt{{{\left| \log\sigma \right|}}} + {{\left| \log\sigma \right|}}\right),\end{align*}

where U is a constant such that $ U \ge \sup_{f \in \mathcal{H}_n} \|f\|_{\infty}$ , and $\sigma$ is a constant such that $4U^2 \ge \sigma^2 \ge \sup_{f \in \mathcal{H}_n} \mathrm{E}[f^2(X)]$ .

Since $\mathcal{H}_n\subset \mathcal{G}$ , we can choose U to be the constant envelope of $\mathcal{G}$ (thus, independent of n). Furthermore, we see that

\begin{align*}\sup_{f \in \mathcal{H}_n}\mathrm{E}\big[ f^2(X)\big]& \le \|K\|_{\infty} \sup_{x \in \mathcal{M}} \int_{\mathcal{M}} K\left(\frac{\| x-y\|}{h_n} \right)(x^{\alpha}-y^{\alpha})^2 p(y)\mu(\mathrm{d} y).\end{align*}

By (33) of Lemma 1, we deduce that there is $c>0$ such that

(53) \begin{equation}\sup_{f \in \mathcal{H}_n}\mathrm{E}\big[ f^2(X)\big] {\le} \|K\|_{\infty}\|p\|_{\infty} \mu(\mathcal{M})c h_n^{d+2}, \end{equation}

which goes to 0 when $n\to +\infty$ . Choose $\sigma^2 \,:\!=\, \sigma_n^2=\|K\|_{\infty}\|p\|_{\infty} \mu(\mathcal{M})c h_n^{d+2} $ . For n large enough, $\sigma_n \le 2U$ . Hence, using assumption (5) on the sequence $(h_n)_{n\geqslant 1}$ , we deduce

\begin{align*} \frac{1}{nh_n^{d+2}} \mathrm{E} {{\left[{{\left\| \sum_{i=1}^n f(X_i)-\mathrm{E}[f(X)] \right\|}}_{\mathcal{H}_n}\right]}} &=O\left( \sqrt{ \frac{ \log h_n^{-1}}{nh_n^{d+2}}} {+} \frac{ \log h_n^{-1}}{nh_n^{d+2}} \right) \\ &= O\left( \sqrt{ \frac{ \log h_n^{-1}}{nh_n^{d+2}}}\right).\end{align*}

This concludes the proof.

The conclusion of the above proposition means that

\begin{equation*}\frac{1}{nh_n^{d+2}}\mathrm{E} \big[Y^{\alpha}_n\big ]= O\left( \sqrt{ \frac{ \log h_n^{-1}}{nh_n^{d+2}}}\right).\end{equation*}

5.3.2. Control the second-order term $\mathrm{E}{{[Y^{\alpha,\beta}_n]}}$

The way to bound the second-order term $Y^{\alpha,\beta}_n$ , for $\alpha,\beta \in [\![ 1,m ]\!]$ , is similar to the previous step, but instead of considering $\mathcal{H}_n$ , we consider the following Vapnik–Chervonenkis-type family of functions:

\begin{equation*}\mathcal{I}_n\,:\!=\,\left\{ \xi_{n,y,z}\,:\,x \mapsto K\left(\frac{\| x-y\|}{h_n} \right) (x^{\alpha}-y^{\alpha})(x^{\beta}-q^{\beta}) \,:\, y \in \mathcal{M}, z\in \mathcal{M} , \right\}.\end{equation*}

We note that, for any random variable X,

\begin{equation*}\mathrm{E}{{\Big[\sup_{g\in \mathcal{I}_n}{{\big| g^2(X) \big|}}\Big]}}\leqslant \mathrm{diam}(\mathcal{M})^2\mathrm{E}{{\Big[\sup_{f\in \mathcal{H}_n}{{\big| f^2(X) \big|}}\Big]}}.\end{equation*}

Using (53), we deduce that

\begin{equation*}\sup_{g \in \mathcal{I}_n} \mathrm{E}[g^2(X)] = O\big( h_n^{d+2}\big)\end{equation*}

and

\begin{align*} \frac{1}{nh_{n}^{d+2}} \mathrm{E} \sup_{g \in \mathcal{I}_n} \left| \sum_{i=1}^{n} \left( g(X_i)-\mathrm{E}[g(X_i)] \right)\right| = O\left( \sqrt{ \frac{ \log h_n^{-1}}{nh_n^{d+2}}}\right).\end{align*}

