2.1. Positive reach

The notion of positive reach is foundational in convex geometry. As indicated in the introduction, the reach of a subset M is defined to be

(2.2) \begin{equation}\tau (M) \;:\!=\; \hbox{sup}\{r\geq 0\ |\ \forall y\in M^{\oplus r}\,\, \exists !\ x\in M\ \hbox{nearest to }{y}\}.\end{equation}

A PR set is any set M with $\tau (M)> 0$ . Compact submanifolds are PR. Figure 1 gives an example of a PR set that is not a submanifold. The quintessential property of PR sets is the existence, for $0<r<\tau$ , of the ‘unique closest point’ projection

(2.3) \begin{equation}\pi_M \;:\; M^{\oplus r}\longrightarrow M,\ \||y - \pi_M(y)|| = d_H(y,M),\end{equation}

with $\pi_M(y)$ the unique nearest point to y in M. PR sets are necessarily closed, and thus compact if bounded.

Figure 1.

This space has positive reach $\tau$ in ${\mathbb R}^2$ , but a neighborhood of point A indicates it is not a submanifold (with boundary).

As already indicated, we use the notation ${Y\overset{\simeq}{\longrightarrow}X}$ , if $Y\subset X$ , to denote the fact that X deformation retracts onto Y. ‘Thin-enough’ offsets of PR sets deformation retract onto M.

The original interest in sets of positive reach lies in the fact that they have suitable small parallel neighborhoods with no self-intersections which allow one to compute their volume. This leads to a Steiner-type formula and a definition of curvature measures for these sets (see [Reference Thale38]). If M is a compact Riemannian submanifold in ${\mathbb R}^d$ , as considered in [Reference Niyogi, Smale and Weinberger33], then $\tau (M)$ is positive and is the largest number having the property that the open normal bundle about M of radius r is smoothly embedded in ${\mathbb R}^d$ for every $r < \tau$ . It is enough, however, for M to be $C^2$ to ensure that $\tau (M)>0$ (see [Reference Thale38, Proposition 14]), and it is even enough for it to be $C^{1,1}$ in the case that M is a closed hypersurface (see [Reference Scholtes36, Theorem 1.3], which is an if-and-only-if statement). We recall the definition of $C^{1,1}$ (see [Reference Hörmander24, Definition 2.4.2]).

Crucial to us in this section are the next two results. For general discussion, we refer to [Reference Federer17, Reference Fu19] for Proposition 2.1, and [Reference Barb4, Reference Scholtes36] for Proposition 2.2. Throughout, a manifold is assumed to be compact, and without boundary unless we specify the contrary. For $x\in{\mathbb R}^d$ , let $d_M(x )=d(x , M ) = \hbox{inf} \{d(x,y), y \in M\}$ be the distance function to M. This function is 1-Lipschitz, and it is continuously differentiable when restricted to the interior of $M^{\oplus r}\setminus M$ if $r<\tau$ (see [Reference Federer17, Theorem 4.8]). Elementary point-set topology shows that the interior of the r-offset of M is $\hbox{int}(M^{\oplus r}) = \bigcup_{x\in M}\mathring{B}(x,r) = d^{-1}_M[0,r)$ , and the topological boundary is $d_M^{-1}(r)$ .

We next describe tubular neighborhoods of a $C^{1,1}$ -submanifold.

If M is a compact PR set, its offset $M^{\oplus r}$ is also compact and PR for $r<\tau$ , with reach $\tau-r$ , where $\tau$ is the reach of M. This assertion is not entirely obvious, since, in general, the reach is not always well-behaved for nested compact sets. By this we mean that if $(K_2,K_1)$ is a pair of nested compact sets in ${\mathbb R}^d$ , $K_1\subset K_2$ , then both cases $\tau_1<\tau_2$ or $\tau_2<\tau_1$ can occur, where $\tau_i$ is the reach of $K_i$ . As an example of the former case, take $K_1$ to be the circle and $K_2$ to be the closed disk; for the latter case, take $K_1$ to be a point in a finite-reach $K_2$ . The case of $(K_2,K_1) = (M^{\oplus r}, M)$ is therefore special.

Proof. Essentially, the point is that any ray from $y\not\in M^{\oplus r}$ to M must cut the boundary $\partial M^{\oplus r}$ at a point lying at a distance of r to M. Suppose $r>0$ , so that $M^{\oplus r}$ is a codimension-0 manifold with boundary $\partial M^{\oplus r}$ in ${\mathbb R}^d$ . Write $\tau_r$ for its reach. We will first prove that if y in the complement of $M^{\oplus r}$ has a unique projection onto M, then necessarily it has a unique projection onto $M^{\oplus r}$ (this will prove that $\tau_r>0$ and that $\tau - r\leq \tau_r$ ). Reciprocally, we will argue that if y has a unique projection onto M, then it also has a unique projection onto $M^{\oplus r}$ .

To prove the first claim, write $y_1 = [y,\pi_M(y)]\cap\partial M^{\oplus r}$ . We claim that $y_1$ is the unique closest point to y in $M^{\oplus r}$ . Indeed, if there is $z_1$ on that boundary that is closer to y, then

\begin{align*}d(y,\pi_M(z_1))\leq d(y,z_1) + d(z_1,\pi_M(z_1)) = d(y,z_1)+r\leq d(y,y_1) + d(y_1,M)= d(y,\pi_M(y)),\end{align*}

and so $d(y,\pi_M(z_1))=d(y,\pi_M(y))$ (since $d(y,\pi_M(y))$ is smallest distance from y to M), and by uniqueness, $\pi_M(z_1)=\pi_M(y)$ . This implies that $d(y,z_1) + d(z_1,M)=d(y,M)$ , and so $y,z_1,\pi_M(y)$ are aligned. This can only happen if $y_1=z_1$ .

Suppose now that y has a unique projection onto $M^{\oplus r}$ , which we label y’. We can check that it also has a unique projection onto M. Let z be that projection. By a similar argument as previously, z must be $\pi_M(y')$ (so unique) and y, y’,z are aligned. This shows reciprocally that $\tau_r+r\leq \tau$ .

The above arguments show that $\tau = \tau_r+r$ , and in fact they can be used to show in this case that $M^{\oplus r'}= (M^{\oplus r})^{\oplus (r'-r)} $ for all $r\leq r'< \tau$ .

2.2. Manifolds with boundary

In order to apply our ideas to PR sets, we need to extend Theorem 2.1 from closed Riemannian submanifolds to submanifolds with boundary. Note that the reach of $\partial M$ (manifold boundary) and M are not comparable in general. Indeed, take M to be the $y=\sin (x)$ curve on $[0,\pi]$ , with boundary the endpoints. Then $\tau (M) < \tau (\partial M)$ . Take now a closed disk M in ${\mathbb R}^2$ . Then $\tau (\partial M)<\tau (M)=\infty$ . If M is of codimension 0, then $\tau (\partial M)\leq\tau (M)$ always.

In [Reference Wang and Wang39], the authors managed to extend Theorem 2.1 to smooth submanifolds with boundary and showed that in this case the bound $\sqrt\frac{3}{5}\tau$ in Theorem 2.1 can be replaced by $\frac{\delta}{2}$ , where $\delta = \min (\tau (M), \tau (\partial M))$ . We revisit this result in the codimension-0 case and establish the following ‘twice the density, half the reach’ criterion.

Notice that M need not be connected. Notice also that we have weakened the regularity on $\partial M$ from smooth to $C^{1,1}$ . According to [Reference Scholtes36, Theorem 1.3] (see also [Reference Federer17, Remark 4.20]), this condition is enough to ensure that $\tau (\partial M)>0$ . Finally, notice that we use closed balls in our statement, and that they may have radius larger than $\epsilon$ , but not exceeding $\frac{\tau}{2}$ .

Proof. The proof is an adaptation of Lemma 4.1 of [Reference Niyogi, Smale and Weinberger33] and Lemma 4.3 of [Reference Wang and Wang39] for smooth submanifolds. For completeness, we will reconstruct the part of the argument that we need.

Firstly, since the reach of a disjoint finite union of PR sets is the least of their reaches and their pairwise distances, we can assume without loss of generality that M is connected from the start. Let M be connected of codimension 0. Then its boundary is a connected codimension-1 closed submanifold (i.e. a closed hypersurface). It divides Euclidean space into two regions: M and its complement. We let $\tau^-$ denote the reach of a component of $\partial M$ in M (the interior region), and $\tau^+$ its reach within the open (exterior) region. Clearly $\tau\;:\!=\;\tau (M)=\tau^+$ .

