2. Robustness of AR(1) with ARCH(1) errors

According to Definition 1.2, we consider the perturbed AR(1)-ARCH(1) real-valued $\{{X_n^{(\theta)}}\}_{n \in {\mathbb{N}}}$ defined, for ${X_0^{(\theta)}}$ a given real-valued random variable, by

(2.1) \begin{equation} \textrm{for all } n\geq 1,\quad {X_n^{(\theta)}} \;:\!=\; F_{\xi}\big({X_{n-1}^{(\theta)}},\vartheta^{(\gamma)}_n\big),\end{equation}

where $F_{\xi}(x,v) = \alpha x + v (\beta + \lambda x^2)^{1/2}$ with constants $\alpha \in {\mathbb{R}}$ , $\beta>0$ , $\lambda> 0$ , and $\{\vartheta^{(\gamma)}_n\}_{n \in {\mathbb{N}}}$ has a common PDF $\mathfrak{p}_{\gamma}$ and is independent of ${X_0^{(\theta)}}$ . Therefore, we have $\theta=(\xi,\gamma)$ with $\xi\;:\!=\;(\alpha,\beta,\lambda) \in {\mathbb{R}}\times (0,+\infty)^2$ and $\gamma \in \Gamma$ , where $\Gamma$ is some metric space (typically $\Gamma\subset{\mathbb{R}}$ ). Thus, $\Theta$ is a subset of ${\mathbb{R}}\times (0,+\infty)^2 \times\Gamma$ . Here, $d(x;\;x_0)=|x-x_0|$ and $x_0\;:\!=\;0$ so that $V_a(x) = (1+|x|)^a$ . The Markov kernel ${P_{\theta}}$ of $\big\{{X_n^{(\theta)}}\big\}_{n \in {\mathbb{N}}}$ is given by ${P_{\theta}}(x,A) \;:\!=\; \int_{\mathbb{R}} \textbf{1}_A(y) p_{\theta}(x,y)\, {\textrm{d}} y$ ( $A\in{\mathcal{X}}$ ), with

(2.2) \begin{equation} p_{\theta}(x,y) \;:\!=\; \big(\beta+\lambda x^2\big)^{-1/2} \mathfrak{p}_{\gamma}\bigg(\frac{y-\alpha x}{(\beta+\lambda x^2)^{1/2}}\bigg).\end{equation}

Next, we report the following observations with respect to basic quantities required in Assumptions 1.2 and 1.3. First, it can be checked (see Lemma B.1) that

(2.3) \begin{equation} L_{F_\xi}(\vartheta_1) = \max\!\big(|\alpha-\sqrt{\lambda} \vartheta_1|,\,|\alpha+\sqrt{\lambda} \vartheta_1|\big). \end{equation}

Hence, the real number $\kappa_{a}$ in Assumption 1.3 is

(2.4) \begin{equation} \kappa_{a} = \sup_{\theta\in \Theta} \bigg(\int_{{\mathbb{R}}} \max\!\big(|\alpha-\sqrt{\lambda} v|,\,|\alpha+\sqrt{\lambda} v|\big)^a \mathfrak{p}_{\gamma}(v)\, {\textrm{d}} v \bigg)^{1/a}.\end{equation}

Second, the real number $M_a$ in Assumption 1.2 is given by

(2.5) \begin{equation} M_a \;:\!=\; \sup_{\theta\in \Theta} \sqrt{\beta}\, \mathbb{E}\big[\big| \vartheta^{(\gamma)}_1 \big|^a \big]^{1/a} = \sup_{\theta\in \Theta} \sqrt{\beta} \bigg(\displaystyle\int_{{\mathbb{R}}} |v|^a \mathfrak{p}_{\gamma}(v) \, {\textrm{d}} v \bigg) ^{1/a}.\end{equation}

Note that if $\beta$ lies in a compact set, then $M_a < \infty$ under the following uniform moment condition for the PDF of $\vartheta^{(\gamma)}_1$ : $\sup_{\gamma \in \Gamma} \int_{{\mathbb{R}}} |v|^a \mathfrak{p}_{\gamma}(v) \, {\textrm{d}} v < \infty$ .

Let us formulate the assumptions under which the conclusions of Theorem 1.1 hold true for $\big\{X^{(\theta)}_n\big\}_{n \in {\mathbb{N}}}$ . Let $\theta_0 = (\alpha_0,\beta_0,\lambda_0,\gamma_0)\in {\Theta}^{^{^{\!\!\!\!\kern-1pt\circ}}}$ . We denote by ${\mathbb{L}}^1({\mathbb{R}})$ the usual Lebesgue space, and by $\|\!\cdot\!\|_{{\mathbb{L}}^1({\mathbb{R}})}$ its norm.

Proof. Let $f \in {\mathcal{B}}_0$ be such that $|f |_0\le 1$ . We have, for all $x \in {\mathbb{X}}$ ,

\begin{equation*} \frac{|({P_{\theta}} f)(x)- (P_{\theta_0}f)(x)|}{V_a(x)} = \frac{\big|\int_{{\mathbb{R}} } (p_{\theta} (x,y) - p_{\theta_0}(x,y))f(y) \, {\textrm{d}} y \big|}{V_a(x)} \le \frac{\int_{{\mathbb{R}} } \big|p_{\theta} (x,y) - p_{\theta_0}(x,y) \big|\, {\textrm{d}} y}{V_a(x)}. \end{equation*}

Let $\varepsilon >0$ . Since $\lim_{x\rightarrow +\infty} V_a(x) = +\infty$ and the last term is bounded from above by $2/V_a(x)$ , there exists $B>0$ such that

(2.6) \begin{equation} |x| > B \Rightarrow \textrm{for all } \theta\in\Theta,\quad \frac{|({P_{\theta}} f)(x)- (P_{\theta_0})f(x)|}{V_a(x)} < \frac{\varepsilon}{2}. \end{equation}

It follows that the conclusion of the lemma holds true provided that, under the condition $\lim_{\gamma \rightarrow \gamma_0} \| \mathfrak{p}_{\gamma} - \mathfrak{p}_{\gamma_0} \|_{{\mathbb{L}}^1({\mathbb{R}})}=0$ , we have

(2.7) \begin{equation} \textrm{for all } A > 0, \quad \lim_{\theta \rightarrow \theta_0}\sup_{|x| \le A} \frac{\int_{{\mathbb{R}}}|p_{\theta}(x,y)- p_{\theta_0}(x,y)| \, {\textrm{d}} y}{V_a(x)} =0. \end{equation}

Indeed, (2.6) and (2.7) with $A=B$ ensure that $\| {P_{\theta}} - P_{\theta_0}\|_{0,a} < \varepsilon$ when $\theta$ is sufficiently close to $\theta_0$ .

Let us prove (2.7). It follows from (2.2) that

(2.8) \begin{align} & \int_{{\mathbb{R}} } | p_{\theta} (x,y) - p_{\theta_0}(x,y))| \, {\textrm{d}} y \nonumber \\[5pt] & \quad \le \int_{{\mathbb{R}} } (\beta+\lambda x^2)^{-1/2}\bigg| \mathfrak{p}_{\gamma} \bigg(\frac{y-\alpha x}{(\beta+\lambda x^2)^{1/2}}\bigg) - \mathfrak{p}_{\gamma_0} \bigg(\frac{y-\alpha_0 x}{(\beta_0+\lambda_0 x^2)^{1/2}}\bigg) \bigg| \, {\textrm{d}} y \end{align}

(2.9) \begin{align} + \int_{{\mathbb{R}} } \mathfrak{p}_{\gamma_0}\bigg(\frac{y-\alpha_0 x}{(\beta_0+\lambda_0 x^2)^{1/2}}\bigg) \big| (\beta+\lambda x^2)^{-1/2} - (\beta_0+\lambda_0 x^2)^{-1/2} \big| \, {\textrm{d}} y. \end{align}

First, using the change of variables $z = (y - \alpha x)/(\beta+\lambda x^2)^{1/2}$ in the integral (2.8) and the triangle inequality, we obtain

(2.10)

where

\begin{align*} b_{\beta,\lambda}(x) \;:\!=\; \bigg(\frac{\beta+\lambda x^2}{\beta_0+\lambda_0 x^2}\bigg)^{1/2}, \qquad a_{\alpha}(x) \;:\!=\; x \frac{\alpha-\alpha_0}{(\beta_0+\lambda_0 x^2)^{1/2}}. \end{align*}

The first integral in (2.10) does not depend on x and is equal to $\|\mathfrak{p}_{\gamma} - \mathfrak{p}_{\gamma_0}\|_{{\mathbb{L}}^1({\mathbb{R}})}$ , which converges to 0 when $\gamma \rightarrow \gamma_0$ from the assumption. Now let $A>0$ be fixed. It follows from Lemma B.2 that $\lim_{(\beta,\lambda) \rightarrow (\beta_0,\lambda_0)}\sup_{|x|\le A} |b_{\beta,\lambda}(x) - 1| = 0$ and $\lim_{\alpha \rightarrow \alpha_0}\sup_{|x|\le A} a_{\alpha}(x)= 0$ . Then, under Assumption 2.2, Lemma B.3 allows us to conclude that the second integral in (2.10) is such that

\begin{align*} \lim_{(\alpha,\beta,\lambda) \rightarrow (\alpha_0,\beta_0,\lambda_0)} \sup_{|x|\le A} \int_{{\mathbb{R}} }| \mathfrak{p}_{\gamma_0}(z) - \mathfrak{p}_{\gamma_0}(b_{\beta,\lambda}(x) z + a_{\alpha}(x))| \, {\textrm{d}} z = 0. \end{align*}

Second, let us consider the integral (2.9). We must show that the supremum of this integral on $x\in [\!-\!A,A]$ converges to 0 when $(\beta,\lambda) \rightarrow (\beta_0,\lambda_0)$ . We obtain, for any $x \in {\mathbb{R}}$ such that $|x|\le A$ ,

with $b_\beta(A) \;:\!=\; (\beta/(\beta_0+\lambda_0 A^2))^{1/2} \leq \min_{|x|\leq A} b_{\beta,\lambda}(x)$ and $\int_{{\mathbb{R}}} \mathfrak{p}_{\gamma_0}(z)\,{\textrm{d}} z = 1$ . We know that $\lim_{(\beta,\lambda) \rightarrow (\beta_0,\lambda_0)}\sup_{|x|\le A} |b_{\beta,\lambda}(x) - 1|=0$ from Lemma B.2, so the expected convergence holds.