Therefore, we conclude that

\begin{equation*}\frac{1}{nh_n^{d+2}}\mathrm{E}\big[Y^{\alpha,\beta}_n \big]= O\left( \sqrt{ \frac{ \log h_n^{-1}}{nh_n^{d+2}}}\right).\end{equation*}

5.3.3. Control the third-order term $\mathrm{E}{{[Y^{(3)}_n]}}$

This step is essentially the same as the two previous steps, except that the family of functions considered is a little bit different:

\begin{equation*}\mathcal{K}_n\,:\!=\, \left\{ x \mapsto K\left(\frac{\| x-y\|}{h_n} \right) \|x-y\|^3 \,:\, y \in \mathcal{M} \right\}.\end{equation*}

With the same arguments as before, we obtain

\begin{equation*}\frac{1}{nh_n^{d+2}}\mathrm{E}{{\big[Y^{(3)}_n \big]}}= O\left( \sqrt{ \frac{ \log h_n^{-1}}{nh_n^{d+2}}}\right).\end{equation*}

Thanks to Step I, Step II,and Lemma 3, we have shown that

(54) \begin{equation}\mathrm{E}[Z_n]= O\left( \sqrt{ \frac{ \log h_n^{-1}}{nh_n^{d+2}}} \text{ } {+} h_n \right).\end{equation}

5.4. Step III: Conclusion

Recall that the set of functions $\mathcal{F}$ is defined by (47). Since p is bounded on $\mathcal{M}$ , by (35) of Lemma 1, there exists $c>0$ such that for all $f \in \mathcal{F}$ , for all $x \in \mathcal{M}$ ,

\begin{align*}&\mathrm{E}\left[ K\left( \frac{ \|X-x\|}{h_n}\right)^2 ( f(X)-f(x))^2 \right] \\ &\quad\le \|K\|_{\infty} \, \mathrm{E}\left[ K\left( \frac{ \|X-x\|}{h_n}\right) \| X -x\|^2 \right]\stackrel{}{\le} {\|K\|_{\infty}} c h_n^{d+2}.\end{align*}

In other words,

\begin{equation*}\sup_{f \in \mathcal{F}} \sup_{x \in \mathcal{M}} \mathrm{E}\left[ K\left( \frac{ \|X-x\|}{h_n}\right)^2 ( f(X)-f(x))^2\right]\le \|K\|_{\infty} ch_n^{d+2}.\end{equation*}

Thus, by choosing $\sigma\,:\!=\, \sigma_n = \sqrt{ \|K\|_{\infty}ch_n^{d+2}}$ , and using Massart’s version of the Talagrand inequality (cf. Corollary 1) with the functions of the form $y\mapsto K{{\left(\frac{{{\left\| y-x\right\|}}_2}{h_n}\right)}}{{\left(f(y)-f(x)\right)}}$ , for all n sufficiently large and any positive number $t_n>0$ , with probability at least $1-\mathrm{e}^{-t_n}$ ,

(55) \begin{equation}\sup_{f \in \mathcal{F} }\sup_{y \in \mathcal{M}} nh_n^{d+2}{{| \mathcal{A}_{h_n,n}(f)(x) -\mathrm{E}{{[\mathcal{A}_{h_n,n}(f)(x)]}} |}} \le 9\big( nh_n^{d+2}\mathrm{E}[Z_n] +\sigma_n\sqrt{nt_n } +bt_n \big),\end{equation}

where, in this case, the constant envelope b is equal to

\begin{equation*}b\,:\!=\, \|K\|_\infty \mathrm{diam}{\mathcal{M}}.\end{equation*}

Choose $t_n=2 \log n$ . By the Borel–Cantelli lemma, with probability 1,

\begin{equation*}\sup_{f \in \mathcal{F} }\sup_{x \in \mathcal{M}} \big|\mathcal{A}_{h_n,n}(f)(x) -\mathrm{E}[\mathcal{A}_{h_n,n}(f)(x)]\big| =O\left( \sqrt{ \frac{ \log h_n^{-1}}{nh_n^{d+2}}} {+} h_n {+} \sqrt{ \frac{\log n}{n h_n^{d+2}}} \right).\end{equation*}