Now $\partial M$ is $C^{1,1}$ by hypothesis, and being a closed hypersurface, it is necessarily orientable. It has a continuous normal vector field into the exterior region, defining a (trivial) ${\mathbb R}_+$ -bundle $T(\partial M)^+$ of $T(\partial M)$ . We write $T_p^{\perp,+}(\partial M)$ for the fiber at $p\in\partial M$ , which is a half-line extending into the exterior region, perpendicular to $T_p(\partial M)$ . Linear deformation retraction along this direction as in Lemma 2.2, keeping M fixed, shows that ${M\overset{\simeq}{\longrightarrow}M^{\oplus r}}$ as long as $r<\tau^+=\tau $ (where normal directions never intersect). We have that

\begin{align*}M^{\oplus r} = \bigcup_{x\in M}B(x,r)\simeq M,\ r<\tau. \end{align*}

We want to show that this retraction of $M^{\oplus r}$ onto M (along fibers of $T^+(\partial M)$ ) restricts to a deformation retraction onto M of the middle space $S^{\oplus r}$ in the sequence of inclusions below:

\begin{align*}M\subset S^{\oplus r}=\bigcup_{x\in S}B(x,r)\subset M^{\oplus r},\ {\epsilon}\leq r<\frac{\tau}{2}. \end{align*}

That is, we only take the union of balls centered at points of S. This covers M since $r\geq \frac{\epsilon}{2}$ and any point of M is within distance $\epsilon/2$ of S. Let us see how the deformation retraction of the bigger space $M^{\oplus r}$ onto M may fail to restrict to a retraction on $S^{\oplus r}$ : let $v\in T_p^{\perp,+}(\partial M)$ , and suppose $v\in B(q,r)$ , with $q\in S$ but $q\not\in B(p,r)$ . So the line segment [v, p] is not in the ball B(q, r), and the linear retraction will leave that ball eventually. This, however, will not cause a problem as long as the segment falls in another ball and does not leave the entire union $\bigcup_{x\in S}B(x,r)$ . This happens if both v and p are in some other ball B(x, r), $x\in S$ (because balls are convex). By the density condition, we can also demand that x be at a distance of at most $\frac{\epsilon}{2}$ from p. To recapitulate, for every such p, v, by picking an $x\in S$ within a distance of $\frac{\epsilon}{2}$ from p and a distance of r from v, we guarantee that the deformation retraction of $M^{\oplus r}$ restricts to $S^{\oplus r}$ (see Figure 2).

Figure 2.

$M\subset{\mathbb R}^d$ is represented by the shaded area, $p,q\in\partial M$ , and $x,q\in S$ . The points q, p are on a circle tangent to $T_p(M)$ , having radius $\tau$ and center on the vertical dashed line representing the normal direction $T_p^{\perp,+}(M)$ , pointing into the exterior region, while x is anywhere in $M\cap S$ at a distance of at most $\frac{\epsilon}{2}$ from p. An extreme disposition of such points (meaning when v is as far as possible from x) happens when v, p, x are aligned. This figure is the analog of Figure 2 of [Reference Niyogi, Smale and Weinberger33] and Figure 1 of [Reference Wang and Wang39].

With the above target in mind, consider the following configuration of points: $p\in\partial M$ , $v\in T_p^{\perp,+}(\partial M)\cap B(p,\tau)$ , $q\in S$ , and $v\in B(q,r)$ but $p\not\in B(q,r)$ . How far can v be from p, among all choices of such points q? The answer can be extracted from the key Lemma 4.1 of [Reference Niyogi, Smale and Weinberger33]. The worst-case scenario, corresponding to when v would be farthest from p, is when q and p lie on the circle of radius $\tau$ , with center in $T_p^{\perp,+}$ as in Figure 2, and p,q,v make up an isosceles triangle with $|p-q|=|q-v|=r$ . Lemma 4.1 of [Reference Niyogi, Smale and Weinberger33] applied to this situation gives that $\displaystyle d(v,p)< \frac{r^2}{\tau^+} = \frac{r^2}{\tau} <\tau$ .

Next, we look for an $x\in B(p,\frac{\epsilon}{2})\cap S\neq\emptyset$ which is within distance r of v. By the triangle inequality, $\displaystyle d(x,v)\leq d(x,p)+d(p,v)\leq \frac{\epsilon}{2} + \frac{r^2}{\tau}$ , and so in order for $v\in B(x,r)$ , it is enough to require that

(2.4) \begin{equation} \frac{\epsilon}{2} + {r^2}{\tau}< r\ \Longleftrightarrow\ r^2-r\tau + \frac{\epsilon\tau}{2}<0.\end{equation}

Any value of r between the roots of the polynomial on the right-hand side does the job, and it is immediate that $r\in \left[\epsilon, \frac{\tau}{2}\right[$ satisfies this condition. The proof is complete.

Note that in the case of Theorem 2.1 in [Reference Niyogi, Smale and Weinberger33], that is, when one is considering M that is closed (i.e. with no boundary), the point x to be chosen cannot simply be ‘anywhere’ around $p\in M$ , as in Figure 2, but must lie on M (which would be the boundary in that figure), and thus the authors get a different bound on r.

We finally come to the proof of the main reconstruction result of this section, which yields Theorem 1.2 as a consequence. We thank the referee for suggesting this statement, which is simpler than the one we originally gave.

Proof. Notice that by Proposition 2.3, the result is true if M is a codimension-0 submanifold whose boundary is $C^{1,1}$ of reach $\tau>0$ (let us refer to this submanifold as a ‘good object’). The idea now is very simple and relies on the fact that, if M itself is not such a good object but is of positive reach, then any slight thickening of it will be a good object.

Assume that S is $\epsilon/2$ -dense in M, with $\epsilon < \frac{\tau}{2}$ . We first collect the following facts:

1. 1. For $0<\delta<\tau$ , $M^{\oplus\delta}$ deformation retracts onto M (Lemma 2.2).

2. 2. For $0<\delta<\tau$ , S is $\left(\frac{\epsilon}{2}+\delta\right)$ -dense in $M^{\oplus\delta}\supset M$ .

3. 3. The offset $M^{\oplus\delta}$ is a codimension-0 submanifold of ${\mathbb R}^d$ , with $C^{1,1}$ boundary (Proposition 2.1). Its reach is $\tau'=\tau-\delta$ (Lemma 2.3).

Assume that $\displaystyle\delta < \frac{\tau-2\epsilon}{5}$ . This is equivalent to $\epsilon + 2\delta < \frac{\tau - \delta}{2}$ ; that is, twice the density of S in $M^{\oplus\delta}$ is less than half the reach of $M^{\oplus\delta}$ . By Proposition 2.3, for all r that we can insert between these two numbers, we get a homotopy reconstruction of $M^{\oplus\delta}$ ; more precisely,

(2.5) \begin{equation}\displaystyle\epsilon+2\delta\leq r< \frac{\tau-\delta}{2}\ \Longrightarrow\ \bigcup_{x\in S }B(x,r)\simeq M^{\oplus\delta}.\end{equation}

Returning to the hypotheses of Theorem 2.2, choose any r such that $\epsilon< r<\frac{\tau}{2}$ . Pick $\delta$ such that $\displaystyle 0<\delta < \min\left\{\frac{r - \epsilon}{2}, \tau -2r\right\}$ . For this $\delta$ , $\epsilon+2\delta\leq r$ and $r< \frac{\tau-\delta}{2}$ , and thus by (2.5), $\bigcup_{x\in S }B(x,r)$ deformation retracts onto $M^{\oplus\delta}$ . Since the latter deformation retracts onto M, the composition of both retractions shows that ${M\overset{\simeq}{\longrightarrow}\bigcup_{x\in S }B(x,r)}$ . The proof is complete.

.

4.0.1. Stationary m-dependent sequence on a compact set.

Recall that the sequence is m-dependent for some $m\geq 0$ if the two $\sigma$ -fields $\sigma (X_i,\,\,i\leq k)$ and $\sigma (X_{i}, i\geq k+m+1)$ are independent for every k. In particular, 0-dependent is the same as independent.

Example 4.1. (An m-dependent sequence.) Let $(T_i)_{i\in \mathrm{I\!N}}$ be a sequence of i.i.d. random variables with values in $\mathrm{I\!R}^d$ . Let h be a real-valued function defined on ${\mathrm{I\!R}^{dm}}$ . The stationary sequence $(X_n)_{n\in \mathrm{I\!N}}$ defined by $X_n=h(T_n,T_{n+1},\cdots,T_{n+m})$ is a stationary sequence of m-dependent random variables.

For $m\in \mathrm{I\!N}\setminus\{0\}$ , $\epsilon>0$ , and $Y_{1,m}=(X_1,\cdots, X_m)^t$ , as in the introduction, define the concentration coefficient of the vector $Y_{1,m}$ by

(4.1) \begin{equation}\rho_{m}(\epsilon)=\inf_{x\in {\mathrm{I\!M}}_{dm}}\mathrm{I\!P}\!\left(\|Y_{1,m}-x\|\leq \epsilon\right).\end{equation}

The following proposition gives conditions on $\rho_m(\epsilon)$ under which the asymptotically dense property evoked in Definition 1.1 is satisfied.

Proposition 4.1. Let be a stationary sequence of m-dependent $\mathrm{I\!R}^d$ -valued random vectors. Suppose that $X_1$ has compact support $\mathrm{I\!M}$ . Let $\epsilon_0>0$ be fixed. Suppose that for any $0<\epsilon<\epsilon_0$ , there exists a strictly positive constant $\kappa_{\epsilon}$ such that

\begin{align*}\rho_{m+1}(\epsilon) \geq \kappa_{\epsilon}.\end{align*}

Then it holds for any $0<\epsilon<\epsilon_0$ and any $n\geq m+1$ that

\begin{eqnarray*}&&\mathrm{I\!P}\!\left(d_H({\mathbb X}_n , {\mathrm{I\!M}}) > \epsilon\right)\leq \frac{(1- \kappa_{\frac{\epsilon}{2}})^{[\frac{1}{2}[\frac{n}{m+1}]]}}{\kappa_{\frac{\epsilon}{4}}},\end{eqnarray*}

where $[\cdot]$ denotes the integer part. Consequently, for any $\alpha\in ]0,1[$ and any $n\geq n_0(\epsilon,\alpha)$ , where

\begin{align*}n_0(\epsilon,\alpha)= \frac{2(m+1)}{\kappa_{\frac{\epsilon}{2}}} \left(\log\left(\frac{1}{\alpha }\right)+ \log\left(\frac{1}{ \kappa_{\frac{\epsilon}{4}} } \right)\right)+ 3(m+1),\end{align*}

we have $d_H({\mathbb X}_n , {\mathrm{I\!M}}) \leq \epsilon$ with probability at least $1-\alpha$ .