Remark 2.1. If the PDF $\mathfrak{p}_{\gamma_0}$ for the unperturbed AR(1)-ARCH(1) process is continuous on ${\mathbb{R}}$ , then Assumption 2.1(a) (stated to prove Assumption 1.1) can be omitted in Proposition 2.1. Actually, under the condition $\int_{\mathbb{R}} L_{F_{\xi_{0}}}(v)^a \mathfrak{p}_{\gamma_0}(v)\, {\textrm{d}} v < 1$ , which is contained in Assumption 2.1(c), Assumption 1.1 holds with any real number $\rho_a$ (and the associated constant $C_a$ ) such that

\begin{align*} \bigg(\int_{\mathbb{R}} L_{F_{\xi_{0}}}(v)^a \mathfrak{p}_{\gamma_0}(v)\, {\textrm{d}} v\bigg)^{1/a} <\rho_a <1.\end{align*}

Indeed, the kernel $p_{\theta_0}(x,y)$ given by (2.2) is continuous, so Remark 4.3 and Proposition 4.2 ensure that, under the conditions in Assumption 2.1(b) and (c), $P_{\theta_0}$ satisfies (1.3) for any $\rho_a$ satisfying the above condition. In other words, if the PDF $\mathfrak{p}_{\gamma_0}$ is continuous on ${\mathbb{R}}$ , then only the conditions in Assumption 2.1(b) and (c) with $\Theta=\{\theta_0\}$ are useful in obtaining Assumption 1.1.

3. Robustness of IFS under roundoff error

From [Reference Breyer, Roberts and Rosenthal7, Reference Roberts, Rosenthal and Schwartz42], the effect of roundoff errors using a Markov chain with transition kernel P means considering a Markov chain with a (perturbed) transition kernel of the form $\tilde P(x,A) \;:\!=\; P(x,h^{-1}(A))$ , where $h \;:\; {\mathbb{X}}\rightarrow{\mathbb{X}}$ is such that h(x) is close to x. Let us consider an ${\mathbb{X}}$ -valued IFS as defined in Definition 1.1. Let $(h_\theta)_{\theta\in \Theta}$ be a family of functions on ${\mathbb{X}}$ such that $h_\theta\rightarrow \textrm{id}$ when $\theta \rightarrow \theta_0$ in a sense to be specified later, where id denotes the identity map on ${\mathbb{X}}$ , $\Theta$ is a subset of a metric space, and $\theta_0 \in {\Theta}^{^{^{\!\!\!\!\kern-1pt\circ}}}$ . Then the associated roundoff IFS $\big\{X^{(\theta)}_n\big\}_{n \in {\mathbb{N}}}$ is defined as

\begin{equation*} \textrm{for}\;X^{(\theta)}_0\in {\mathbb{X}}\ \textrm{and}\;\textrm{all}\;n \ge 1,\ X^{(\theta)}_n = F_{\theta}\big( X^{(\theta)}_{n-1}, \vartheta_{n}\big), \end{equation*}

where $F_{\theta}(x,v) \;:\!=\; {h_{\theta}}(F(x,v))$ and $F_{\theta_0}(x,v) = \textrm{id}(F(x,v))= F(x,v)$ . The perturbed/roundoff transition kernels associated with $\big\{X^{(\theta)}_n\big\}_{n \in {\mathbb{N}}}$ (or $({h_{\theta}})_{\theta\in \Theta}$ ) are given by

(3.1) \begin{equation} \textrm{for all } f \in {\mathcal{B}}_0 \textrm{ and } x \in {\mathbb{X}}, \quad ({P_{\theta}} f)(x) = P(f \circ {h_{\theta}})(x) = \int_{{\mathbb{V}}} f(({h_{\theta}}\circ F)(x,v))\, {\textrm{d}}\mathfrak{p}(v). \end{equation}

When the Markov kernel $P_{\theta_0}$ is assumed to be V-geometrically ergodic, the first natural question is to know whether ${P_{\theta}}$ remains V-geometrically ergodic for $\theta$ close to $\theta_0$ . The simplest way used in [Reference Roberts, Rosenthal and Schwartz42] to study this question is to assume that $h_\theta \rightarrow \textrm{id}$ uniformly on ${\mathbb{R}}^q$ when $\theta\rightarrow \theta_0$ (i.e., for all $x\in {\mathbb{R}}^q$ , $\|h(x) - x\| \leq \varepsilon(\theta)$ with $\lim_{\theta\rightarrow \theta_0} \varepsilon(\theta) = 0$ ). However, as mentioned in [Reference Breyer, Roberts and Rosenthal7], this assumption is too restrictive in practice since the roundoff errors for some $x\in{\mathbb{R}}^q$ are obviously proportional to x. In [Reference Breyer, Roberts and Rosenthal7], the authors introduced the weaker assumption $\|h(x) - x\| \leq \varepsilon(\theta)\|x\|$ with $\lim_{\theta\rightarrow \theta_0} \varepsilon(\theta) = 0$ , and proved that the V-geometric ergodicity property is stable for the roundoff Markov kernels under some mild assumptions on the function V. Below, as a by-product of Theorem 1.1, we again find this result in the specific instance of the roundoff process associated with a VAR model $\{X_n\}_{n\in{\mathbb{N}}}$ , but more importantly the sensitivity of the probability distribution of $X^{(\theta)}_n$ and of the stationary distribution of $\big\{X^{(\theta)}_n\big\}_{n \in {\mathbb{N}}}$ when $\theta\rightarrow \theta_0$ is addressed too. These two issues are not investigated in [Reference Breyer, Roberts and Rosenthal7].

Let $\{X_n\}_{n\in{\mathbb{N}}}$ be an ${\mathbb{R}}^q$ -valued VAR model as defined in Example 1.1. To simplify, we assume that, for some $p\geq 1$ , $\Theta$ is an open subset of ${\mathbb{R}}^p$ containing $\theta_0\;:\!=\;0$ (the null vector of ${\mathbb{R}}^p$ ), and we consider a family $(h_\theta)_{\theta\in \Theta}$ of functions on ${\mathbb{X}}\;:\!=\;{\mathbb{R}}^q$ such that $h_{0}=\textrm{id}$ . Thus, the roundoff process $\big\{X^{(\theta)}_n\big\}_{n \in {\mathbb{N}}}$ associated with $F_\theta(x,y) \;:\!=\; h_\theta(Ax+v)$ is the Markov chain with transition kernel ${P_{\theta}}$ (see (3.1)),

(3.2) \begin{equation} \textrm{for all } f \in {\mathcal{B}}_0 \textrm{ and } x \in {\mathbb{X}}, \quad ({P_{\theta}} f)(x) = \int_{{\mathbb{R}}^q} f({h_{\theta}} (Ax + v)) \, {\textrm{d}} \mathfrak{p}(v). \end{equation}

If $g \;:\; {\mathbb{R}}^q\rightarrow {\mathbb{R}}^q$ is differentiable and $z\in{\mathbb{R}}^q$ , we denote by $\nabla g(z)$ the Jacobian matrix of g at z, and we set $\|\nabla g\|_\infty \;:\!=\; \sup_{z\in{\mathbb{R}}^q} \|\nabla g(z)\|$ , where $\|\!\cdot\!\|$ here denotes the induced matrix-norm of Example 1.1. For the sake of simplicity the norms chosen on ${\mathbb{R}}^q$ and ${\mathbb{R}}^p $ are both denoted by $\|\!\cdot\!\|$ . We introduce the following assumptions in order to apply Theorem 1.1 to $\big\{X^{(\theta)}_n\big\}_{n \in {\mathbb{N}} }$ .

Proof. Recall that $\theta_0=0$ here. We know that Assumption 1.1 holds (see Remark 4.2). Next, for any $\theta\in \Theta$ and $z\in{\mathbb{R}}^q$ , set $\Gamma_\theta(z) = |\det \nabla g_\theta(z)|$ . Then, using (3.2), ${P_{\theta}}$ has the form

\begin{equation*} \textrm{for all}\;f \in {\mathcal{B}}_0\;\textrm{and}\;x\in{\mathbb{R}}^q,\; ({P_{\theta}} f)(x) = \int_{{\mathbb{R}}^q} f(z) \mathfrak{p}(g_\theta(z)-Ax) \Gamma_\theta(z)\, {\textrm{d}} z \end{equation*}

from the change of variable $z=h_\theta(Ax+v)$ . Recall that ${P_{\theta}}$ is the transition kernel of the ${\mathbb{R}}^q$ -valued IFS $\big\{X^{(\theta)}_n\big\}_{n\in{\mathbb{N}}}$ associated with $F_\theta(x,v) \;:\!=\; h_\theta(Ax+v)$ . Then, Assumption 3.2 is just Assumption 1.2 (here $x_0=0$ ), while Assumption 1.3 is implied by Assumption 3.3 from Taylor’s inequality applied to $h_\theta$ .