Furthermore, under assumption (5) on the sequence $(h_n)_{n\geqslant 1}$ , $\lim_{n \rightarrow +\infty} {nh_{n}^{d+2}}= +\infty$ , hence $\log h_n^{-1} = O(\log n)$ . Thus, with probability 1,

\begin{equation*}\sup_{f \in \mathcal{F} }\sup_{x \in \mathcal{M}} \big|\mathcal{A}_{h_n,n}(f)(x) -\mathrm{E}[\mathcal{A}_{h_n,n}(f)(x)]\big| =O\left( \sqrt{ \frac{ \log h_n^{-1}}{nh_n^{d+2}}} {+} h_n \right).\end{equation*}

This concludes the proof of Proposition 3. Hence, Theorem 1 is proved.

5.5. Proof of Corollary 1

Using the results of the above sections, we can now prove Corollary 1. First, we see that by the proofs of Propositions Proof of Proposition 1, Proof of Proposition 2, and (54), there is a constant $C>0$ such that for all $h>0$ , $n \in \mathbb{N}$ ,

(56) \begin{equation}\sup_{f \in \mathcal{F}}\sup_{x\in \mathcal{M}} {{\left| \mathrm{E}{{[\mathcal{A}_{h,n}(f)(x)]}}- \mathcal{A}(f)(x) \right|}} \le Ch \end{equation}

and

(57) \begin{equation}\mathrm{E}[Z_n] \le C\left(\sqrt{ \frac{\log h_n^{-1}}{nh_n^{d+2}}}+h_n \right). \end{equation}

Then, by choosing $t_n\,:\!=\, \delta^2 nh_n^{d+2}$ in (55) with

\begin{equation*}\delta \in \left[h_n\vee \sqrt{\frac{\log h_n^{-1}}{nh_n^{d+2}}},1\right],\end{equation*}

we know that, with probability at least $1 -\mathrm{e}^{-\delta^2 nh_n^{d+2}}$ ,

\begin{equation*} \sup_{f \in \mathcal{F} }\sup_{y \in \mathcal{M}} {{\left| \mathcal{A}_{h_n,n}(f)(x) -\mathrm{E}{{\left[\mathcal{A}_{h_n,n}(f)(x)\right]}} \right|}} \le \frac{9\left( nh_n^{d+2}\mathrm{E}[Z_n] +\sigma_n\sqrt{nt_n } +bt_n \right) }{nh_n^{d+2}}.\end{equation*}

In addition, by (57), we have

\begin{align*}\frac{ nh_n^{d+2}\mathrm{E}[Z_n] +\sigma_n\sqrt{nt_n } +bt_n }{nh_n^{d+2}} & \le C\left(\sqrt{ \frac{\log h_n^{-1}}{nh_n^{d+2}}}+h_n \right) + \frac{ \sigma_n\sqrt{nt_n } +bt_n }{nh_n^{d+2}}\\&= C\left(\sqrt{ \frac{\log h_n^{-1}}{nh_n^{d+2}}}+h_n \right)+\sqrt{\|K \|_{\infty}c}\delta+ \|K\|_{\infty} (\text{dim}\mathcal{M}) \delta^2\\& \le \big( 2C+ \sqrt{\|K \|_{\infty}c}+\|K\|_{\infty} (\text{dim}\mathcal{M}) \big) \delta .\end{align*}

In addition, from (56), we have

\begin{align*}\sup_{f \in \mathcal{F} }\sup_{x \in \mathcal{M}} {{\left| \mathcal{A}_{h_n,n}(f)(x) -\mathcal{A}(f)(x) \right|}} \le & \sup_{f \in \mathcal{F} }\sup_{x \in \mathcal{M}} {{\left| \mathcal{A}_{h_n,n}(f)(x) -\mathrm{E}[\mathcal{A}_{h_n,n}(f)(x)] \right|}} \\ & +\sup_{f \in \mathcal{F} }\sup_{x \in \mathcal{M}} {{\left| \mathrm{E}[\mathcal{A}_{h_n,n}(f)(x)] - \mathcal{A}(f)(x) \right|}}\\ \le & \sup_{f \in \mathcal{F} }\sup_{x \in \mathcal{M}} {{\left| \mathcal{A}_{h_n,n}(f)(x) -\mathrm{E}[\mathcal{A}_{h_n,n}(f)(x)] \right|}}+Ch_n\\ \le & \sup_{f \in \mathcal{F} }\sup_{x \in \mathcal{M}} {{\left| \mathcal{A}_{h_n,n}(f)(x) -\mathrm{E}[\mathcal{A}_{h_n,n}(f)(x)] \right|}}+C \delta .\end{align*}