The requirements of Proposition 4.1 prove that the sequence is asymptotically dense in $\mathrm{I\!M}$ with threshold $n_0(\epsilon,\alpha)$ as above.

4.0.2. Stationary m-approximable random variables on a compact set.

In this section we discuss, in the spirit of [Reference Hörmann and Kokoszka25], some examples of stationary compactly supported random variables $(X_n)_{n\in {\mathbb Z}}$ that can be approximated by m-dependent stationary sequences. More precisely, the article [Reference Hörmann and Kokoszka25] introduced the notion of an $L^p$ -m-approximable sequence. This notion is related to m-dependence (see [Reference Hörmann and Kokoszka25, Definition 2.1]) and is different from mixing (see Paragraph 4.0.3 below for a definition of mixing). The idea is to construct, for $m\in \mathrm{I\!N}$ , a stationary sequence $(X_n^{(m)})_{n\in {\mathbb Z}}$ that is m-dependent and compactly supported, for which the Hausdorff distance between the two sets ${{\mathbb X}}_n$ and ${{{\mathbb X}}_n^{(m)}}\;:\!=\;\{X_1^{(m)},\cdots, X_n^{(m)}\}$ is suitably controlled. In our case, it is not necessary that $X_1$ and $X_1^{(m)}$ have the same distribution; rather, what we need is that $X_1$ and $X_1^{m}$ have the same compact support. This will give us more choices for the construction of the sequence $(X_n^{(m)})_{n\in {\mathbb Z}}$ , which can be obtained by the method of coupling or by a truncation argument (see [Reference Hörmann and Kokoszka25] for more details). For our purpose, we shall use a truncation.

More precisely, we will consider the sequence

(4.2) \begin{equation}X_n=f(\epsilon_n,\epsilon_{n-1},\cdots),\end{equation}

where $(\epsilon_i)_{i\in {\mathbb Z}}$ is an i.i.d. sequence with values in some measurable space S and f is a real bounded function defined on $S^{\infty}$ . The sequence $(X_n^{(m)})_{n\in {\mathbb Z}}$ constructed from $(X_n)_{n\in {\mathbb Z}}$ by truncation is

(4.3) \begin{equation}X_n^{(m)}=f(\epsilon_n,\cdots,\epsilon_{n-m},0,\cdots).\end{equation}

Clearly $(X_n^{(m)})_{n\in {\mathbb Z}}$ is a stationary, m-dependent sequence and has the same compact support as $(X_n)_{n\in {\mathbb Z}}$ as soon as f is bounded. We will thus assume that f is bounded; that is, $\|f\|_{\infty}=\sup_{x\in S^{\infty}}|f(x)|<\infty.$ As before, $\mathrm{I\!M}$ will be the common support of these two sequences.

Now we need an additional assumption on f in order to ensure good control of the Hausdorff distance between the two sets ${{\mathbb X}}_n$ and ${{{\mathbb X}}_n^{(m)}}$ . We suppose that f is a real-valued bounded function and that it satisfies the following Lipschitz-type assumption (stated in [Reference Hörmann and Kokoszka25]): there exists a decreasing sequence $(c_m)_{m\in \mathrm{I\!N}}$ , tending to 0 as m tends to infinity, such that

(4.4) \begin{equation}|f(a_{m+1},\cdots,a_1,x_0,\cdots)-f(a_{m+1},\cdots,a_1,y_0,\cdots)| \leq c_m |f(x_0,x_{-1},\cdots)-f(y_0,y_{-1},\cdots)|,\end{equation}

for any numbers $a_l,x_i,y_i\in S$ , $l\in\{1,m+1\}$ , and $ i\leq 0$ . This assumption is satisfied, for instance, by some autoregressive models.

The next lemma proves that the truncated sequence $(X_n^{(m)})_{n\in {\mathbb Z}}$ is a Hausdorff approximation of the original sequence $(X_n)_{n\in {\mathbb Z}}$ .

Lemma 4.1. Let $\epsilon>0$ be fixed, let $(\epsilon_i)_{i\in {\mathbb Z}}$ be an i.i.d. sequence, let f be a bounded function satisfying (4.4), and let $(X_n)_{n\in {\mathbb Z}}$ and $(X_n^{(m)})_{n\in {\mathbb Z}}$ be the associated sequences as in (4.2) and (4.3), respectively. Let $m \in \mathrm{I\!N}$ be such that

\begin{align*}2c_m \|f\|_{\infty} < \epsilon,\end{align*}

where $\|f\|_{\infty}$ is the supremum of f. Then $d_H({{\mathbb X}}^{(m)}_n, {{\mathbb X}}_n)<\epsilon$ , a.s.

In view of Lemma 4.1, the condition $\lim_{m\rightarrow \infty}c_m=0$ is enough to approximate, in the Hausdorff sense, the sequence $(X_n)_{n\in {\mathbb Z}}$ by the truncated sequence $(X_n^{(m)})_{n\in {\mathbb Z}}$ .

Proof. We recall that

\begin{align*}d_H({{\mathbb X}}^{(m)}_n, {{\mathbb X}}_n)=\max\left(\max_{1\leq i\leq n}\min_{1\leq j\leq n}|X_i-X_j^{(m)}|, \max_{1\leq i\leq n}\min_{1\leq j\leq n}|X_j-X_i^{(m)}| \right).\end{align*}

Hence,

\begin{eqnarray*} && \mathrm{I\!P}(d_H({{\mathbb X}}^{(m)}_n, {{\mathbb X}}_n)\geq \epsilon) \\[5pt] && \leq \mathrm{I\!P}(\max_{1\leq i\leq n}\min_{1\leq j\leq n}|X_i-X_j^{(m)}|\geq \epsilon) + \mathrm{I\!P}( \max_{1\leq i\leq n}\min_{1\leq j\leq n}|X_j-X_i^{(m)}|\geq \epsilon) \\[5pt] && \leq 2 n \max_{1\leq i \leq n}\mathrm{I\!P}( |X_i-X_i^{(m)}|\geq \epsilon).\end{eqnarray*}

Now,

\begin{align*}|X_i-X_i^{(m)}| \leq c_m|f(\epsilon_{n-m-1},\epsilon_{n-m-2},\cdots)-f(0,0,\cdots)| \leq 2c_m \|f\|_{\infty}.\end{align*}

Consequently, the event $( |X_i-X_i^{(m)}|\geq \epsilon)$ implies that $\epsilon \leq 2c_m \|f\|_{\infty}$ , and then the probability $\mathrm{I\!P}( |X_i-X_i^{(m)}|\geq \epsilon)$ vanishes whenever m satisfies $2c_m \|f\|_{\infty}<\epsilon$ . We conclude that, for such m, $\mathrm{I\!P}(d_H({{\mathbb X}}^{(m)}_n, {{\mathbb X}}_n)\geq \epsilon)=0$ .

We can now apply Proposition 4.1 to the m-dependent sequence $(X_n^{(m)})_{n\in {\mathbb Z}}$ , combined with Lemma 4.1, to establish asymptotic $(\epsilon,\alpha)$ -density for $(X_n)_{n\in{\mathbb Z}}$ . Doing this, we obtain the result below under a suitable control of the following concentration coefficient, related to the truncated sequence $(X_n^{(m)})_{n\in {\mathbb Z}}$ (as defined in (4.3)):

(4.5) \begin{equation} \rho_{m+1}^{(m)}(\epsilon)= \inf_{x \in \mathrm{I\!R}^{m+1}}\mathrm{I\!P}(\|Y_{1,m+1}^{(m)}-x\|\leq \epsilon),\end{equation}

with $Y_{1,m+1}^{(m)}=(X_1^{(m)},\cdots,X_{m+1}^{(m)})^t$ .

Proposition 4.2. Let $(X_n)_{n\in {\mathbb Z}}$ and f be as in the statement of Lemma 4.1. Let $\epsilon_0>0$ , $\epsilon \in ]0,\epsilon_0[$ be fixed, and let $m \in \mathrm{I\!N}$ be such that $2c_m \|f\|_{\infty} < \epsilon$ . Suppose that the concentration coefficient $\rho_{m+1}^{(m)}(\epsilon)$ related to the truncated sequence $(X_n^{(m)})_{n\in {\mathbb Z}}$ , and defined in (4.5), satisfies

\begin{align*} \rho_{m+1}^{(m)}(\epsilon)\geq {\kappa_{\epsilon}}, \end{align*}

for some ${\kappa_{\epsilon}}>0$ . Then for any $n\geq m+1$ ,

\begin{align*}\mathrm{I\!P}(d_H({\mathbb X}_n, \mathrm{I\!M})\geq 2\epsilon)\leq \frac{(1- \kappa_{\frac{\epsilon}{2}})^{[\frac{1}{2}[\frac{n}{m+1}]]}}{\kappa_{\frac{\epsilon}{4}}}, \end{align*}

and a conclusion similar to that of Proposition 4.1 is true for such m.