Next, we prove Assumption 1.4. For every $r>0$ , let $B(0,r) = \{z\in {\mathbb{R}}^q \;:\; \|z\| \leq r\}$ . Let $f\in {\mathcal{B}}_0$ be such that $|f|_0\leq 1$ , and let $x\in{\mathbb{R}}^q$ . Fix $\varepsilon >0$ . First, let $K\equiv K(\varepsilon)>0$ be such that $(1+K)^{-a} < \varepsilon/2$ . Then

(3.3) \begin{equation} \textrm{for all } x\in {\mathbb{R}}^q\setminus B(0,K), \quad \frac{| ({P_{\theta}} f)(x) - (P_0 f)(x)|}{V(x)} \leq \frac{2}{V(x)} < \varepsilon. \end{equation}

Now we assume that $x\in B(0,K)$ . Note that

(3.4) \begin{equation} | ({P_{\theta}} f)(x) - (P_0 f)(x) | \leq \int_{{\mathbb{R}}^q} |\mathfrak{p}(g_\theta(z)-Ax) \Gamma_\theta(z) - \mathfrak{p}(z-Ax)|\, {\textrm{d}} z \end{equation}

since $g_0=\textrm{id}$ . Set $d\;:\!=\;2/(1-c)$ , where c is given in Assumption 3.4(b)(i). Note that $\|Ax\| \leq K$ and that Assumption 3.4(b)(i) provides, for all $z\in{\mathbb{R}}^q$ , $\|g_\theta(z)\| \geq (1-c)\|z\|$ . Then we have, for every $z\in{\mathbb{R}}^q$ such that $\|z\| \geq dK$ ,

\begin{align*} \|g_\theta(z) - Ax\| \geq \|g_\theta(z)\| - \|Ax\| \geq (1-c)\|z\| - K \geq (1-c)\|z\| - \frac{1}{d}\|z\| \geq \frac{1-c}{2}\|z\|.\end{align*}

It follows from Assumption 3.4(a) that we have, for every $\theta\in \Theta$ ,

\begin{align*} \|z\| \geq B \equiv B(\varepsilon) \;:\!=\; \max\!(dM, dK) \Rightarrow \mathfrak{p}(g_\theta(z) - Ax) \leq \mathfrak{p}(d^{-1}z).\end{align*}

Since the function $z\mapsto \mathfrak{p}(d^{-1}z)$ is Lebesgue integrable on ${\mathbb{R}}^q$ , we can choose $C\equiv C(\varepsilon)>0$ such that $\int_{\|z\| \geq C} \mathfrak{p}(d^{-1}z)\, {\textrm{d}} z \leq \varepsilon/2(\gamma+1)$ , where $\gamma \;:\!=\; \sup_{\theta\in \Theta} \sup_{z\in{\mathbb{R}}^q} \Gamma_\theta(z)$ . Note that $\gamma < \infty$ from the first condition of Assumption 3.4(b)(iii) and from the continuity of the function $\det\!(\!\cdot\!)$ . Set $D = \max\!(B,C)$ . We deduce from the triangle inequality that, for every $\theta\in \Theta$ ,

(3.5) \begin{equation} \int_{\|z\| \geq D}|\mathfrak{p}(g_\theta(z)-Ax) \Gamma_\theta(z) - \mathfrak{p}(z-Ax)|\, {\textrm{d}} z \leq (\gamma+1) \int_{{\|z\| \geq C}} \mathfrak{p}(d^{-1}z)\, {\textrm{d}} z \leq \frac{\varepsilon}{2}. \end{equation}

Now we investigate the integrand in (3.4) for $z\in B(0,D)$ (recall that $x\in B(0,K)$ ). First, setting $m\;:\!=\; \sup_{u\in{\mathbb{R}}^q} \mathfrak{p}(u)$ , we have, for every $z\in B(0,D)$ and $x\in B(0,K)$ ,

(3.6) \begin{equation} |\mathfrak{p}(g_\theta(z)-Ax) \Gamma_\theta(z) - \mathfrak{p}(z-Ax)| \leq \gamma |\mathfrak{p}(g_\theta(z)-Ax) - \mathfrak{p}(z-Ax)| + m|\Gamma_\theta(z)-1|. \end{equation}

We have, for all $z\in B(0,D)$ , $\|g_\theta(z)\| \leq (1+c)D$ (using Assumption 3.4(b)(i)). From the standard statement for uniform convergence of differentiable functions, we deduce from the conditions in Assumption 3.4(b)(ii) and (b)(iii) that $\lim_{\theta\rightarrow 0} g_\theta = \textrm{id}$ uniformly on B(0, D). Let $\ell_D$ denote the volume of B(0, D) with respect to Lebesgue’s measure on ${\mathbb{R}}^q$ . From the previous uniform convergence and from the uniform continuity of $\mathfrak{p}$ on $B(0,(1+c)D+K)$ , there exists an open neighbourhood ${\mathcal{V}}_0$ of $\theta=0$ in ${\mathbb{R}}^p$ such that

\begin{align*} \textrm{for all } \theta\in {\mathcal{V}}_0,\ z \in B(0,D),\ \textrm{and } x\in B(0,K),\quad |\mathfrak{p}(g_\theta(z)-Ax) - \mathfrak{p}(z-Ax)| < \frac{\varepsilon}{4\gamma \ell_D}.\end{align*}

Moreover, there exists an open neighbourhood ${\mathcal{V}}'_{\!\!0} \subset {\mathcal{V}}_0$ of $\theta=0$ in ${\mathbb{R}}^p$ such that

\begin{align*} \textrm{for all } \theta\in {\mathcal{V}}'_{\!\!0}, \textrm{ and } z \in B(0,D),\quad |\Gamma_\theta(z)-1| < \frac{\varepsilon}{4m \ell_D}\end{align*}

from Assumption 3.4(b)(iii) and from the uniform continuity of the function $\det\!(\!\cdot\!)$ on every compact subset of the set ${\mathcal{M}}_q({\mathbb{R}})$ of real $q\times q$ matrices. It then follows from (3.6) that

\begin{align*} \textrm{for all } \theta\in {\mathcal{V}}'_{\!\!0}, z \in B(0,D), \textrm{ and } x\in B(0,K), \quad |\mathfrak{p}(g_\theta(z)-Ax) \Gamma_\theta(z) - \mathfrak{p}(z-Ax)| \leq \frac{\varepsilon}{2 \ell_D}.\end{align*}

Integrating this inequality on B(0, D) gives

(3.7) \begin{equation} \textrm{for all } \theta\in {\mathcal{V}}'_{\!\!0} \textrm{ and } x\in B(0,K), \quad \int_{\|z\| \leq D} |\mathfrak{p}(g_\theta(z)-Ax) \Gamma_\theta(z) - \mathfrak{p}(z-Ax)|\, {\textrm{d}} z \leq \frac{\varepsilon}{2}. \end{equation}

We deduce from (3.4), (3.5), and (3.7) that

\begin{align*} \textrm{for all } \theta\in {\mathcal{V}}'_{\!\!0} \textrm{ and } x\in B(0,K),\quad \frac{| ({P_{\theta}} f)(x) - (P_0 f)(x)|}{V(x)} \leq | ({P_{\theta}} f)(x) - (P_0 f)(x) | \leq \varepsilon.\end{align*}

This inequality, combined with (3.3), gives Assumption 1.4.

4. $\boldsymbol{V}_{\boldsymbol{a}}$ -geometric ergodicity of IFS

For ${a}\ge 1$ , define for any $x\in{\mathbb{X}}$ , $p(x) \;:\!=\; 1 + d(x,x_0)$ , so that $V_a(x) \;:\!=\; p(x)^a$ , and let us introduce the following space ${\mathcal{L}}_{a}$ :

\begin{equation*} {\mathcal{L}}_{a} \;:\!=\; \bigg\{f \;:\; {\mathbb{X}}\rightarrow{\mathbb{C}} \;:\; m_{a}(f) \;:\!=\; \sup_{(x,y)\in {\mathbb{X}}^2,\,x\neq y}\bigg\{\frac{|f(x)-f(y)|}{d(x,y)\, (p(x) + p(y))^{a-1}} \bigg\} < \infty \bigg\}. \end{equation*}

Such Lipschitz-weighted spaces were introduced in [Reference Le Page30] to obtain the quasi-compactness of Lipschitz kernels (see also [Reference Benda6, Reference Duflo13, Reference Hennion20, Reference Hennion and Hervé21, Reference Milhaud and Raugi38]). Note that, for $f\in{\mathcal{L}}_{a}$ , we have, for all $x\in{\mathbb{X}}$ , $|f(x)|\leq |f(x_0)| + 2^{a-1}\,m_{a}(f)\,V_a(x)$ , so that $|f|_{a} < \infty$ for any $f\in{\mathcal{L}}_{a}$ . Hence, ${\mathcal{L}}_a \subset {\mathcal{B}}_a$ . Moreover, ${\mathcal{L}}_a$ is a Banach space when equipped with the norm, for all $f\in{\mathcal{L}}_{a}$ , $\|f\|_{a} \;:\!=\; m_{a}(f) + |f|_{a}$ .

Let $\{X_n\}_{n\in{\mathbb{N}}}$ be an IFS of Lipschitz maps as in Definition 1.1. For all $x\in {\mathbb{X}}$ and $v\in {\mathbb{V}}$ , we set $F_v x \;:\!=\; F(x,v)$ . Recall that we set $L_F(v) \;:\!=\;L(F_v)$ in Section 1. Since F is fixed in this section, we simply write L(v) for $L_F(v)$ . Similarly, for every $(v_1,\ldots,v_n)\in{\mathbb{V}}^n$ ( $n\in{\mathbb{N}}^*$ ), define

(4.1) \begin{equation} F_{v_n:v_1}\;:\!=\;F_{v_n}\circ \cdots\circ F_{v_1}, \qquad L(v_n\;:\;v_1) \;:\!=\; L(F_{v_n:v_1}). \end{equation}

By hypothesis we have $L(v)<\infty$ , and thus $L(v_n\;:\;v_1) <\infty$ . Note that, for each $a\ge 1$ , the limit $\hat\kappa_a \;:\!=\; \lim_{n\rightarrow+\infty}\mathbb{E}[L(\vartheta_n\;:\;\vartheta_1)^{{a}}]^{{1}/({na})}$ exists in $[0,+\infty]$ , since the sequence $(\mathbb{E}[L(\vartheta_n\;:\;\vartheta_1)^{{a}}])_{n\in{\mathbb{N}}^*}$ is sub-multiplicative. In this section we first present a standard contraction/moment condition, Condition 4.1 (counterpart of Condition 1.1 in Section 1), for P given in (1.2) to have a geometric rate of convergence on ${\mathcal{L}}_a$ (see Proposition 4.1). Then the passage to $V_a$ -geometric ergodicity is addressed in Proposition 4.2.