Therefore, by letting

(58) \begin{equation}C'\,:\!=\, 9[2C+\sqrt{\|K \|_{\infty}c}+\|K\|_{\infty} (\text{dim}\mathcal{M})]+C,\end{equation}

where C is the constant appearing in (56) and (57), we have

\begin{align*}&\mathbf{P}\Big( \sup_{f \in \mathcal{F} }\sup_{x \in \mathcal{M}} {{\left| \mathcal{A}_{h_n,n}(f)(x) -\mathcal{A}(f)(x) \right|}} > C' \delta \Big)\\&\quad\le \mathbf{P} \Big( \sup_{f \in \mathcal{F} }\sup_{x \in \mathcal{M}} {{\left| \mathcal{A}_{h_n,n}(f)(x) -\mathrm{E}[\mathcal{A}_{h_n,n}(f)(x)] \right|}} > C'\delta- C\delta\Big)\\ &\quad = \mathbf{P} \Big( \sup_{f \in \mathcal{F} }\sup_{x \in \mathcal{M}} {{\left| \mathcal{A}_{h_n,n}(f)(x) -\mathrm{E}[\mathcal{A}_{h_n,n}(f)(x)] \right|}} >9\left( 2C+ \sqrt{\|K \|_{\infty}c}+\|K\|_{\infty} (\text{dim}\mathcal{M})\right) \delta \Big)\\ &\quad\le \exp{{\big( -\delta^2 n h_n^{d+2}\big)}}.\end{align*}

This proves Corollary 1.

7.1. Proof of Theorem 3

Let K be a kernel that satisfies Assumption 1. Let $n\geqslant 1$ be fixed. Recall that the generator $\mathcal{A}_{h_n,n}$ can be related to a random walk $X^{(n)}$ on the sample points ${{\left\{X_1,\ldots,X_n\right\}}}$ , which solves the SDE

\begin{equation*} X^{(n)}_t= X^{(n)}_0+\int_0^t \int_{\mathbb N} \int_{{\mathbb R}_+} \mathbf{1}_{i\leqslant n} \mathbf{1}_{\theta\leqslant \frac{1}{nh_n^{d+2}}K{{\big(\frac{\|X_i-X^{(n)}_{s_-}\|_2}{h_n}\big)}}} \big(X_i-X^{(n)}_{s_-}\big)\ Q(\mathrm{d} s,\mathrm{d} i,\mathrm{d} \theta)\end{equation*}

with initial condition $X^{(n)}_0$ and where $Q(\mathrm{d} s,\mathrm{d} i,\mathrm{d}\theta)$ is a Poisson point measure on ${\mathbb R}_+\times {\mathbb N}\times {\mathbb R}_+$ independent of $X_0^{(n)}$ , and of intensity $\mathrm{d} s\otimes \mathrm{n}(\mathrm{d} i)\otimes \mathrm{d}\theta$ , with $\mathrm{d} s$ and $\mathrm{d}\theta$ Lebesgue measures on ${\mathbb R}_+$ and $\mathrm{n}(\mathrm{d} i)$ the counting measure on ${\mathbb N}$ .

Proposition 6. For a fixed $n\geqslant 1$ , a random variable $X^{(n)}_0$ and a Poisson point measure $Q(\mathrm{d}s,\mathrm{d}i,\mathrm{d}\theta)$ , there exists a unique strong solution of the SDE (13).