Proof. This follows from Lemma 4.1 together with Proposition 4.1 and the fact that

\begin{eqnarray*}&& \mathrm{I\!P}(d_H({\mathbb X}_n, \mathrm{I\!M})\geq 2\epsilon)\leq \mathrm{I\!P}(d_H({{\mathbb X}}_n, {\mathbb X}^{(m)}_n)+ d_H({{\mathbb X}}^{(m)}_n, \mathrm{I\!M}) \geq 2\epsilon) \\[5pt] &&\leq \mathrm{I\!P}(d_H({{\mathbb X}}_n, {{\mathbb X}}^{(m)}_n)\geq\epsilon)+ \mathrm{I\!P}(d_H({{\mathbb X}}^{(m)}_n, \mathrm{I\!M}) \geq\epsilon).\end{eqnarray*}

4.0.3. Stationary $\beta$ -mixing sequence on a compact set.

Recall that the stationary sequence $(X_n)_{n\in \mathrm{I\!N}}$ is $\beta$ -mixing if $\beta_n$ tends to 0 when n tends to infinity, where the coefficients $(\beta_n)_{n>0}$ are defined as follows (see [Reference Bradley5, Reference Yu40] for the expression for $\beta_n$ ):

(4.6) \begin{equation}\beta_n=\sup_{l\geq 1}\mathrm{I\!E} \left\{\sup\left| \mathrm{I\!P}\!\left(B|\sigma(X_1,\cdots,X_l)\right) -\mathrm{I\!P}(B)\right|,\,\, B\in \sigma(X_i,\,\, i\geq l+n)\right\}.\end{equation}

The following corollary gives conditions on the behavior of the two sequences $(\rho_n(\epsilon))_{n>0}$ and $(\beta_n)_{n>0}$ under which the asymptotically dense property of Definition 1.1 is satisfied.

Proposition 4.3. Let $(X_n)_{n\geq 0}$ be a stationary $\beta$ -mixing sequence. Suppose that $X_1$ is supported on a compact set $\mathrm{I\!M}$ . Then it holds, for any $\epsilon>0$ and any sequences $k_n$ and $r_n$ such that $k_nr_n\leq n$ ,

\begin{align*}\mathrm{I\!P}\!\left(d_H({\mathbb X}_n , {\mathrm{I\!M}}) > \epsilon\right) \leq\frac{k_n^2\beta_{r_n}+ k_n\exp\!\left(-[\frac{k_n}{2}]\rho_{r_n}(\epsilon/2)\right)}{k_n\rho_{r_n}(\epsilon/4)}.\end{align*}

Suppose moreover that for some $\beta>1$ , and any $\epsilon>0$ small enough,

\begin{align*}\lim_{m\rightarrow \infty} \rho_m(\epsilon)\frac{e^{m^{\beta}}}{m^{1+\beta}}=\infty,\,\,\,{{\textit{and}}}\,\,\,\lim_{m\rightarrow\infty} \frac{e^{2m^{\beta}}}{m^2}\beta_m=0.\end{align*}

Then $(X_n)_{n\geq 0}$ is asymptotically dense in $\mathrm{I\!M}$ .

In the proof of Proposition 4.3 given in Section 7, we construct two sequences $(k_n)_n$ and $(r_n)_n$ for which

\begin{align*}\lim_{n\rightarrow \infty} \frac{k_n^2\beta_{r_n}+ k_n\exp\!\left(-[\frac{k_n}{2}]\rho_{r_n}(\epsilon/2)\right)}{k_n\rho_{r_n}(\epsilon/4)} =0.\end{align*}

The threshold $n_0(\epsilon,\alpha)$ is not explicitly calculated, but it is that integer for which

\begin{align*} \frac{k_n^2\beta_{r_n}+ k_n\exp\!\left(-[\frac{k_n}{2}]\rho_{r_n}(\epsilon/2)\right)}{k_n\rho_{r_n}(\epsilon/4)} \leq \alpha,\end{align*}

for all $n\geq n_0(\epsilon,\alpha)$ .

4.0.4. Stationary weakly dependent sequence on a compact set.

We suppose here that is a stationary sequence such that $X_1$ takes values in a compact support $\mathrm{I\!M}$ . We suppose also that this sequence is weakly dependent in the sense of [Reference Doukhan and Louhichi14]. More precisely, we suppose that it satisfies the following definition.

Definition 4.1. We say that the sequence is $(\mathrm{I\!L}_{\infty}, \Psi)$ -weakly dependent if there exists a non-increasing function $\Psi$ such that $\lim_{r\rightarrow\infty}\Psi(r)=0$ , and such that for any measurable functions f and g bounded (respectively by $\|f\|_{\infty}$ and $\|g\|_{\infty}$ ) and for any $ i_1\leq \cdots \leq i_k<i_{k}+r\leq i_{k+1}\leq \cdots\leq i_n$ one has

(4.7) \begin{eqnarray}\left|\mathrm {Cov}\left(\frac{f(X_{i_1},\cdots,X_{i_k})}{\|f\|_{\infty}}, \frac{g(X_{i_{k+1}},\cdots, X_{i_n})}{\|g\|_{\infty}}\right) \right| \leq \Psi(r).\end{eqnarray}

See [Reference Dedecker12, Definition 2.2] for a more general setting.

The dependence condition in Definition 4.1 is weaker than the Rosenblatt strong mixing dependence [Reference Rosenblatt35]. Let us briefly explain this. The $\alpha$ -mixing coefficient between the two sigma-fields ${\mathcal A}$ and ${\mathcal B}$ is defined as

\begin{align*}\alpha(\mathcal A, \mathcal B)=\sup_{A\in {\mathcal A} ,B\in \mathcal B}|\mathrm{I\!P}(A\cap B)-\mathrm{I\!P}(A)\mathrm{I\!P}(B)|.\end{align*}

The sequence

is strongly mixing if its coefficient $\alpha_n$ defined, for $n\geq 1$ , by

tends to 0 as n tends to infinity, with ${\mathcal P}_k=\sigma(X_i,\, i\leq k)$ and ${\mathcal F}_{k+n}=\sigma(X_i,\, i\geq k +n)$ . An equivalent formula for $\alpha_n$ , using the covariance between some functions, is stated in [Reference Bradley6, Theorem 4.4]:

\begin{align*}\alpha_n=\frac{1}{4}\sup\left\{\frac{\mathrm {Cov}(f,g)}{\|f\|_{\infty}\|g\|_{\infty}},\,\, f\in L_{\infty}({\mathcal P}_k), g\in L_{\infty}({\mathcal F}_{k+n})\right\},\end{align*}

where $L_{\infty}({\mathcal A})$ denotes the set of bounded ${\mathcal A}$ -measurable functions for some $\sigma$ -fields $\mathcal A$ . It follows from this formula that strongly mixing sequences are $(\mathrm{I\!L}_{\infty}, \Psi)$ -weakly dependent, as stated in Definition 4.1 (with $\Psi(r)=\alpha_r$ for $r>0$ ). The converse, however, is not necessarily true (see [Reference Dedecker12]).

We can now state our result for stationary weakly dependent sequences.

Proposition 4.4. Let be a stationary, $(\mathrm{I\!L}_{\infty}, \Psi)$ -weakly dependent sequence in the sense of Definition 4.1. Suppose that $X_1$ is supported on a compact set $\mathrm{I\!M}$ . Then it holds, for any $\epsilon>0$ and any sequences $k_n$ and $r_n$ such that $k_nr_n\leq n$ , that

\begin{align*}\mathrm{I\!P}\!\left(d_H({\mathbb X}_n , {\mathrm{I\!M}}) > \epsilon\right) \leq\frac{k_n^2\Psi({r_n})+ k_n\exp\!\left(-[\frac{k_n}{2}]\rho_{r_n}(\epsilon/2)\right)}{k_n\rho_{r_n}(\epsilon/4)}.\end{align*}

Suppose moreover that, for some $\beta>1$ , and any positive $\epsilon$ small enough,

\begin{align*}\lim_{m\rightarrow \infty} \rho_m(\epsilon)\frac{e^{m^{\beta}}}{m^{1+\beta}}=\infty,\,\,\,{\mbox{and that}}\,\,\,\lim_{m\rightarrow\infty} \frac{e^{2m^{\beta}}}{m^2}\Psi(m)=0.\end{align*}

Then this sequence is asymptotically dense in $\mathrm{I\!M}$ .

Here again the threshold $n_0(\epsilon,\alpha)$ is not explicitly calculated but can be given by an inequality, as in the case of $\beta$ -mixing.