Condition 4.1. For some ${a}\in[1,+\infty)$ ,

(4.2) \begin{align} \mathbb{E}[d(x_0, F(x_0,\vartheta_1)^{{a}})] & < \infty, \end{align}

(4.3) \begin{align} \widehat{\kappa}_a & < 1. \end{align}

Note that (4.3) is equivalent to

(4.4) \begin{equation} \textrm{there exists } N\in{\mathbb{N}}^* \textrm{ such that } \mathbb{E}[L(\vartheta_N\;:\;\vartheta_1)^{{a}}] <1 ,\end{equation}

and Condition 1.1 in Section 1 corresponds to (4.2) and to (4.4) with $N=1$ .

The properties in the next proposition can be derived from the results of [Reference Duflo13, Chapter 6]; see also [Reference Benda6] for the existence and uniqueness of the invariant distribution. For convenience, in Appendix C the properties (4.5) and (4.6) are proved with explicit constants under the assumptions (4.2) and (4.4) with $N=1$ (i.e. $\mathbb{E}[L(\vartheta_1)^{a}] < 1$ ).

Proposition 4.1. ([Reference Duflo13, Chapter 6].) Under Condition 4.1, P has a unique invariant distribution on $({\mathbb{X}},{\mathcal{X}})$ , denoted by $\pi$ , and we have $\pi(d(x_0,\cdot)^a)<\infty$ . Moreover, the Markov kernel P continuously acts on ${\mathcal{L}}_{a}$ , and for any $\kappa\in(\widehat{\kappa}_a,1)$ , there exist positive constants $c\equiv c_\kappa$ and $c'\equiv c'_{\!\!\kappa}$ such that:

(4.5) \begin{align} \textit{for all } f\in {\mathcal{L}}_{a} \textit{ and } n\geq1,\quad |P^nf-\pi(f)\textbf{1}_{\mathbb{X}}|_{a} \leq c \kappa^n m_a(f);\; \end{align}

(4.6) \begin{align} \textit{for all } f\in {\mathcal{L}}_{a} \textit{ and } n\geq1,\quad\|P^nf-\pi(f)\textbf{1}_{\mathbb{X}}\|_{a}\leq c'\kappa^n\|f\|_a. \end{align}

In particular, if $\kappa_{1,a} \;:\!=\; \mathbb{E}[L(\vartheta_1)^{a}]^{1/a} < 1$ , then

(4.7) \begin{equation} \textit{for all } f\in {\mathcal{L}}_{a} \textit{ and } n\geq1,\quad |P^nf-\pi(f)\textbf{1}_{\mathbb{X}}|_{a} \leq c_1 {\kappa_{1,a}}^n m_a(f), \end{equation}

where the constant $c_1$ is defined by $c_1 \;:\!=\; \xi^{(a-1)/a} \|\pi\|_1 (1 + \|\pi\|_a)^{a-1}$ , with

\begin{align*} \xi \;:\!=\; \sup_{n\geq 1}\,\sup_{x\in{\mathbb{X}}} \frac{(P^nV_a)(x)}{V_a(x)} < \infty, \qquad \|\pi\|_b \;:\!=\; \bigg(\int_{\mathbb{X}} V_b(y)\, {\textrm{d}}\pi(y)\bigg)^{{1}/{b}} \textit{ for } b = 1,a.\end{align*}

Under Condition 4.1, Property (4.5) with $f\;:\!=\;V_a$ and $n\;:\!=\;1$ gives $PV_a\leq \xi_1 V_a$ for some $\xi_1\in(0,+\infty)$ , so that P continuously acts on ${\mathcal{B}}_a$ . But it is worth noticing that Property (4.5) (or (4.7)) does not provide the $V_a$ -geometric ergodicity (1.3) since (4.5) (or (4.7)) is only established for $f\in{\mathcal{L}}_a$ . Under Condition 4.1, it was proved in [Reference Alsmeyer2, Proposition 5.2] that, if $\{X_n\}_{n\in{\mathbb{N}}}$ is Harris recurrent and the support of $\pi$ has a non-empty interior, then $\{X_n\}_{n\in{\mathbb{N}}}$ is $V_a$ -geometrically ergodic. Under Condition 4.1, the Markov chain $\{X_n\}_{n\in{\mathbb{N}}}$ is shown to be $V_a$ -geometrically ergodic in [Reference Wu47, Proposition 7.2] provided that P and $P^N$ for some $N\geq 1$ are Feller and strongly Feller, respectively. An alternative approach is proposed in Proposition 4.2 below. The bound (4.8) is the same as in [Reference Wu47, Proposition 7.2], but the Feller-type assumptions of [Reference Wu47] are replaced with the following: $P^\ell \;:\; {\mathcal{B}}_0\rightarrow{\mathcal{B}}_a$ for some $\ell\geq1$ is compact (see Remark 4.2 for comparisons).

Proposition 4.2. Let us assume that Condition 4.1 holds, and that $P^\ell \;:\; {\mathcal{B}}_0\rightarrow {\mathcal{B}}_a$ for some $\ell\geq1$ is compact. Then P is $V_a$ -geometrically ergodic, and the spectral gap $\rho_{V_a}(P)$ of P on ${\mathcal{B}}_a$ (i.e. the infimum bound of the positive real numbers $\rho_a$ such that Property (1.3) holds) satisfies the following bound:

(4.8) \begin{equation} \rho_{V_a}(P) \leq \widehat{\kappa}_a. \end{equation}

Remark 4.2. Assume that every closed ball of ${\mathbb{X}}$ is compact. Let $\{X_n\}_{n\in{\mathbb{N}}}$ be a Markov chain such that its transition kernel P satisfies the hypothesis that there exist a positive measure $\eta$ on $({\mathbb{X}},{\mathcal{X}})$ and a measurable function $K \;:\; {\mathbb{X}}^2\rightarrow [0,+\infty)$ such that

(4.9) \begin{equation} \textrm{for all } x\in{\mathbb{X}}, \quad P(x,{\textrm{d}} y) = K(x,y)\, {\textrm{d}}\eta(y). \end{equation}

If $P^{\ell}$ is strongly Feller for some $\ell\geq1$ , then $P^{2\ell}$ is compact from ${\mathcal{B}}_0$ to ${\mathcal{B}}_a$ (see, e.g., [Reference Guibourg, Hervé and Ledoux18, Lemma 3]). Hence, if P admits a kernel as in (4.9), then assuming that $P^N$ is strongly Feller for some N in [Reference Wu47, Proposition 7.2] is more restrictive than the compactness hypothesis of Proposition 4.2. A detailed comparison with the approach in [Reference Wu47, Proposition 7.2] is provided in [Reference Guibourg, Hervé and Ledoux18] for general Markov kernels. Finally, note that the transition kernel P of a VAR process (see Example 1.1) is always strongly Feller. Indeed, let $f\in{\mathcal{B}}_0$ be such that $\|f\|_0\leq 1$ . Then we have

\begin{align*} \textrm{for all } (x,x')\in{\mathbb{R}}^q\times{\mathbb{R}}^q, \quad |(Pf)(x') - (Pf)(x)| \leq \int_{{\mathbb{R}}^q} |\mathfrak{p}(y-A(x'-x)) - \mathfrak{p}(y)|\, {\textrm{d}} y.\end{align*}

Since $t\mapsto \mathfrak{p}(\cdot-t)$ is continuous from ${\mathbb{R}}^q$ to the Lebesgue space ${\mathbb{L}}^1({\mathbb{R}}^q)$ , it follows that P is strongly Feller. Thus the $V_a$ -geometric ergodicity of P claimed in Example 1.1 follows from Proposition 4.2. See also [Reference Wu47, Section 8].

Remark 4.3. If $\{X_n\}_{n\in{\mathbb{N}}}$ is an IFS of Lipschitz maps as in Definition 1.1 such that its transition kernel P satisfies the assumption in (4.9) with K continuous in the first variable, then P is strongly Feller, thus $P^{2}$ is compact from ${\mathcal{B}}_0$ to ${\mathcal{B}}_a$ , so that the conclusions of Proposition 4.2 hold true under Condition 4.1. Indeed, we have, for all $(x,x')\in{\mathbb{X}}^2$ and for any $f \in {\mathcal{B}}_0$ ,

\begin{align*} |(Pf)(x') - (Pf)(x)| \leq \int_{\mathbb{X}} |K(x',y) - K(x,y)|\, {\textrm{d}}\eta(y).\end{align*}

Since $K(\cdot,\cdot)\geq0$ , $\int K(\cdot,y)\,{\textrm{d}}\eta(y) =1$ , and $\lim_{x^{\prime}\rightarrow x} K(x',y) = K(x,y)$ , we deduce from Scheffé’s theorem that $\lim_{x^{\prime}\rightarrow x} \int_{\mathbb{X}} |K(x',y) - K(x,y)|\, {\textrm{d}}\eta(y) = 0$ . This proves the desired statement. Note that the previous argument even shows that $\{Pf,\, |f|_0\leq 1\}$ is equicontinuous, so that the compactness of $P \;:\; {\mathcal{B}}_0\rightarrow {\mathcal{B}}_1$ can be directly proved from Ascoli’s theorem.

5. Further applications

Theorem 1.1 was applied in Section 2 to real-valued AR(1) models with ARCH(1) errors (see Proposition 2.1), and in Section 3 to the roundoff errors of a VAR model (see Proposition 3.1). Although these applications have been presented for specific IFSs, it is worth noticing that they give a general road map for investigating the issues in P₁–P₃ of Section 1 for other instances of ${\mathbb{R}}^q$ -valued IFSs, provided that the probability distribution of the noise $\mathfrak{p}_\gamma$ in Definition 1.2 admits a PDF with respect to Lebesgue’s measure on ${\mathbb{V}}={\mathbb{R}}^q$ , and that the change of variable $v \mapsto z = F_\xi(x,v)$ is valid for every $x\in{\mathbb{R}}^q$ , where $F_\xi(\cdot,\cdot)$ is the perturbed function involved in Definition 1.2. In Section 5.1 we propose two examples to support this claim. Finally, in Section 5.2 we discuss the robustness of IFSs of Lipschitz maps under perturbation caused by some thresholding and truncation.