For any real-valued measurable bounded function f on $\mathcal{M}$ , we have that

(68) \begin{equation} M^{n,f}_t= f\big(X^{(n)}_t\big) -f\big(X^{(n)}_0\big) -\int_0^t \mathcal{A}_{h_n,n}(f)\big(X^{(n)}_s\big)\,\mathrm{d} s\end{equation}

is a square-integrable martingale with predictable quadratic variation:

(69) \begin{equation} \langle M^{n,f}\rangle_t = \int_0^t \frac{1}{nh_n^{d+2}} \sum_{i=1}^n K{{\left(\frac{\|X_i-X^{(n)}_{s}\|_2}{h_n}\right)}}\big(f(X_i)-f\big(X^{(n)}_{s}\big)\big)^2\, \mathrm{d} s.\end{equation}

Let $T>0$ be a positive fixed time. Note that when we have Feller processes, the convergence of the generators implies the convergence of the stochastic processes (see, for example, [Reference Kallenberg23, Theorem 17.25]). However, without a continuity assumption on the kernel K, the Feller property may not be satisfied for our random walks and we follow a more classical line of proof. The proof of Theorem 3 is now divided into several steps. First, we prove that the sequence of processes $(X^{(n)})_{n\geqslant 0}$ is tight in the path space ${\mathbb D}([0,T],\mathcal{M})$ . By Prohorov’s theorem (e.g., [Reference Billingsley5]), the sequence is therefore sequentially relatively compact. The convergence of the generators $\mathcal{A}_{h_n,n}$ to $\mathcal{A}$ defined in (2) will yield that any limiting value is a solution of the martingale problem associated with $\mathcal{A}$ , which is well posed.

Proof. We check conditions (T1) and (T2) in Aldous’s criteria for tightness (see, for example, [Reference Aldous1, Reference Joffe and Métivier22]). Because $\mathcal{M}$ is compact, only (T2) needs to be considered. Thanks to the Markov inequality, it is sufficient to show that for any $\varepsilon>0$ , there exists $\delta>0$ such that for any couple of stopping times $(S_n,T_n)_{n\geqslant 0}$ satisfying $0 \leqslant S_n\leqslant T_n\leqslant (S_n+\delta)\wedge T$ , we have for all n sufficiently large that

(70) \begin{equation} \mathrm{P}{{\big(\rho\big(X^{(n)}_{S_n},X^{(n)}_{T_n}\big)>\varepsilon \big)}} \leqslant \varepsilon.\end{equation}

By the Markov inequality and Theorem 5, it is sufficient to study $\mathrm{E}{{\big[{{\big\| X^{(n)}_{S_n}-X^{(n)}_{T_n}\big\|}}^2_2\big]}}$ . We observe that

\begin{align}&\mathrm{E}{{\Big[{{\Big\| X^{(n)}_{S_n}-X^{(n)}_{T_n}\Big\|}}^2_2\Big]}} \leqslant \mathrm{E}{{\left[\int_{S_n}^{T_n} \sum_{i=1}^n \frac{1}{nh_n^{d+2}}K{{\left(\frac{\|X_i-X^{(n)}_{s}\|_2}{h_n}\right)}} {{\Big\| X_i-X^{(n)}_{s}\Big\|}}^2_2 \mathrm{d} s\right]}}\nonumber\\&\quad\leqslant \delta\,\mathrm{E}{{\left[\sup_{x\in\mathcal{M}} \sum_{i=1}^n \frac{1}{nh_n^{d+2}}K{{\left(\frac{\|X_i-x\|_2}{h_n}\right)}} {{\left\| X_i-x\right\|}}^2_2 \right]}}\nonumber\\&\quad\leqslant \delta\,\sup_{x\in\mathcal{M}} \frac{1}{h_n^{d+2}} \int K{{\left(\frac{\|x-y\|_2}{h_n}\right)}} {{\left\| x-y\right\|}}^2_2 p(y)\mu(\mathrm{d} y)\nonumber\\ &\qquad +\delta\frac{1}{h_n^{d+2}}\mathrm{E}{{\left[\sup_{x\in\mathcal{M}} {{\left| \frac{1}{n}\sum_{i=1}^n K{{\left(\frac{\|X_i-x\|_2}{h_n}\right)}} {{\left\| X_i-x\right\|}}^2_2 -\mathrm{E}{{\left[K{{\left(\frac{\|X_1-x\|_2}{h_n}\right)}} {{\left\| X_1-x\right\|}}^2_2 \right]}} \right|}}\right]}}\nonumber.\end{align}

Since p is bounded on the compact $\mathcal{M}$ by continuity, using (35) of Lemma 1, and using the estimate of $Y^{(\alpha,\beta)}_n$ in Section 5.3.2 based on the Vapnik–Chervonenkis theory, we deduce that there is a constant $C>0$ , which does not depend on n nor $\varepsilon$ , such that

(71) \begin{equation} \mathrm{E}{{\Big[{{\Big\| X^{(n)}_{S_n}-X^{(n)}_{T_n}\Big\|}}^2_2\Big]}}\leqslant C \delta.\end{equation}

Choosing $\delta=\varepsilon^2/C$ yields (70).