4.1. Comparison with the i.i.d. case

We can compare the bounds of Proposition 4.1 (Propositions 4.3 and 4.4 respectively) with what has already been obtained in the i.i.d. case (see [Reference Chazal, Glisse, Labruyère and Michel7, Reference Cuevas10, Reference Cuevas and Rodrguez-Casal11]). That is, we restrict to the case when $m=0$ (respectively, $k_n=n$ and $r_n=1$ , $\beta_n=\Psi(n)=0$ for $n\geq 1$ ). Suppose that we are in this situation and that, moreover, $\rho_1(\epsilon)$ has a strictly positive lower bound, say $\kappa_{\epsilon}$ . Then all the conclusions of the three propositions above give the same upper bound for $\mathrm{I\!P}\!\left(d_H({\mathbb X}_n , {\mathrm{I\!M}}) > \epsilon\right)$ , which is

(4.8) \begin{equation}\frac{\exp\!(\!-[\frac{n}{2}]\kappa_{\frac{\epsilon}{2}})}{\kappa_{\frac{\epsilon}{4}}}.\end{equation}

Now we suppose, as already done in the i.i.d. case, that the (a, b)-standard assumption is satisfied, i.e, $\kappa_{\epsilon}= a \epsilon^{b},$ for some $a>0$ , $b>0$ , and for positive $\epsilon$ small enough. Recall that the (a, b)-standard assumption was used, in the i.i.d. context, for set estimation problems under Hausdorff distance ([Reference Cuevas10, Reference Cuevas and Rodrguez-Casal11]) and also for a statistical analysis of persistence diagrams ([Reference Chazal, Glisse, Labruyère and Michel7, Reference Fasy16]). Then, clearly, an upper bound for (4.8) is

\begin{align*}C \frac{\exp\!(\!-c n \epsilon^b)}{\epsilon^b},\end{align*}

for some positive constants C and c (independent of n and of $\epsilon$ ), as has already been found in the i.i.d. case (see for instance the upper bound (3.2) in [Reference Cuevas10]). Finally, we have to check that the requirements of Propositions 4.1, 4.3, and 4.4 are satisfied under the (a, b)-standard assumption (i.e. when $\rho_1(\epsilon)\geq a \epsilon^{b}$ ). Since we are in the case when $m=0$ in Proposition 4.1, and when $\beta_n=0, \Psi_n=0, \, n\geq 1$ in Propositions 4.3 and 4.4, we have only to check that the (a, b)-standard assumption ensures, for i.i.d. random variables, that $\lim_{m\rightarrow \infty} \rho_m(\epsilon)\frac{e^{m^{\beta}}}{m^{1+\beta}}=\infty$ . We deduce from the inequality

\begin{align*} \|(X_1,\cdots,X_m)^t -(x_1,\cdots,x_m)^t\|^2 \leq m \max_{1\leq i\leq m}\|X_i-x_i\|^2 \end{align*}

that

\begin{align*} \mathrm{I\!P}\!\left(m \max_{1\leq i\leq m}\|X_i-x_i\|^2 \leq \epsilon^2\right)\leq \mathrm{I\!P}\!\left(\|(X_1,\cdots,X_m)^t -(x_1,\cdots,x_m)^t\|^2\leq \epsilon^2\right). \end{align*}

Now,

\begin{align*} \mathrm{I\!P}\!\left(m \max_{1\leq i\leq m}\|X_i-x_i\|^2 \leq \epsilon^2\right) = \left(\mathrm{I\!P}\!\left(\|X_1-x_1\| \leq \epsilon/{\sqrt{m}}\right)\right)^m. \end{align*}

Hence, $\rho_m(\epsilon)\geq \rho_1^m(\epsilon/{\sqrt{m}}).$ Combining this bound with the (a, b)-standard assumption, we get

\begin{eqnarray*}&& a^m \frac{\epsilon^{bm}}{m^{bm/2}}\frac{e^{m^{\beta}}}{m^{1+\beta}}\leq \rho_m(\epsilon)\frac{e^{m^{\beta}}}{m^{1+\beta}}.\end{eqnarray*}

The left term tends to infinity as n goes to infinity (since $\beta>1$ ); hence $\displaystyle\lim_{m\rightarrow \infty} \rho_m(\epsilon)\frac{e^{m^{\beta}}}{m^{1+\beta}}=\infty$ .

As an important conclusion, the previous three propositions generalize the i.i.d. case well, even without the (a, b)-standard assumption.

5.1. Stationary Markov chains on a ball of $\mathrm{I\!R}^d$

5.1.1. Random difference equations.

Let $(X_n)_{n\geq 0}$ be a Markov chain defined, for $n\geq 0$ , by

(5.1) \begin{equation}X_{n+1}=A_{n+1}X_{n}+ B_{n+1},\end{equation}

where $A_{n+1}$ is a $(d\times d)$ -matrix, $X_n\in \mathrm{I\!R}^d,\,\, B_n\in \mathrm{I\!R}^d$ , and $(A_n,B_n)_{n\geq 1}$ is an i.i.d. sequence independent of $X_0$ . Recall that for a matrix M, $\|M\|$ is the operator norm, defined by $\|M\|=\sup_{x\in \mathrm{I\!R}^d,\,\|x\|=1}\|Mx\|$ . It is well known that for any $n\geq 1$ , $X_n$ is distributed as $\sum_{k=1}^n A_1\cdots A_{k-1}B_k + A_1\cdots A_{n}X_0$ ; see for instance [Reference Kesten27]. It is also well known that the conditions (see [Reference Goldie and Maller21, Reference Kesten28])

(5.2) \begin{equation}\mathrm{I\!E}(\ln^+\|A_1\|)<\infty,\,\, \mathrm{I\!E}(\ln^+\|B_1\|)<\infty,\,\, \lim_{n\rightarrow \infty}\frac{1}{n}\ln\|A_1\cdots A_n\|<0\,\,\, \text{a.s.}\end{equation}

ensure the existence of a stationary solution to (5.1), and that $\|A_1\cdots A_n\|$ approaches 0 exponentially fast. If in addition $\mathrm{I\!E}\|B_1\|^{\beta}<\infty$ for some $\beta>0$ , then the series $R\;:\!=\;\sum_{i=1}^{\infty}A_1\cdots A_{i-1}B_i$ converges a.s. and the distribution of $X_n$ converges to that of R, independently of $X_0$ . The distribution of R is then that of the stationary measure of the chain.

Compact state space. If $\|B_1\|\leq c<\infty$ for some fixed c, then this stationary Markov chain is $\mathrm{I\!M}$ -compactly supported. In particular, if $\|A_1\|\leq \rho <1$ for some fixed $\rho$ , then $\mathrm{I\!M}$ is included in the ball $B_d(0, \frac{c}{1-\rho})$ of $\mathrm{I\!R}^d$ .

Transition kernel. Suppose that, for any $x \in \mathrm{I\!M}$ , the random vector $A_1x+B_1$ has a density $f_{A_1x+B_1}$ with respect to the Lebesgue measure (here $\nu$ is the Lebesgue measure) satisfying $\inf_{x,\,\,y\in \mathrm{I\!M}}f_{A_1x+B_1}(y)\geq \kappa$ ; then $k(x,y)= f_{A_1x+B_1}(y)\geq \kappa>0.$

We collect all of the above results in the following corollary.

Corollary 5.1. Suppose that in the model (5.1), the conditions (5.2) are satisfied, and moreover $\|B_1\|\leq c<\infty$ . If the density of $A_1x+B_1$ , denoted by $f_{A_1x+B_1}$ , satisfies $\inf_{x,\,\,y\in \mathrm{I\!M}}f_{A_1x+B_1}(y)\geq \kappa>0$ for some positive $\kappa$ , then Assumptions ( ${\mathcal A}_1$ ) and ( ${\mathcal A}_2$ ) are satisfied with $b=d$ and $\nu$ the Lebesgue measure on $\mathrm{I\!R}^d$ .

Example: the AR(1) process in $\mathrm{I\!R}$ .

We consider a particular case of the Markov chain as defined in (5.1) with $d=1$ , where, for each n, $A_n=\rho$ with $|\rho|<1$ . We obtain the standard first-order linear autoregressive process, i.e.,

\begin{align*}X_{n+1}=\rho X_n + B_{n+1}.\end{align*}

We suppose that

• $B_1$ has a density function $f_B$ supported on $[\!-c,c]$ for some $c>0$ with $\kappa\;:\!=\;\inf_{x\in [-c,c]}f_B(x)>0$ , and

• $X_0 \in [\frac{-c}{1-|\rho|}, \frac{c}{1-|\rho|}]$ .

This Markov chain evolves in a compact state space which is a subset of $ [\frac{-c}{1-|\rho|}, \frac{c}{1-|\rho|}]$ . Thanks to Corollary 5.1, $(X_n)_n$ admits a stationary measure $\mu$ . We have, moreover,

\begin{align*}k(x,y)=f_{B_1}(y-\rho x) \geq \kappa,\,\, \forall\,\, x\in \mathrm{I\!M},\,\, \forall\,y\in \mathrm{I\!M}.\end{align*}

Assumptions ( ${\mathcal A}_1$ ) and ( ${\mathcal A}_2$ ) are then satisfied with $b=1$ and $\nu$ the Lebesgue measure on $\mathrm{I\!R}$ .

Example: the AR(k) process in $\mathrm{I\!R}$ .

The AR(k) process is defined by

\begin{align*}Y_n= \alpha_1 Y_{n-1} + \alpha_2 Y_{n-2}+\cdots+ \alpha_k Y_{n-k} + \epsilon_n,\end{align*}

where $\alpha_1,\cdots,\alpha_k \in \mathrm{I\!R}$ . Since this model can be written in the form of (5.1) with $d=1$ ,

\begin{align*}X_n= (Y_n,Y_{n-1},\cdots,Y_{n-k+1})^t,\,\,\,\, B_n= (\epsilon_n,0,\cdots,0)^t,\,\,\, A_n=\left( \begin{array}{c@{\quad}c@{\quad}c} \alpha_1 & \cdots & \alpha_k \\[5pt] I_{k-1} & & 0 \\[5pt] \end{array} \right),\end{align*}

all of the above results for random difference equations apply under the corresponding assumptions. In particular, the process AR(2) is stationary as soon as $|\alpha_2|<1$ and $\alpha_2+ |\alpha_1|<1$ .