5.1. A general non-linear time series model

Denoting by $\textrm{GL}_q({\mathbb{R}})$ the set of invertible real $q\times q$ matrices, consider an IFS $\{X_n\}_{n\in{\mathbb{N}}}$ of the form

(5.1) \begin{equation} \textrm{for all } n\geq 1,\quad X_n = \psi(X_{n-1}) + B(X_{n-1}) \vartheta_n,\end{equation}

where $\psi \;:\; {\mathbb{R}}^q\rightarrow {\mathbb{R}}^q$ , $B\;:\; {\mathbb{R}}^q\rightarrow \textrm{GL}_q({\mathbb{R}})$ , and the random variables $\{\vartheta_n\}_{n\geq 1}$ have common PDF $\mathfrak{p}$ . If $B(x)=I_q$ for any $x \in {\mathbb{R}}^q$ , where $I_q$ is the identity $q\times q$ matrix, this Markov chain is called a functional-coefficient AR model. The Markov model (5.1) encompasses a very large class of non-linear time series models (see, e.g., [Reference Meyn and Tweedie36, Chapter 2], [Reference Tsay46, Chapter 4]), and [Reference Cline8, Reference Cline9, Reference Cline and Pu10, Reference Meitz and Saikkonen35, and references therein].

As a generalization of Section 2, consider the following general parametric perturbation of the ${\mathbb{R}}^q$ -valued IFS $\{X_n\}_{n\in{\mathbb{N}}}$ defined in (5.1):

\begin{equation*} \textrm{for all}\;n\geq 1,\; {X_n^{(\theta)}} = \psi_\xi\big({X_{n-1}^{(\theta)}}\big) + B_\xi\big({X_{n-1}^{(\theta)}}\big) \vartheta^{(\gamma)}_n, \end{equation*}

with some parametrized maps $\psi_\xi \;:\; {\mathbb{R}}^q\rightarrow {\mathbb{R}}^q$ and $B_\xi \;:\; {\mathbb{R}}^q\rightarrow \textrm{GL}_q({\mathbb{R}})$ , and with an i.i.d. sequence $\big\{\vartheta^{(\gamma)}_n\big\}_{n\geq 1}$ of ${\mathbb{R}}^q$ -valued random variables with common parametric PDF denoted by $\mathfrak{p}_{\gamma}$ (hence $\theta=(\xi,\gamma)$ ). Then, noticing that for every $x\in{\mathbb{R}}^q$ the change of variable $v\mapsto z\;:\!=\;\psi_\xi(x) + B_\xi(x) v$ is valid and leads to ${P_{\theta}}(x,A) \;:\!=\; \int_{\mathbb{R}} \textbf{1}_A(z) p_{\theta}(x,z)\, {\textrm{d}} z$ ( $A\in{\mathcal{X}}$ ), with

(5.2) \begin{equation} p_{\theta}(x,z) \;:\!=\; |\det B_\xi(x)|^{-1} \mathfrak{p}_\gamma( B_\xi(x)^{-1}(z- \psi_\xi(x))), \end{equation}

the following remarks are relevant to investigating Assumptions 1.1–1.4 of Theorem 1.1.

Remark 5.1. If the PDF $\mathfrak{p}_{\gamma_0}$ of the unperturbed IFS (corresponding to some $\theta_0=(\xi_0,\gamma_0)$ ), as well as the functions $\psi_{\xi_0}$ and $B_{\xi_0}$ , are continuous on ${\mathbb{R}}^q$ , then it follows from Remark 4.3 and Proposition 4.2 that $P_{\theta_0}$ is $V_a$ -geometrically ergodic provided that the unperturbed IFS satisfies Condition 1.1. More precisely, in this case, Assumption 1.1 holds with any real number $\rho_a$ (and the associated constant $C_a$ ) such that $\mathbb{E}\big[ L_{F_{\xi_{0}}}\big(\vartheta^{(\gamma_0)}_1\big)^a\big] ^{1/a} <\rho_a <1$ , where $F_{\xi_{0}}(x,v) =\psi_{\xi_0}(x) + B_{\xi_0}(x) v$ .

Remark 5.2. The moment/contractive conditions (1.4) and (1.5) related to $\theta_0=(\xi_0,\gamma_0)$ in Remark 5.1 involve expectations which depend on the above function $F_{\xi_0}$ and on the PDF $\mathfrak{p}_{\gamma_0}$ . Hence, the conditions in Assumptions 1.2 and 1.3 consist in assuming that these expectations are respectively bounded and strictly less than 1 in a uniform way on the parameters $\theta\;:\!=\;(\xi,\gamma)$ near $\theta_0=(\xi_0,\gamma_0)$ (reducing the set $\Theta$ if necessary).

Remark 5.3. Thanks to (5.2), Assumption 1.4 holds provided that, for every $A>0$ ,

\begin{align*} \lim_{\theta\rightarrow \theta_0} \sup_{\|x\| \leq A} \int_{{\mathbb{R}}^q} \frac{\big|p_{\theta}(x,z) - p_{\theta_0}(x,z)\big|}{(1+\|x\|)^a} \, {\textrm{d}} z = 0,\end{align*}

since the previous integral is less than $2/(1+A)^a$ for $\|x\| > A$ . Moreover, the above integral on ${\mathbb{R}}^q$ can be decomposed on some ball of ${\mathbb{R}}^q$ and on its complement in order to use the uniform continuity and decay properties of the kernel $p_{\theta}(\cdot,\cdot)$ (see the proof of Proposition 5.1 in Appendix D).

Next, as a generalization of Section 3, consider the IFS defined by (5.1) under roundoff error. If $(h_\theta)_{\theta\in \Theta}$ is the roundoff family with $h_\theta$ close to $h_0=\textrm{id}$ when $\theta\rightarrow 0$ , then the roundoff transition kernel ${P_{\theta}}(x,A) = P\big(x,h_\theta^{-1}(A)\big)$ is written as ${P_{\theta}}(x,A) \;:\!=\; \int_{\mathbb{R}} \textbf{1}_A(z) p_{\theta}(x,z)\, {\textrm{d}} z$ with

(5.3) \begin{equation} p_{\theta}(x,z) \;:\!=\; \Gamma_\theta(z) \mathfrak{p}(B(x)^{-1}(g_\theta(z) - \psi(x)))\end{equation}

from the change of variable $v\mapsto z \;:\!=\; h_\theta(\psi(x) + B(x)v)$ , where $g_\theta$ denotes the inverse function of $h_\theta$ and $\Gamma_\theta(z) \;:\!=\; |\det B_\xi(x)|^{-1} |\det \nabla g_\theta(z)|$ . Using the kernels in (5.3), Remarks 5.1–5.3 then apply.

5.2. Robustness of IFS under thresholding/truncation

Here we consider ${\mathbb{X}}\;:\!=\;{\mathbb{R}}^d$ ( $d\geq 1$ ) equipped with the Euclidean norm $\|\!\cdot\!\|$ , and ${\mathbb{V}}\;:\!=\;{\mathbb{R}}^q$ ( $q\geq 1$ ) equipped with some norm still denoted by $\|\!\cdot\!\|$ for the sake of simplicity. Let $\{X_n\}_{n\in{\mathbb{N}}}$ be an IFS of Lipschitz maps,

(5.4) \begin{equation} \textrm{for } X_0 \in {\mathbb{R}}^d \textrm{ and all } n\geq 1,\quad X_n \;:\!=\; F(X_{n-1},\vartheta_n),\end{equation}

with $F\;:\;{\mathbb{R}}^d\times {\mathbb{R}}^q \rightarrow {\mathbb{R}}^d$ and $\{\vartheta_n\}_{n\geq 1}$ satisfying the assumptions of Definition 1.1. Suppose that the probability distribution of $\vartheta_1$ is absolutely continuous with respect to Lebesgue’s measure on ${\mathbb{R}}^q$ , with PDF denoted by $\mathfrak{p}$ . Assume that $\{X_n\}_{n\in {\mathbb{N}}}$ is V-geometrically ergodic. Then, a natural question is: what happens if we consider a perturbation of the IFS (5.4) caused by some thresholding and/or truncation? Such an issue may arise as soon as a numerical implementation of the model is considered. Thus, let us investigate the robustness of the IFS (5.4) when thresholding the function F on the infinite set ${\mathbb{X}}$ and truncating the PDF $\mathfrak{p}$ on ${\mathbb{R}}^q$ .

More precisely, for any $\xi\in(0,+\infty)$ let $\Phi_\xi \;:\; {\mathbb{R}}^d\rightarrow {\mathbb{R}}^d$ be the following thresholding function at level $\xi$ :

\begin{equation*} \textrm{for all } x\in{\mathbb{R}}^d,\quad \Phi_\xi(x) = \min\bigg(\frac{\xi}{\|x\|} ,\, 1\bigg) x = \begin{cases} x & \textrm{if} \ \|x\| \leq \xi , \\[5pt] \xi\dfrac{x}{\|x\|} & \textrm{if} \ \|x\| > \xi. \end{cases}\end{equation*}

Moreover, for any $\gamma\in(0,+\infty)$ , define the truncated PDF $\mathfrak{p}_\gamma$ at level $\gamma$ , for all $v\in{\mathbb{R}}^q$ , by

\begin{equation*} \mathfrak{p}_\gamma(v) = c_\gamma \mathfrak{p}(v) \textbf{1}_{B(0,\gamma)}(v)\;\textrm{with}\; c_\gamma \;:\!=\; \bigg(\!\int_{B(0,\gamma)} \mathfrak{p}(v)\, {\textrm{d}} v\bigg)^{-1}, \end{equation*}

where $B(0,\gamma)$ denotes the ball centred at 0 with radius $\gamma$ in ${\mathbb{R}}^q$ . Then, according to Definition 1.2, we consider the perturbed IFS $\{{X_n^{(\theta)}}\}_{n\in{\mathbb{N}}}$ defined by ${X_0^{(\theta)}} \in {\mathbb{X}}$ and