Consider now a limiting value $Y\in \mathbb{D}([0,T],\mathcal{M})$ of the tight sequence $(X^{(n)})_{n\geqslant 0}$ . There exists a subsequence converging in distribution to Y. Using the Skorokhod representation theorem (see [Reference Billingsley6, Theorem 29.6 p. 399]), we can assume that this convergence holds almost surely, and with an abuse of notation we write $(X^{(n)})_{n\geqslant 0}$ for the subsequence converging to Y.

Proof. By assumption, $Z_0$ has the distribution $\nu$ , so it is sufficient to show that for all $0 \le t_0 < t_1 < t_2 < \cdots < t_k \leqslant s < t$ and functions $g_1,g_2,g_2,\ldots,g_k,g \in \mathcal{C}(\mathcal{M}), f \in \mathcal{C}^3(\mathcal{M})$ , we have

\begin{equation*}\mathrm{E}\left[ \left( \prod_{i=0}^k g_i(Y_{t_i})\right)\left( f(Y_t)-f(Y_s)-\int_{s}^t \mathcal{A}f(Y_u)\,\mathrm{d}u \right)\right]=0.\end{equation*}

Let us denote by $\Psi$ the application

\begin{equation*}\Psi\ :\ y\in \mathbb{D}([0,T],\mathcal{M}) \mapsto \left(\prod_{i=0}^k g_i\big(y_{t_i}\big) \right) \left( f(y_{t})-f(y_{s})-\int_{s}^t \mathcal{A}f(y_{u})\,\mathrm{d}u \right).\end{equation*}

We have

\begin{align*} \left| \mathrm{E}\big[ \Psi\big(X^{(n)}\big) \big]\right| & \leqslant \left| \mathrm{E}\Bigg[ \left(\prod_{i=0}^k g_i\big(X^{(n)}_{t_i}\big) \right) \left( f\big(X^{(n)}_{t}\big)-f\big(X^{(n)}_{s}\big)-\int_{s}^t \mathcal{A}_{h_n,n}f(X^{(n)}_{u})\,\mathrm{d}u \right)\Bigg]\right| \\ & \quad + \left| \mathrm{E}\Bigg[ \left(\prod_{i=0}^k g_i\big(X^{(n)}_{t_i}\big) \right) \ \int_{s}^t \big(\mathcal{A}_{h_n,n}f(X^{(n)}_{u})-\mathcal{A} f(X^{(n)}_u)\,\mathrm{d}u \Bigg]\right|\\ & = O\left( (t-s) \sqrt{ \frac{ \log h_n^{-1}}{nh_n^{d+2}}} + h_n \right) \end{align*}

by Proposition 6 and Theorem 1. The application $\Psi$ is not continuous on $\mathbb{D}([0,T],\mathcal{M})$ in general, but since the limiting process is continuous almost surely, we have that $\Psi(X^{(n)})$ converges to $\Psi(Y)$ almost surely. We can then conclude the proof as $\mathrm{E}[\Psi(Y)]=0$ by using the dominated convergence theorem.

Proof of Theorem 3. From Lemmas Proof of Lemma 4 and Proof of Lemma 5, the limiting processes are all solutions of the same martingale problem associated with $(\mathcal{A},\nu)$ . The well-posedness of the latter martingale problem is a consequence of Theorem 1.2.9 in [Reference Hsu19]. Hence the limiting processes all have the same distribution and the sequence $(X^{(n)})_{n\geqslant 0}$ converges in distribution to the limit stated in Theorem 3.