5.2. The Möbius Markov chain on the circle

Our aim is to give an example of a Markov chain on the unit circle, known as the Möbius Markov chain, which satisfies the requirements of Proposition 5.1. This Markov chain $(X_n)_{n\in \mathrm{I\!N}}$ is introduced in [Reference Kato26] and is defined as follows:

• $X_0$ is a random variable which takes values on the unit circle.

• For $n\geq 1$ ,

  \begin{align*}X_n=\frac{X_{n-1} + \beta}{\beta X_{n-1}+1}\epsilon_n,\end{align*}
  where $\beta\in ]-1,1[$ and $(\epsilon_n)_{n\geq 1}$ is a sequence of i.i.d. random variables which are independent of $X_0$ and distributed as the wrapped Cauchy distribution with a common density with respect to the arc length measure $\nu$ on the unit circle $\partial B(0,1)$ ,
  \begin{align*}f_{\varphi}(z)= \frac{1}{2\pi}\frac{1-\varphi^2}{|z-\varphi|^2},\,\, \varphi\in [0,1[ \,\, {\mbox{fixed}},\,\,\, z\in \partial B(0,1).\end{align*}

The following proposition holds.

Proposition 5.2. Let $(X_n)_{n\geq 0}$ be the Möbius Markov chain on the unit circle as defined above. Then this Markov chain admits a unique invariant distribution, denoted by $\mu$ . If $X_0$ is distributed as $\mu$ , then the set ${\mathbb X}_n =\{X_1,\cdots,X_n\}$ converges in probability, as n tends to infinity, in the Hausdorff distance to the unit circle $\partial B(0,1)$ ; more precisely, for any $\alpha\in ]0,1[$ , any positive $\epsilon$ sufficiently small, and any $n\geq \frac{2}{\kappa v \epsilon}\left(\ln(\frac{1}{\alpha})+\ln(\frac{4}{\epsilon\kappa v}) \right) $ , we have

\begin{align*}d_H({\mathbb X}_n , \partial B(0,1)) \leq \epsilon\end{align*}

with probability at least $1-\alpha$ . Here v is a finite positive constant and $\kappa= \frac{1}{2\pi}\frac{1-\varphi}{1+\varphi}.$

The Möbius Markov chain of Proposition 5.2 is then asymptotically dense in the unit circle with a threshold $n_0(\epsilon,\alpha)$ given by

\begin{align*}n_0(\epsilon,\alpha) = \frac{2}{\kappa v \epsilon}\left(\ln \left(\frac{1}{\alpha}\right)+\ln\left(\frac{4}{\epsilon\kappa v}\right) \right),\end{align*}

$\kappa$ being as in the statement of the proposition, while the positive constant v is defined by (5.5) below.

Proof. We have to prove that all the requirements of Proposition 5.1 are satisfied. Our main reference for this proof is [Reference Kato26], where it is shown that this Markov chain has a unique invariant measure $\mu$ on the unit circle. This measure $\mu$ has full support on $\partial B(0,1)$ (so that Assumption $({\mathcal A}_1)$ is satisfied with $\mathrm{I\!M}=\partial B(0,1)$ ). The task now is to check Assumption $({\mathcal A}_2)$ . We have also, for $x\in \partial B(0,1)$ ,

(5.3) \begin{eqnarray}&& K(x,dz)= \mathrm{I\!P}(X_1\in dz)|X_0=x)= k(x,z) \nu(dz),\end{eqnarray}

where $\nu$ is the arc length measure on the unit circle, and for $x,z\in \partial B(0,1)$ ,

\begin{align*}k(x,z)= \frac{1}{2\pi}\frac{1-|\phi_1(x)|^2}{|z-\phi_1(x)|^2},\end{align*}

with

\begin{align*}\phi_1(x)= \frac{\varphi x + \beta \varphi}{\beta x + 1}.\end{align*}

Since $\frac{x+\beta}{\beta x+ 1}\in \partial B(0,1)$ whenever $x\in \partial B(0,1)$ , we obtain $\displaystyle|\phi_1(x)|^2= \varphi^2.$ Now, for $x,z\in \partial B(0,1)$ ,

\begin{align*}|z-\phi_1(x)|\leq |z|+ |\phi_1(x)|\leq 1+ \varphi.\end{align*}

Hence,

(5.4) \begin{equation}k(x,z)\geq \frac{1}{2\pi} \frac{1-\varphi^2 }{(1+ \varphi)^2}= \frac{1}{2\pi}\frac{1-\varphi}{1+\varphi}>0.\end{equation}

We now have to check that, for some $\epsilon_0>0$ ,

(5.5) \begin{equation}v\;:\!=\;\inf_{u\in \partial B(0,1)} \inf_{0<\epsilon<\epsilon_0} \left(\epsilon^{-1}\int_{\partial B(0,1) \cap B(u,\epsilon)}\nu(dx_1)\right)>0.\end{equation}

For this let $u\in \partial B(0,1)$ , and define $\displaystyle\overset{\frown}{AB} = \int_{\partial B(0,1) \cap B(u,\epsilon)}\nu(dx_1).$

We have $\|u-A\|=\|u-B\|=\epsilon$ (see Figure 3). Let $\alpha = \widehat{AOB}$ ; then on the one hand $\overset{\frown}{AB}=\alpha$ . On the other hand, since the triangle OAu is isosceles, with an angle of $\alpha/2$ in O, we have $\epsilon = 2\sin (\alpha/4)$ . We thus obtain

\begin{align*}\lim_{\epsilon\rightarrow 0}\frac{1}{\epsilon}{\overset{\frown}{AB}}= \lim_{\epsilon\rightarrow 0}\frac{\alpha}{\epsilon}= \lim_{\alpha\rightarrow 0}\frac{\alpha}{2\sin (\alpha/4)}=2.\end{align*}

From this, (5.5) is satisfied.

Figure 3.

The ball $B(u,\epsilon)$ intersects the unit circle at two points A and B.

Assumption $({\mathcal A}_2)$ is satisfied thanks to (5.3), (5.4), and (5.5). The proof of Proposition 5.2 is then complete by Proposition 5.1.

7.1. Proof of Proposition 4.1

Let $\epsilon_0>0$ and $\epsilon \in ]0,\epsilon_0[$ be fixed. In this proof we set $m'=m+1$ , $r=m'$ , and $ k= k_n=[n/m']$ . Proposition 3.1, applied with these values of r and k, gives

(7.1) \begin{eqnarray}&&\mathrm{I\!P}\!\left(d_H({\mathbb X}_n , {\mathrm{I\!M}}) > \epsilon\right)\leq \mathrm{I\!P}\!\left(d_H(\{Y_{1,m'},\cdots,Y_{k_n,m'}\}, {\mathrm{I\!M}}_{dm'}) > \epsilon\right) {\nonumber} \\[5pt] && \leq \frac{\sup_{x\in {\mathrm{I\!M}}_{dm'}}\mathrm{I\!P}\!\left(\min_{1\leq i\leq k_n}\|Y_{i,m'}-x\|> \epsilon/2\right)}{1-\sup_{x\in {\mathrm{I\!M}}_{dm'}}\mathrm{I\!P}\!\left(\|Y_{1,m'}-x\|> \epsilon/4\right)},\end{eqnarray}

where $Y_{i,m'}= (X_{(i-1)m'+1},\cdots,X_{im'})^t.$ The sequence is stationary and assumed to be m-dependent. Consequently, the two families $\{Y_{1,m'}, Y_{3,m'}, Y_{5,m'},\cdots\}$ and $\{Y_{2,m'}, Y_{4,m'}, Y_{6,m'},\cdots\}$ each consist of i.i.d. random vectors. Since we are assuming that $\rho_{m'}(\epsilon)\geq \kappa_{\epsilon}$ , we have

(7.2) \begin{eqnarray}&&\sup_{x\in {\mathrm{I\!M}}_{dm'}}\mathrm{I\!P}\!\left(\min_{1\leq i\leq k_n}\|Y_{i,m'}-x\|> \frac{\epsilon}{2}\right)\leq \sup_{x\in {\mathrm{I\!M}}_{dm'}}\mathrm{I\!P}\!\left(\min_{1\leq 2i\leq k_n}\|Y_{2i,m'}-x\|> \frac{\epsilon}{2}\right) {\nonumber}\\[5pt] &&\leq \sup_{x\in {\mathrm{I\!M}}_{dm'}}\left(\mathrm{I\!P}\!\left(\|Y_{1,m'}-x\|> \frac{\epsilon}{2}\right)\right)^{[k_n/2]}\leq \left(1- \rho_{m'}(\frac{\epsilon}{2})\right)^{[k_n/2]}\leq \left(1- \kappa_{\frac{\epsilon}{2}}\right)^{[k_n/2]}\end{eqnarray}

and

(7.3) \begin{eqnarray}&& 1-\sup_{x\in {\mathrm{I\!M}}_{dr}}\mathrm{I\!P}\!\left(\|Y_{1,m'}-x\|> \frac{\epsilon}{4}\right)\geq \kappa_{\frac{\epsilon}{4}}.\end{eqnarray}

Collecting the bounds (7.1), (7.2), and (7.3), we find that for any $\epsilon>0$ ,

\begin{align*}\mathrm{I\!P}\!\left(d_H({\mathbb X}_n , {\mathrm{I\!M}}) > \epsilon\right) \leq \frac{\left(1- \kappa_{\frac{\epsilon}{2}}\right)^{[k_n/2]}}{\kappa_{\frac{\epsilon}{4}}} \leq \frac{\exp\!\left(-\kappa_{\frac{\epsilon}{2}}[k_n/2]\right)}{\kappa_{\frac{\epsilon}{4}}}.\end{align*}

Let $\alpha \in ]0,1[$ be such that $\displaystyle\frac{\exp\!\left(-\kappa_{\frac{\epsilon}{2}}[k_n/2]\right)}{\kappa_{\frac{\epsilon}{4}}} \leq \alpha,$ which is equivalent to

\begin{align*}[k_n/2] \geq \frac{1}{\kappa_{\frac{\epsilon}{2}}} \log\left(\frac{1}{\alpha \kappa_{\frac{\epsilon}{4}} }\right).\end{align*}

Then, for any $n\geq \frac{2m'}{\kappa_{\frac{\epsilon}{2}}} \log\left(\frac{1}{\alpha \kappa_{\frac{\epsilon}{4}} }\right)+3m'$ , we have

\begin{align*}[k_n/2]\geq k_n/2-1\geq \frac{n}{2m'}-3/2 \geq \frac{1}{\kappa_{\frac{\epsilon}{2}}} \log\left(\frac{1}{\alpha \kappa_{\frac{\epsilon}{4}} }\right),\end{align*}

and therefore $\displaystyle\mathrm{I\!P}\!\left(d_H({\mathbb X}_n , {\mathrm{I\!M}}) > \epsilon\right) \leq \alpha.$ The proof of Proposition 4.1 is complete.