(5.5) \begin{equation} \textrm{for all } n\geq 1,\quad {X_n^{(\theta)}} \;:\!=\; F_{\xi}\big({X_{n-1}^{(\theta)}},\vartheta^{(\gamma)}_n\big), \quad \textrm{ with } F_{\xi}(x,v) \;:\!=\; \Phi_\xi(F(x,v)),\end{equation}

where the sequence $\big\{\vartheta^{(\gamma)}_n\big\}_{n\geq 1}$ of ${\mathbb{R}}^q$ -valued i.i.d. random variables is assumed to admit the common PDF $\mathfrak{p}_\gamma$ . Note that the stability of quantitative bounds for Markov chains via truncation rather than thresholding is studied in [Reference Medina-Aguayo, Rudolf and Schweizer34]. However, it is worth mentioning that we cannot set $\Phi_\xi(x)=0$ for $x \in {\mathbb{R}}^d$ such that $\|x\| > \xi$ , as in [Reference Medina-Aguayo, Rudolf and Schweizer34, Section 3.2, Theorem 9], since the resulting perturbed process is no longer an IFS of Lipschitz maps. Morevover, note that the study of $\{{X_n^{(\theta)}}\}_{n\in{\mathbb{N}}}$ does not fit into the framework of Section 3. Indeed, the family $\{F_\xi, \xi>0 \}$ does not satisfy the assumptions of Section 3 since $\Phi_{\xi}$ is neither bijective nor differentiable. By contrast, each function $\Phi_{\xi}$ is 1-Lipschitz (i.e. $L(\Phi_{\xi}) = 1$ ), and this property is well suited to our perturbation approach. Therefore, Proposition 5.1 is stated in the general framework of Definition 1.1 up to the condition of absolute continuity of the probability distribution of $\vartheta_1$ . The proof of Proposition 5.1 is postponed to Appendix D.

Proposition 5.1. Assume that the unperturbed IFS $\{X_n\}_{n\in{\mathbb{N}}}$ given in (5.4) satisfies Definition 1.1, with $\vartheta_1$ having a PDF on ${\mathbb{R}}^q$ . Moreover, suppose that Assumption 1.1 holds for some $a\geq 1$ , and that $\widetilde{M}_a \;:\!=\; \mathbb{E}\big[ \|F(0,\vartheta_1)\|^a] ^{1/a} < \infty$ , $\widetilde{\kappa}_a \;:\!=\; \mathbb{E}[ L_{F}(\vartheta_1)^a] ^{1/a} < 1$ , and $\mathbb{E}[\|\vartheta_1\|^a] < \infty$ . Let $\kappa_a\in(\widetilde{\kappa}_a,1)$ , and let $\Theta \;:\!=\; (0,+\infty)\times(\gamma_0,+\infty)$ , with $\gamma_0>0$ defined by the condition, for all $\gamma > \gamma_0$ , $c_\gamma \leq (\kappa_a/\widetilde{\kappa}_a)^a$ . Then, the perturbed IFS $\{{X_n^{(\theta)}}\}_{n\in{\mathbb{N}}}$ defined by (5.5) with $\theta\in\Theta$ satisfies P ₁ –P ₃ of Theorem 1.1 with $\Delta_\theta \rightarrow 0$ when $\xi\rightarrow +\infty$ and $\gamma\rightarrow +\infty$ . More precisely, for every $\varepsilon\in (0,2)$ define $A_\varepsilon = 2^a\varepsilon^{-a} -1 $ . Then we have $\Delta_\theta \leq \varepsilon$ , provided that $\theta\;:\!=\;(\xi,\gamma)\in \Theta$ is such that

\begin{align*} |c_\gamma-1| + \bigg(1+\bigg(\frac{\kappa_a}{\widetilde{\kappa}_a}\bigg)^a\bigg) \bigg(\frac{\mathbb{E}[\|\vartheta_1\|^a]}{\gamma^a} + \frac{(2A_\varepsilon\widetilde{\kappa}_a)^a + (2\widetilde{M}_a)^a}{\xi^a}\bigg)\leq \varepsilon.\end{align*}

Appendix A. Proof of (1.7)

Suppose that Assumptions 1.2 and 1.3 are fulfilled. Then we prove the drift inequality (1.7) in Section 1. In fact, for any $\kappa \in (\kappa_{a},1)$ , we prove that the following strengthened inequality holds:

(A.1) \begin{equation} \textrm{for all } \theta \in \Theta, \quad {P_{\theta}} V_a \le \delta_{a} V_a + K_a \textbf{1}_{[\!-\!r_a,r_a]} ,\end{equation}

where the constants $\delta_{a}<1$ and $K_a>0$ are given in (1.7), and where $r_a \;:\!=\; (1+M_a+\kappa_a-\kappa)/(\kappa-\kappa_a)$ . We have, for any $\theta \in \Theta$ and any $ x\in{\mathbb{X}}$ ,

\begin{align*} \bigg(\frac{({P_{\theta}} V_a)(x)}{V_a(x)}\bigg)^{1/a} & = \bigg(\mathbb{E}\bigg[\bigg(\frac{1+d(F_{\xi}\big(x,\vartheta^{(\gamma)}_1\big);\;x_0)} {1+d(x;\;x_0)}\bigg)^a\bigg] \bigg)^{1/a} \\[5pt] & \leq \bigg(\mathbb{E}\bigg[\bigg( \frac{1 + d\big(F_{\xi}\big(x,\vartheta^{(\gamma)}_1\big);\;F_{\xi}\big(x_0,\vartheta^{(\gamma)}_1\big)\big) + d\big(F_{\xi}\big(x_0,\vartheta^{(\gamma)}_1\big);\;x_0\big)}{1+d(x;\;x_0)}\bigg)^a\bigg]\bigg)^{1/a} \\[5pt] & \leq \bigg(\mathbb{E}\bigg[\bigg(\frac{1}{1+d(x;\;x_0)} + L_{F_\xi}\big(\vartheta^{(\gamma)}_1\big) + \frac{d\big(F_{\xi}\big(x_0,\vartheta^{(\gamma)}_1\big);\;x_0\big)}{1+d(x;\;x_0)}\bigg)^a\bigg]\bigg)^{1/a} \\[5pt] & \leq \frac{1}{1+d(x;\;x_0)} + \mathbb{E}\big[ L_{F_\xi}\big(\vartheta^{(\gamma)}_1\big)^a \big]^{1/a} + \frac{ \mathbb{E}\big[ d(F_{\xi}\big(x_0,\vartheta^{(\gamma)}_1\big);\;x_0)^a\big]^{1/a}}{1+d(x;\;x_0)} \end{align*}

using Holder’s inequality. It follows from Assumptions 1.2 and 1.3 that

(A.2) \begin{equation} \textrm{for all } \theta \in \Theta \textrm{ and } x \in {\mathbb{X}}, \quad \bigg(\frac{({P_{\theta}} V_a)(x)}{V_a(x)}\bigg)^{1/a} \le \frac{1}{1+d(x;\;x_0)} + \kappa_{a} + \frac{M_a}{1+d(x;\;x_0)}. \end{equation}

For any $\kappa \in (\kappa_{a},1)$ , set $r_a \;:\!=\; (1+M_a+\kappa_a-\kappa)/(\kappa-\kappa_a)>0$ . Then we have, for every $x\in{\mathbb{X}}$ such that $d(x;\;x_0) >r_a$ ,

\begin{align*} \frac{1+M_a}{1+d(x;\;x_0)} \le \frac{1+M_a}{1+r_a} = \kappa - \kappa_{a}.\end{align*}

It follows that, for every $\theta \in \Theta$ and for every $x\in{\mathbb{X}}$ such that $d(x;\;x_0) >r_a$ ,

(A.3) \begin{equation} ({P_{\theta}} V_a)(x)\le {\kappa}^a V_a(x). \end{equation}

Moreover, for every $\theta \in \Theta$ and for every $x\in{\mathbb{X}}$ such that $d(x;\;x_0) \le r_a$ , we deduce from (A.2) that

(A.4) \begin{equation} ({P_{\theta}} V_a)(x)\le {F_a} V_a(x) \le {F_a} (1+r_a)^a,\end{equation}

where $F_a \;:\!=\; (1+ \kappa_a + M_a)^a$ . Finally, combining (A.3) and (A.4) provides (A.1), and then (1.7) with $\delta_a\;:\!=\;{\kappa_a}^a <1$ and $K_a\;:\!=\; {F_a} (1+r_a)^a >0$ .

Appendix B. Complements to Proposition 2.1

First, we prove (2.3).

(B.1) \begin{equation} L(v) \;:\!=\; \sup_{(x,y)\in {\mathbb{R}}^2,\,x\neq y} \frac{|F(x,v) - F(y,v)|}{|x-y|} = \max\!\big(|\alpha-\sqrt{\lambda} v|;\;|\alpha+\sqrt{\lambda} v|\big). \end{equation}

Proof. Let $v\in{\mathbb{R}}$ be fixed, and define, for all $x\in{\mathbb{R}}$ , $F_v(x) \;:\!=\; F(x,v)$ . Then,

If $v=0$ , (B.1) is obvious. Assume that $v>0$ . Then $F'_{\!\!v}$ is strictly increasing, so that

\begin{align*} \inf_{x\in{\mathbb{R}}} F'_{\!\!v}(x) = \lim_{x\rightarrow -\infty} F'_{\!\!v}(x) = \alpha - \sqrt{\lambda} v \leq \alpha + \sqrt{\lambda} v = \lim_{x\rightarrow +\infty} F'_{\!\!v}(x) = \sup_{x\in{\mathbb{R}}} F'_{\!\!v}(x).\end{align*}

Then $L(v) \le \max\!(|\alpha - \sqrt{\lambda} v|;\; |\alpha + \sqrt{\lambda} v|)$ follows from Taylor’s inequality. If $v<0$ , then $F'_{\!\!v}$ is strictly decreasing, so that

\begin{align*} \inf_{x\in{\mathbb{R}}} F'_{\!\!v}(x) = \lim_{x\rightarrow +\infty} F'_{\!\!v}(x) = \alpha + \sqrt{\lambda} v \leq \alpha - \sqrt{\lambda} v = \lim_{x\rightarrow -\infty} F'_{\!\!v}(x) = \sup_{x\in{\mathbb{R}}} F'_{\!\!v}(x),\end{align*}

and the same conclusion holds. That $L(v) \ge \max\!(|\alpha - \sqrt{\lambda} v|;\; |\alpha + \sqrt{\lambda} v|)$ follows from the inequality $L(v)\ge |F'_{\!\!v}(x)|$ for any $x\in {\mathbb{R}}$ , which is easily deduced from the definition of L(v) in (B.1). Hence, we obtain $L(v) \ge \lim_{x \pm \infty} |F'_{\!\!v}(x)|$ . The proof of (B.1) is complete.