7.2. Convergence of the kNN random walk

We prove Theorem 4. For the sake of notation, the random walk $X^{(n),k\mathrm{NN}}$ is now denoted by $X^{(n)}$ . Let us recall its SDE:

\begin{equation*} X^{(n)}_t= X^{(n)}_0+\int_0^t \int_{\mathbb N} \int_{{\mathbb R}_+} \mathbf{1}_{i\leqslant n} \mathbf{1}_{\theta\leqslant \frac{1}{nR_{n,k_n}^{d+2}(X^{(n)}_{s_-})}} \mathbf{1}_{[0,1]}{{\left(\frac{\|X_i-X^{(n)}_{s_-}\|_2}{R_{n,k_n}{{\big(X_{s^-}^{(n)}\big)}}}\right)}}\big(X_i-X^{(n)}_{s_-}\big)\ Q(\mathrm{d} s,\mathrm{d} i,\mathrm{d} \theta),\end{equation*}

with

\begin{equation*} R_{n,k}(x)=\inf\bigg\{ r\geqslant 0,\ \sum_{i=1}^n \mathbf{1}_{\|x-X_i\|_2\leqslant r}\geqslant k\bigg\}.\end{equation*}

The related martingale is thus

\begin{equation*} M^{n,f}_t= f\big(X^{(n)}_t\big) -f\big(X^{(n)}_0\big) -\int_0^t \mathcal{A}_{n}^{k\mathrm{NN}}(f)(X^{(n)}_s)\,\mathrm{d} s\end{equation*}

with predictable quadratic variation

\begin{equation*} \langle M^{n,f}\rangle_t = \int_0^t \sum_{i=1}^n \mathbf{1}_{[0,1]}{{\left(\frac{\|X_i-X^{(n)}_{s}\|_2}{R_{n,k_n}{{\big(X_s^{(n)}\big)}}}\right)}}\big(f(X_i)-f\big(X^{(n)}_{s}\big)\big)^2\,\mathrm{d} s.\end{equation*}

As in the proof of Lemma 4, to obtain the tightness of the distribution, thanks to Aldous’s criterion, it is sufficient to show that for any $\varepsilon>0$ , there exists $\delta>0$ such that for any couple of stopping times $(S_n,T_n)_{n\geqslant 0}$ satisfying $0\leqslant S_n\leqslant T_n\leqslant (S_n+\delta)\wedge T$ , we have for all n sufficiently large that

\begin{equation*} \mathrm{P}{{\big(\rho\big(X^{(n)}_{S_n},X^{(n)}_{T_n}\big)>\varepsilon \big)}} \leqslant \varepsilon.\end{equation*}

We observe that, using the quantities $h_n(x)$ and $\gamma_n$ defined by (62) in Section 6,

\begin{align*} \mathrm{P}{{\left(\rho\big(X^{(n)}_{S_n},X^{(n)}_{T_n}\big)>\varepsilon \right)}} \leqslant& \mathrm{P}{{\Big(\rho\big(X^{(n)}_{S_n},X^{(n)}_{T_n}\big)>\varepsilon, \sup_{x\in \mathcal{M}}{{\left| R_{n,k_n}{{\left(x\right)}}-h_n(x) \right|}}\leqslant\gamma_n \Big)}} \\ & +\mathrm{P}{{\Big( \sup_{x\in \mathcal{M}}{{\left| R_{n,k_n}{{\left(x\right)}}-h_n(x) \right|}}>\gamma_n \Big)}} \\ \leqslant & \frac{1}{\varepsilon^2}\mathrm{E}{{\left({{\left\| X^{(n)}_{S_n}-X^{(n)}_{T_n}\right\|}}^2_2 \mathbf{1}_{\sup_{x\in \mathcal{M}}{{\left| R_{n,k_n}{{\left(x\right)}}-h_n(x) \right|}}\leqslant\gamma_n }\right)}} +n^{-10}.\end{align*}