7.2. Proof of Proposition 4.3

We use the blocking method of [Reference Yu40] to transform the dependent $\beta$ -mixing sequence $(X_n)_{n\in \mathrm{I\!N}}$ into a sequence of nearly independent blocks. Let $Z_{2i,{r_n}}=(\xi_j,\,\, j\in \{(2i-1){r_n}+1,\cdots, 2i{r_n}\})^t$ be a sequence of i.i.d. random vectors independent of the sequence $(X_i)_{i\in \mathrm{I\!N}}$ such that, for any i, $Z_{2i,{r_n}}$ is distributed as $Y_{2i,{r_n}}$ (which is distributed as $Y_{1,{r_n}}$ ). Lemma 4.1 of [Reference Yu40] proves that the two vectors $(Z_{2i,{r_n}})_i$ and $(Y_{2i,{r_n}})_i$ are related by the following relation:

\begin{align*}\left|\mathrm{I\!E}(h(Z_{2i,{r_n}}, 1\leq 2i\leq k_n))- \mathrm{I\!E}(h(Y_{2i,{r_n}}, 1\leq 2i\leq k_n))\right|\leq k_n\beta_{r_n},\end{align*}

which is true for any measurable function bounded by 1. We then have, using the last bound,

\begin{eqnarray*}&&k_n\sup_{x\in {\mathrm{I\!M}}_{d{r_n}}}\mathrm{I\!P}\!\left(\min_{1\leq i\leq k_n}\|Y_{i,{r_n}}-x\|> \epsilon\right)\leq k_n\sup_{x\in {\mathrm{I\!M}}_{d{r_n}}}\mathrm{I\!P}\!\left(\min_{1\leq 2i\leq k_n}\|Y_{2i,{r_n}}-x\|> \epsilon\right) \\[5pt] && \leq k_n\sup_{x\in {\mathrm{I\!M}}_{d{r_n}}}\left|\mathrm{I\!P}\!\left(\min_{1\leq 2i\leq k_n}\|Y_{2i,{r_n}}-x\|> \epsilon\right)-\mathrm{I\!P}\!\left(\min_{1\leq 2i\leq k_n}\|Z_{2i,{r_n}}-x\|> \epsilon\right)\right|\\[5pt] && + k_n\sup_{x\in {\mathrm{I\!M}}_{d{r_n}}}\mathrm{I\!P}\!\left(\min_{1\leq 2i\leq k_n}\|Z_{2i,{r_n}}-x\|> \epsilon\right)\\[5pt] && \leq k_n^2\beta_{r_n}+ k_n\sup_{x\in {\mathrm{I\!M}}_{d{r_n}}}\mathrm{I\!P}\!\left(\min_{1\leq 2i\leq k_n}\|Z_{2i,{r_n}}-x\|> \epsilon\right)\\[5pt] && \leq k_n^2\beta_{r_n}+ k_n\sup_{x\in {\mathrm{I\!M}}_{d{r_n}}}\Bigl(\mathrm{I\!P}\!\left(\|Y_{1,{r_n}}-x\|> \epsilon\right)\Bigr)^{[k_n/2]}\\[5pt] && \leq k_n^2\beta_{r_n}+ k_n\Bigl(1-\rho_{r_n}(\epsilon)\Bigr)^{[k_n/2]}\\[5pt] && \leq k_n^2\beta_{r_n}+ k_n\exp\!\left(-[\frac{k_n}{2}]\rho_{r_n}(\epsilon)\right)\end{eqnarray*}

and

\begin{align*}1-\sup_{x\in {\mathrm{I\!M}}_{dr_n}}\mathrm{I\!P}\!\left(\|Y_{1,r_n}-x\|> \epsilon/4\right)= \rho_{r_n}(\epsilon/4).\end{align*}

Consequently, Proposition 3.1 gives

(7.4) \begin{eqnarray}\mathrm{I\!P}\!\left(d_H({\mathbb X}_n , {\mathrm{I\!M}}) > \epsilon\right) &\leq &\frac{\sup_{x\in {\mathrm{I\!M}}_{dr_n}}\mathrm{I\!P}\!\left(\min_{1\leq i\leq k_n}\|Y_{i,r_n}-x\|> \epsilon/2\right)}{1-\sup_{x\in {\mathrm{I\!M}}_{dr_n}}\mathrm{I\!P}\!\left(\|Y_{1,r_n}-x\|> \epsilon/4\right)} {\nonumber}\\[5pt] &\leq & \frac{k_n^2\beta_{r_n}+ k_n\exp\!\left(-[\frac{k_n}{2}]\rho_{r_n}(\epsilon/2)\right)}{k_n\rho_{r_n}(\epsilon/4)}.\end{eqnarray}

We now have to construct two sequences $k_n$ and $r_n$ such that $ k_nr_n \leq n$ and

(7.5) \begin{eqnarray}&& \lim_{n\rightarrow \infty} k_n^2\beta_{r_n}=0,\,\, \lim_{n\rightarrow\infty} k_n\rho_{r_n}(\epsilon)=\infty,\,\, \lim_{n\rightarrow \infty}k_n\exp\!\left(-\frac{k_n}{2}\rho_{r_n}(\epsilon)\right)=0.\end{eqnarray}

We have supposed that $\lim_{m\rightarrow \infty} \rho_m(\epsilon)\frac{e^{m^{\beta}}}{m^{1+\beta}}=\infty$ for some $\beta>1$ . Define $\gamma =1/\beta\in ]0,1[$ and

\begin{align*}k_n=\left[\frac{n}{(\ln n)^{\gamma}}\right],\,\,\, r_n=[(\ln n)^{\gamma}].\end{align*}

Then, letting $m=r_n=[(\ln n)^{\gamma}]$ , we have $\displaystyle\lim_{n\rightarrow \infty} k_n \frac{\rho_{r_n}(\epsilon)}{\ln n}=\infty,$ and then (since $k_n\leq n$ ),

\begin{align*}\lim_{n\rightarrow \infty} k_n \frac{\rho_{r_n}(\epsilon)}{\ln(k_n)}=\infty.\end{align*}

From the last limit we have that $\lim_{n\rightarrow\infty} k_n\rho_{r_n}(\epsilon)=\infty$ and that, for n large enough and for some $C>2$ , $\displaystyle k_n \frac{\rho_{r_n}(\epsilon)}{\ln(k_n)}\geq C$ , so that

\begin{align*}k_n\exp\!\left(-\frac{k_n}{2}\rho_{r_n}(\epsilon)\right) \leq k_n^{1-C/2}.\end{align*}

Consequently, $\displaystyle \lim_{n\rightarrow \infty}k_n\exp\!\left(-\frac{k_n}{2}\rho_{r_n}(\epsilon)\right)=0$ . Now we deduce from $\lim_{m\rightarrow\infty} \frac{e^{2m^{\beta}}}{m^2}\beta_m=0$ that (letting $m=r_n=[(\ln n)^{\gamma}]$ )

\begin{align*} \lim_{n\rightarrow \infty} k_n^2\beta_{r_n}=0.\end{align*}

The two sequences $k_n$ and $r_n$ so constructed satisfy (7.5), and for these sequences it holds that

\begin{align*}\lim_{n\rightarrow \infty}\frac{k_n^2\beta_{r_n}+ k_n\exp\!\left(-\frac{k_n}{2}\rho_{r_n}(\epsilon/2)\right)}{k_n\rho_{r_n}(\epsilon/4)}=0.\end{align*}

Hence for any $\alpha\in ]0,1[$ there exists an integer $n_0(\epsilon,\alpha)$ such that for any $n\geq n_0(\epsilon,\alpha)$ ,

\begin{align*}\frac{k_n^2\beta_{r_n}+ k_n\exp\!\left(-\frac{k_n}{2}\rho_{r_n}(\epsilon/2)\right)}{k_n\rho_{r_n}(\epsilon/4)}\leq \alpha.\end{align*}

Combining this last inequality with that of (7.4) finishes the proof of Proposition 4.3.