Next, we prove the two following lemmas used in the proof of Proposition 2.1.

Lemma B.2. Let $(\alpha_0,\beta_0,\lambda_0) \in {\mathbb{R}}\times (0,+\infty)^2$ . For any $(\alpha,\beta,\lambda) \in {\mathbb{R}} \times (0,+\infty)^2$ and for any $x \in {\mathbb{R}}$ , define

\begin{align*} b_{\beta,\lambda}(x)\;:\!=\; \bigg(\frac{\beta+\lambda x^2}{\beta_0+\lambda_0 x^2}\bigg)^{1/2}, \qquad a_{{\alpha}}(x) \;:\!=\; x \frac{\alpha-\alpha_0}{\sqrt{\beta_0+\lambda_0 x^2}\,}. \end{align*}

Then, for any $A>0$ ,

\begin{equation*} \lim_{(\beta,\lambda) \rightarrow (\beta_0,\lambda_0)} \sup_{|x|\le A} |b_{\beta,\lambda}(x) - 1| = 0\; and\;\lim_{\alpha \rightarrow \alpha_0}\sup_{|x|\le A} a_{\alpha}(x)= 0. \end{equation*}

Proof. Let $A>0$ . We have, for any $ x \in {\mathbb{R}}$ such that $|x| \le A$ ,

\[ |b_{\beta,\lambda}(x)^2-1| = \bigg|\frac{\beta-\beta_0+(\lambda-\lambda_0) x^2}{\beta_0+\lambda_0 x^2} \bigg| \le \frac{1}{\beta_0} [ |\beta-\beta_0|+|\lambda-\lambda_0| A^2 ]. \]

Therefore, we have $\lim_{(\beta,\lambda) \rightarrow (\beta_0,\lambda_0)} \sup_{|x| \le A}|b_{\beta,\lambda}(x)^2-1|=0$ . Since $1+b_{\beta,\lambda}(x) \geq 1$ , we have $|b_{\beta,\lambda}(x)-1| \leq |b_{\beta,\lambda}(x)^2-1|$ , so that the first convergence is proved. The second one holds since $\sup_{|x|\le A} \big| a_{{\alpha}}(x) \big| \le A |\alpha-\alpha_0|/\sqrt{\beta_0}$ .

The following lemma is an easy extension of the classical continuity property of the map $f\mapsto f(\cdot +\, a)$ from ${\mathbb{R}}$ to ${\mathbb{L}}^1({\mathbb{R}})$ .

Proof. Let ${\mathcal{C}}_K({\mathbb{R}})$ be the set of continuous functions on ${\mathbb{R}}$ with compact support. First, if $g \in {\mathcal{C}}_K({\mathbb{R}})$ , then the desired convergence follows from Lebesgue’s theorem. Second, if $f \in {\mathbb{L}}^1({\mathbb{R}})$ , then we have, for every $g \in {\mathcal{C}}_K({\mathbb{R}})$ and for every $(a,b)\in{\mathbb{R}}^2$ such that $b \geq \frac12$ ,

\begin{align*} & \int_{{\mathbb{R}}}| f(a+b z) - f(z)| \, {\textrm{d}} z \\[5pt] & \le \int_{{\mathbb{R}}}| f(a+b z) - g(a+b z)| \, {\textrm{d}} z + \int_{{\mathbb{R}}}| g(a+b z) - g(z)| \, {\textrm{d}} z + \int_{{\mathbb{R}}}| g(z) - f(z)| \, {\textrm{d}} z \\[5pt] & \quad = \frac{1}{b} \int_{{\mathbb{R}}}| f(y) - g(y)| \, {\textrm{d}} y + \int_{{\mathbb{R}}}| g(a+b z) - g(z)| \, {\textrm{d}} z + \int_{{\mathbb{R}}}| g(z) - f(z)| \, {\textrm{d}} z \\[5pt] & \quad \le \int_{{\mathbb{R}}}| g(a+b z) - g(z)| \, {\textrm{d}} z + 3 \| f -g \|_{{\mathbb{L}}^1({\mathbb{R}})}. \end{align*}

We then conclude by using the density of ${\mathcal{C}}_K({\mathbb{R}})$ in ${\mathbb{L}}^1({\mathbb{R}})$ .

Appendix C. Proof of (4.5) and (4.6) under the assumptions (4.2) and (4.4) with $\boldsymbol{N}=1$

Thoughout this section, the conditions (4.2) and (4.4) with $N=1$ are assumed to hold. Note that (4.4) with $N=1$ is $\kappa_{1,a} = \mathbb{E}[L(\vartheta_1)^{a}]^{1/a} < 1$ . We prove (4.5) and (4.6) of Proposition 4.1 with explicit constants. Under the general assumption $\widehat{\kappa}_a<1$ of Assumption 4.1, the proofs of (4.5) and (4.6) are similar (replace P with $P^N$ where N is such that $\mathbb{E}[L(\vartheta_N\;:\;\vartheta_1)^{{a}}] < 1$ ).

That the constant $\xi$ in Proposition 4.1 is finite can be easily deduced from the drift inequality (A.1) which holds here with $P_{\theta_0}=P$ , $\Theta=\{\theta_0\}$ , and with $\kappa_{1,a}$ in place of $\kappa_a$ . Now let us introduce some notation. If $\mu$ is a probability measure on ${\mathbb{X}}$ and $X_0\sim\mu$ , we make a slight abuse of notation in writing $\{X_n^\mu\}_{n\in{\mathbb{N}}}$ for the associated IFS given in Definition 1.1. We simply write $\{X_n^x\}_{n\in{\mathbb{N}}}$ when $\mu\;:\!=\;\delta_x$ is the Dirac mass at some $x\in{\mathbb{X}}$ . We denote by ${\mathcal{M}}_a$ the set of all the probability measures $\mu$ on ${\mathbb{X}}$ such that $\|\mu\|_a \;:\!=\; \big(\!\int_{\mathbb{X}} V_a(y)\, {\textrm{d}}\mu(y)\big)^{1/a} < \infty$ . Finally, for $n\in{\mathbb{N}}$ and for any probability measures $\mu_1$ and $\mu_2$ on ${\mathbb{X}}$ , define $\Delta_n(\mu_1,\mu_2) \;:\!=\; d\big(X_n^{\mu_1},X_n^{\mu_2}\big)\, \big(p(X_n^{\mu_1}) + p(X_n^{\mu_2})\big)^{a-1}$ .

Lemma C.1. We have, for all $n\geq1$ and $(\mu_1,\mu_2)\in {\mathcal{M}}_{a}\times{\mathcal{M}}_{a}$ ,

(C.1) \begin{equation} \mathbb{E}[\Delta_n(\mu_1,\mu_2)] \leq \xi^{({a-1})/{a}} \kappa_{1,a}^{\, n} \mathbb{E}\big[d\big(X_0^{\mu_1},X_0^{\mu_2}\big)\big](\|\mu_1\|_a + \|\mu_2\|_a)^{a-1}. \end{equation}

Furthermore, we have, for all $f\in{\mathcal{L}}_{a}$ ,

(C.2) \begin{equation} \mathbb{E}\big[|f(X_{n}^{\mu_1}) - f(X_n^{\mu_2})|\big] \leq \xi^{({a-1})/{a}} m_{a}(f) \kappa_{1,a}^{n}\mathbb{E}\big[d\big(X_0^{\mu_1},X_0^{\mu_2}\big)\big] (\|\mu_1\|_a + \|\mu_2\|_a)^{a-1}. \end{equation}

Proof. Note that $X_n^\mu = F_{\vartheta_n:\vartheta_1}X_0^{\mu}$ from Definition 1.1 and the notation introduced in (4.1). If $a\;:\!=\;1$ , then (C.1) follows from the independence of the $\vartheta_n$ and from the definitions of L(v) and $\kappa_{1,a}$ . Now assume that $a\in(1,+\infty)$ . Without loss of generality, we can suppose that the sequence $\{\vartheta_n\}_{n\geq1}$ is independent of $(X_0^{\mu_1},X_0^{\mu_2})$ . Also note that, if $\mu\in {\mathcal{M}}_{a}$ , then we have $\mathbb{E}\big[p(X_n^{\mu})^a\big] = \int_{\mathbb{X}} (P^nV_a)(x) \, {\textrm{d}}\mu(x) \leq \xi \|\mu\|_a^a$ . From Holder’s inequality (use $1 = 1/a + (a-1)/a$ ), we obtain

\begin{align*} \mathbb{E}[\Delta_n(\mu_1,\mu_2)] & = \mathbb{E}[d(F_{\vartheta_n:\vartheta_1}X_0^{\mu_1},F_{\vartheta_n:\vartheta_1}X_0^{\mu_2}) (p(X_n^{\mu_1}) + p(X_n^{\mu_2}))^{a-1}] \\[5pt] & \leq \mathbb{E}[d(X_0^{\mu_1},X_0^{\mu_2})] \mathbb{E}[L(\vartheta_n\;:\;\vartheta_1)(p(X_n^{\mu_1}) + p(X_n^{\mu_2}))^{a-1} ] \\[5pt] & \leq \mathbb{E}[d(X_0^{\mu_1},X_0^{\mu_2})] \mathbb{E}[L(\vartheta_n\;:\;\vartheta_1)^{a}]^{{1}/{a}} \mathbb{E}[(p(X_n^{\mu_1}) + p(X_n^{\mu_2}))^{a}]^{({a-1})/{a}} \\[5pt] & \leq \mathbb{E}[d(X_0^{\mu_1},X_0^{\mu_2})] \mathbb{E}[L(\vartheta_1)^{a}]^{{n}/{a}} \xi^{({a-1})/{a}} (\|\mu_1\|_a + \|\mu_2\|_a)^{a-1}. \end{align*}

This proves (C.1); (C.2) follows from (C.1) and the definition of $m_{a}(f)$ .