Introducing $(a_n,b_n)$ given by (65), we have $[h_n(x)-\gamma_n, \ h_n(x)+\gamma_n]\subset [a_n,b_n]$ for all $x\in \mathcal{M}$ . Thus, on the event ${{\left\{\sup_{x\in \mathcal{M}}{{\left| R_{n,k_n}{{\left(x\right)}}-h_n(x) \right|}}\leqslant\gamma_n \right\}}}$ ,

\begin{multline*}{{\left\| X^{(n)}_{T_n}-X^{(n)}_{S_n}\right\|}}_2\\\begin{aligned} \leqslant & \int_{S_n}^{T_n} \int_{\mathbb N} \int_{{\mathbb R}_+} \mathbf{1}_{i\leqslant n} \mathbf{1}_{\theta\leqslant \frac{1}{nR_{n,k_n}^{d+2}(X_{s_-}^{(n)})}} \mathbf{1}_{[0,1]}{{\left(\frac{\|X_i-X^{(n)}_{s_-}\|_2}{R_{n,k_n}{{\left(X_{s^-}^{(n)}\right)}}}\right)}}{{\big\| X_i-X^{(n)}_{s_-}\big\|}}_2\ Q(\mathrm{d} s,\mathrm{d} i,\mathrm{d} \theta)\\ \leqslant &\int_{S_n}^{T_n}\int_{\mathbb N} \int_{{\mathbb R}_+} \mathbf{1}_{i\leqslant n} \mathbf{1}_{\theta\leqslant \frac{1}{na_n^{d+2}}} \mathbf{1}_{\|X_i-X^{(n)}_{s_-}\|_2\leqslant b_n}{{\big\| X_i-X^{(n)}_{s_-}\big\|}}_2\ Q(\mathrm{d} s,\mathrm{d} i,\mathrm{d} \theta).\end{aligned}\end{multline*}

We deduce that

\begin{multline*}\mathrm{E}{{\left[{{\left\| X^{(n)}_{S_n}-X^{(n)}_{T_n}\right\|}}^2_2 \, \mathbf{1}_{\sup_{x\in \mathcal{M}}{{\left| R_{n,k_n}{{\left(x\right)}}-h_n(x) \right|}}\leqslant\gamma_n }\right]}} \\\begin{aligned}\leqslant & \mathrm{E}{{\left[\int_{S_n}^{T_n} \sum_{i=1}^n \frac{1}{na_n^{d+2}} \mathbf{1}_{\|X_i-X^{(n)}_{s_-}\|_2\leqslant b_n} {{\big\| X_i-X^{(n)}_{s}\big\|}}^2_2 \mathrm{d} s\right]}}\\\leqslant& \frac{\delta}{na_n^{d+2}}\,\mathrm{E}{{\left[\sup_{x\in\mathcal{M}} \sum_{i=1}^n \mathbf{1}_{\|X_i-x\|_2\leqslant b_n} {{\left\| X_i-x\right\|}}^2_2 \right]}}\\\leqslant & \frac{\delta}{a_n^{d+2}} \int_{\mathcal{M}} \mathbf{1}_{\|x-y\|_2\leqslant b_n} {{\left\| x-y\right\|}}^2_2\ p(y) \mu(\mathrm{d} y)\\ & + \frac{\delta}{a_n^{d+2}}\,\mathrm{E}{{\left[\sup_{x\in\mathcal{M}} {{\left| \frac{1}{n}\sum_{i=1}^n \mathbf{1}_{\|X_i-x\|_2\leqslant b_n} {{\left\| X_i-x\right\|}}^2_2 - \mathrm{E}{{\big[ \mathbf{1}_{\|X_1-x\|_2\leqslant b_n} {{\left\| X_1-x\right\|}}_2^2\big]}} \right|}}\right]}}.\end{aligned}\end{multline*}

As in the proof of Lemma 4, we use for the first term on the right-hand side that p is bounded on the compact $\mathcal{M}$ , together with (35) of Lemma 1. For the second term we use the estimate of $\tilde Y^{(\alpha,\beta)}_n$ in Section 6 based on the Vapnik–Chervonenkis theory, and the fact that $[a_n,b_n]\subset [\kappa^{-1} h_n,\kappa h_n]$ . We deduce that there is a constant $C>0$ , which does not depend on n or $\varepsilon$ , such that

(72) \begin{equation} \mathrm{E}{{\big[{{\big\| X^{(n)}_{S_n}-X^{(n)}_{T_n}\big\|}}^2_2\big]}}\leqslant C \delta.\end{equation}

This shows that the sequence of random walks is tight. The identification of the limiting martingale problem follows the proof of Theorem 3 using Theorem 2.