7.3. Proof of Proposition 4.4

We have

(7.6) \begin{eqnarray}&&k_n\mathrm{I\!P}\!\left(\min_{1\leq i\leq k_n}\|Y_{i,{r_n}}-x\|> \epsilon\right)\leq k_n\mathrm{I\!P}\!\left(\min_{1\leq 2i\leq k_n}\|Y_{2i,{r_n}}-x\|> \epsilon\right){\nonumber}\\[5pt] && \leq k_n\left|\mathrm{I\!P}\!\left(\min_{1\leq 2i\leq k_n}\|Y_{2i,{r_n}}-x\|> \epsilon\right)-\prod_{i:\,1\leq 2i\leq k_n} \mathrm{I\!P}\!\left(\|Y_{2i,{r_n}}-x\|> \epsilon\right)\right| {\nonumber}\\[5pt] && + k_n\prod_{i:\,1\leq 2i\leq k_n} \mathrm{I\!P}\!\left(\|Y_{2i,{r_n}}-x\|> \epsilon\right).\end{eqnarray}

For s events $A_1,\cdots,A_s$ (with the convention that, $\prod_{j=1}^{0}\mathrm{I\!P}(A_j)=1$ ), we have

Hence,

Applying the last bound with $A_i= \left(\|Y_{2i,{r_n}}-x\|> \epsilon\right)$ and using (4.7), we get

and

(7.7) \begin{eqnarray}&& \left|\mathrm{I\!P}\!\left(\min_{1\leq 2i\leq k_n}\|Y_{2i,{r_n}}-x\|> \epsilon\right)-\prod_{i:\,1\leq 2i\leq k_n} \mathrm{I\!P}\!\left(\|Y_{2i,{r_n}}-x\|> \epsilon\right)\right| \leq k_n\Psi({r_n}).\end{eqnarray}

Combining (7.6) and (7.7), we deduce that

\begin{eqnarray*}&&k_n\mathrm{I\!P}\!\left(\min_{1\leq i\leq k_n}\|Y_{i,{r_n}}-x\|> \epsilon\right)\leq k_n^2\Psi({r_n})+ k_n\Bigl(1-\rho_{r_n}(\epsilon)\Bigr)^{[k_n/2]}\\[5pt] &&\leq k_n^2\Psi({r_n})+ k_n\exp\Bigl(-[k_n/2]\rho_{r_n}(\epsilon)\Bigr).\end{eqnarray*}

Consequently, as for (7.4), we get

\begin{align*}\mathrm{I\!P}\!\left(d_H({\mathbb X}_n , {\mathrm{I\!M}}) > \epsilon\right) \leq\frac{k_n^2\Psi({r_n})+ k_n\exp\!\left(-[\frac{k_n}{2}]\rho_{r_n}(\epsilon/2)\right)}{k_n\rho_{r_n}(\epsilon/4)}.\end{align*}

We now have to construct two sequences $r_n$ and $k_n$ such that

\begin{align*}\lim_{n\rightarrow \infty} k_n\exp\!(\!-k_n\rho_{r_n}(\epsilon)/2)=0,\,\,\, \lim_{n\rightarrow \infty}k_n^2\Psi({r_n})=0,\,\,\, \lim_{n\rightarrow \infty} k_n\rho_{r_n}(\epsilon)=\infty.\end{align*}

This last construction is possible as argued at the end of the proof of Proposition 4.3.

7.4. Lemmas for Section 5

To prove Proposition 5.1, we need the following two lemmas in order to check the conditions of Proposition 3.1 (with $r=1$ ). Recall that $\mathrm{I\!P}_{x}$ (resp. $\mathrm{I\!P}_{\mu}$ ) denotes the probability when the initial condition $X_0=x$ (resp. $X_0$ is distributed as the stationary measure $\mu$ ).

Lemma 7.1. Let $(X_n)_{n\geq 0}$ be a Markov chain satisfying Assumptions $({\mathcal A}_1)$ and $({\mathcal A}_2)$ . Then, for any $0<\epsilon<\epsilon_0$ and any $x_0\in \mathrm{I\!M}$ , it holds that

\begin{align*}\inf_{x\in \mathrm{I\!M}}\mathrm{I\!P}_{x_0}\left(\|X_1-x\|\leq \epsilon\right) \geq \kappa \epsilon^b V_{d},\,\,\, \inf_{x\in \mathrm{I\!M}}\mathrm{I\!P}_{\mu}\left(\|X_1-x\|\leq \epsilon\right) \geq \kappa\epsilon^bV_{d}.\end{align*}

Proof. Using Assumption $({\mathcal A}_2)$ , we have

\begin{eqnarray*}&& \mathrm{I\!P}_{x_0}\left(\|X_1-x\|\leq \epsilon\right)=\mathrm{I\!P}_{x_0}\left(X_1\in B(x,\epsilon)\cap \mathrm{I\!M}\right)= \int_{B(x,\epsilon) \cap \mathrm{I\!M}} K(x_0,dx_1)\\[5pt] && = \int_{B(x,\epsilon)\cap \mathrm{I\!M}} k(x_0,x_1)\nu(dx_1)\\[5pt] && \geq \kappa \int_{B(x,\epsilon)\cap \mathrm{I\!M} }\nu(dx_1)\geq \kappa \epsilon^b\inf_{0<\epsilon<\epsilon_0}\left(\frac{1}{\epsilon^b}\int_{B(x,\epsilon)\cap \mathrm{I\!M} }\nu(dx_1)\right)\geq \kappa \epsilon^b V_{d}.\end{eqnarray*}

The proof of Lemma 7.1 is then complete since $\mathrm{I\!P}_{\mu}\left(\|X_1-x\|\leq \epsilon\right)=\int \mathrm{I\!P}_{x_0}\left(\|X_1-x\|\leq \epsilon\right) d\mu(x_0)$ .

Lemma 7.2. Let $(X_n)_{n\geq 0}$ be a Markov chain satisfying Assumptions $({\mathcal A}_1)$ and $({\mathcal A}_2)$ . Then, for any $0<\epsilon<\epsilon_0$ and $k\in \mathrm{I\!N}\setminus\{0\}$ , it holds that

\begin{align*}\sup_{x\in \mathrm{I\!M}} \mathrm{I\!P}_{\mu}\left(\min_{1\leq i\leq k}\|X_i-x\| > \epsilon\right)\leq \left(1-\kappa\epsilon^bV_{d}\right)^k.\end{align*}

Proof. Set $\mathcal{F}_{n}=\sigma(X_0,\ldots,X_n)$ . By the Markov property and Lemma 7.1,

Lemma 7.2 is proved using the last bound together with an argument by induction on k.

7.5. Proof of Proposition 5.1

Proposition 3.1 , applied with $r=r_n=1$ and $k=k_n=n$ , gives

\begin{align*}\mathrm{I\!P}_{\mu}\left(d_H({\mathbb X}_n ,\mathrm{I\!M})>\epsilon \right) \leq \frac{\sup_{x\in {\mathrm{I\!M}}_{d}}\mathrm{I\!P}_{\mu}\left(\min_{1\leq i\leq n}\|Y_{i,1}-x\|> \epsilon/2\right)}{1-\sup_{x\in {\mathrm{I\!M}}_{d}}\mathrm{I\!P}_{\mu}\left(\|Y_{1,1}-x\|> \epsilon/4\right)},\end{align*}

with $Y_{i,r}= (X_{(i-1)r+1},\cdots,X_{ir})$ so that $Y_{i,1}=X_i$ . Consequently, noting that $\mathrm{I\!M}_d=\mathrm{I\!M}$ , we have

\begin{align*}\mathrm{I\!P}_{\mu}\left(d_H({\mathbb X}_n ,\mathrm{I\!M})>\epsilon \right) \leq \frac{\sup_{x\in {\mathrm{I\!M}}}\mathrm{I\!P}_{\mu}\left(\min_{1\leq i\leq n}\|X_i-x\|> \epsilon/2\right)}{1-\sup_{x\in {\mathrm{I\!M}}}\mathrm{I\!P}_{\mu}\left(\|X_1-x\|> \epsilon/4\right)}.\end{align*}

Now Lemmas 7.1 and 7.2 give

\begin{eqnarray*}&& \sup_{x\in {\mathrm{I\!M}}}\mathrm{I\!P}_{\mu}\left(\min_{1\leq i\leq n}\|X_i-x\|> \epsilon\right) \leq (1-\kappa \epsilon^bV_{d})^{n}\leq \exp\!(\!-n\kappa \epsilon^bV_{d}), \\[5pt] && 1-\sup_{x\in {\mathrm{I\!M}}}\mathrm{I\!P}_{\mu}\left(\|X_1-x\|> \epsilon\right) \geq \kappa\epsilon^b V_{d}>0.\end{eqnarray*}

Combining the three last bounds, we obtain

\begin{align*}\mathrm{I\!P}_{\mu}\left(d_H({\mathbb X}_n ,\mathrm{I\!M})>\epsilon \right) \leq \frac{4^b\exp\!(-n\kappa \epsilon^bV_{d}/2^b)}{\kappa\epsilon^b V_{d}}.\end{align*}

The proof of the proposition is then complete since $\displaystyle \alpha \geq \frac{4^b\exp\!(\!-n\kappa \epsilon^bV_{d}/2^b)}{\kappa\epsilon^b V_{d}}$ is equivalent to

\begin{align*} n\geq \frac{2^b}{\kappa \epsilon^b V_d}\ln\left(\frac{4^b}{\alpha \kappa \epsilon^b V_d}\right).\end{align*}