Now recall that we consider the case $N=1$ in (4.4). Let us prove the inequality (4.5) in this case (that is, (4.7)). Applying (C.2) to $\mu_1\;:\!=\;\delta_x$ and $\mu_2\;:\!=\;\pi$ gives

\begin{align*} |P^nf(x) - \pi(f)| & \leq \mathbb{E}[\, |f(X_{n}^{x}) - f(X_{n}^{\pi})|\, ] \\[5pt] & \leq \xi^{({a-1})/{a}} m_{a}(f) \kappa_{1,a}^{n} \mathbb{E}[d(x,X_0^{\pi})](\|\delta_x\|_a + \|\pi\|_a)^{a-1}. \end{align*}

Next, observe that $\|\delta_x\|_a=p(x)$ , and

\begin{align*} \mathbb{E}[d(x,X_0^{\pi})] \leq \mathbb{E}[d(x,x_0) + d(x_0,X_0^{\pi})] \leq p(x) + \pi(d(x_0,\cdot)) \leq p(x) \|\pi\|_1.\end{align*}

Hence, $\mathbb{E}[d(x,X_0^{\pi})] (\|\delta_x\|_a + \|\pi\|_a)^{a-1} \leq p(x)^a \|\pi\|_1 (1 + \|\pi\|_a)^{a-1}$ . This proves the expected inequality.

Finally, to prove (4.6), it remains to study $m_a(P^nf)$ for $f\in{\mathcal{L}}_a$ . Applying (C.2) to $\mu_1\;:\!=\;\delta_x$ and $\mu_2\;:\!=\;\delta_y$ for any $(x,y)\in{\mathbb{X}}^2$ gives

\begin{align*} \textrm{for all } f\in {\mathcal{L}}_a,\quad|P^nf(x) - P^nf(y)| \leq \xi^{({a-1})/{a}} m_a(f) \kappa_{1,a}^{n} d(x,y) (p(x) + p(y))^{a-1}.\end{align*}

Thus, $m_a(P^nf) \leq \xi^{({a-1})/{a}} m_a(f) \kappa_{1,a}^{n}$ . Since $m_a(\textbf{1}_{X})=0$ , this gives

\begin{align*} m_a(P^nf-\pi(f)\textbf{1}_{X}) \leq \xi^{({a-1})/{a}} m_a(f) \kappa_{1,a}^{n}.\end{align*}

Combining the last inequality with (4.7) gives (4.6).

Appendix D. Proof of Proposition 5.1

For every $\theta\;:\!=\;(\xi,\gamma)\in \Theta \;:\!=\; (0,+\infty)\times(\gamma_0,+\infty)$ , the expectation

\begin{align*} \mathbb{E}\big[ \big\|F_{\xi}\big(0,\vartheta^{(\gamma)}_1\big)\big\|^a\big] =\int_{{\mathbb{R}}^d} \|\Phi_\xi(F(0,v))\|^a \mathfrak{p}_\gamma(v)\, {\textrm{d}} v \leq\bigg(\frac{\kappa_a}{\widetilde{\kappa}_a}\bigg)^a \int_{{\mathbb{R}}^q} \|F(0,v)\|^a \mathfrak{p}(v)\, {\textrm{d}} v \end{align*}

is finite, so Assumption 1.2 holds with $x_0=0$ and $M_a\;:\!=\; \widetilde{M}_a\kappa_a/\widetilde{\kappa}_a$ . Moreover, note that $L_{\Phi_\xi} \leq 1$ , so that, for all $v\in{\mathbb{V}}$ , $L_{F_\xi}(v) \leq L_{F}(v)$ . Hence, we have, for every $\theta=(\xi,\gamma)\in \Theta$ ,

\begin{align*} \mathbb{E}\big[ L_{F_\xi}\big(\vartheta^{(\gamma)}_1\big)^a\big] =\int_{{\mathbb{R}}^d} L_{F_\xi}(v)^a \mathfrak{p}_\gamma(v)\, {\textrm{d}} v \leq c_\gamma\int_{{\mathbb{R}}^d} L_{F}(v)^a \mathfrak{p}(v)\, {\textrm{d}} v \leq c_\gamma {\widetilde{\kappa}_a}^a \leq \kappa_a^a.\end{align*}

Thus, Assumption 1.3 holds. It remains to check Assumption 1.4 and to specify the error term $\Delta_\theta$ . Let P (respectively ${{P_{\theta}}}$ ) denote the transition kernel of the unperturbed IFS $\{X_n\}_{n\in{\mathbb{N}}}$ (respectively of the perturbed IFS $\{{X_n^{(\theta)}}\}_{n\in{\mathbb{N}}}$ ). Let $\varepsilon > 0$ , and let $f\in{\mathcal{B}}_0$ be such that $|f|_0\leq 1$ . First, note that we have, for every $x\in{\mathbb{R}}^d$ satisfying $\|x\| > A_\varepsilon$ ,

(D.1) \begin{equation} \frac{|({{P_{\theta}}}f)(x) - (Pf)(x)|}{V_a(x)} \leq \frac{2}{V_a(x)} \leq \varepsilon\end{equation}

by the definition of $A_\varepsilon$ in Proposition 5.1. Next, for every $\theta\;:\!=\;(\xi,\gamma)\in \Theta$ and for every $x\in{\mathbb{R}}^d$ , define the following subsets $E_{\theta,x}$ and $G_{\theta,x}$ of ${\mathbb{R}}^q$ :

\begin{align*} E_{\theta,x} & \;:\!=\; \{v\in {\mathbb{R}}^q \;:\; v\in B(0,\gamma),\, \|F(x,v)\| \leq \xi\}, \\[5pt] G_{\theta,x} & \;:\!=\; \{v\in {\mathbb{R}}^q \;:\; v\in B(0,\gamma),\, \|F(x,v)\| > \xi\}.\end{align*}

From the definition of the thresholding function $\Phi_\xi$ , we have, for every $x\in {\mathbb{R}}^d$ ,

\begin{align*} ({{P_{\theta}}}f)(x) & = \int_{{\mathbb{R}}^q} f(F_{\xi}(x,v)) \mathfrak{p}_\gamma(v)\, {\textrm{d}} v \\[5pt] & = c_\gamma \int_{E_{\theta,x}} f(F(x,v)) \mathfrak{p}(v)\, {\textrm{d}} v + c_\gamma \int_{G_{\theta,x}} f(\eta_{x,v}) \mathfrak{p}(v)\, {\textrm{d}} v , \end{align*}

with $\eta_{x,v} \;:\!=\;\xi\|F(x,v)\|^{-1} F(x,v)$ . Hence,

\begin{align*} |({{P_{\theta}}}f)(x) - (Pf)(x)| & \leq |c_\gamma-1| \int_{E_{\theta,x}} \mathfrak{p}(v)\, {\textrm{d}} v + (1+c_\gamma) \int_{{\mathbb{R}}^q \setminus E_{\theta,x}} \mathfrak{p}(v)\, {\textrm{d}} v \\[5pt] & \leq |c_\gamma-1| + \bigg(1+\bigg(\frac{\kappa_a}{\widetilde{\kappa}_a}\bigg)^a\bigg) \bigg({\mathbb{P}}(\|\vartheta_1\| > \gamma) + {\mathbb{P}}(\|F(x,\vartheta_1)\| > \xi)\bigg)\end{align*}

from the definition of ${\mathbb{R}}^q\setminus E_{\theta,x}$ and the condition $c_\gamma \leq (\kappa_a/\widetilde{\kappa}_a)^a$ . Now let $x\in{\mathbb{R}}^d$ be such that $\|x\| \leq A_\varepsilon$ . Then, for all $v\in{\mathbb{R}}^q$ , $\|F(x,v)\| \leq L_{F}(v) A_\varepsilon + \|F(0,v)\|$ from $F(x,v) = (F(x,v)-F(0,v)) + F(0,v)$ and the triangle inequality. Therefore, $\big[ L_{F}(v) A_\varepsilon \leq {\xi}/{2}$ and $\|F(0,v)\| \leq {\xi}/{2}\big] \Rightarrow \|F(x,v)\| \leq \xi$ , from which we deduce that

\begin{align*} {\mathbb{P}}(\|F(x,\vartheta_1)\| > \xi ) & \leq {\mathbb{P}}\bigg(L_{F}(\vartheta_1) > \frac{\xi}{2A_\varepsilon}\bigg) + {\mathbb{P}}\bigg(\|F(0,\vartheta_1)\| > \frac{\xi}{2}\bigg) \\[5pt] & \leq \frac{(2A_\varepsilon)^a}{\xi^a}\mathbb{E}\big[L_{F}(\vartheta_1)^a\big] + \frac{2^a}{\xi^a}\mathbb{E}\big[\|F(0,\vartheta_1)\|^a\big] \\[5pt] & \leq \frac{(2A_\varepsilon \widetilde{\kappa}_a)^a + (2\widetilde{M}_a)^a}{\xi^a} \end{align*}

from the Markov inequality. Consequently, we obtain that, for every $x\in{\mathbb{R}}^d$ such that $\|x\| \leq A_\varepsilon$ ,

\begin{align*} \frac{|({{P_{\theta}}}f)(x) - (Pf)(x)|}{V_a(x)} & \leq |({{P_{\theta}}}f)(x) - (Pf)(x)| \\[5pt] & \leq |c_\gamma-1| + \bigg(1+\bigg(\frac{\kappa_a}{\widetilde{\kappa}_a}\bigg)^a\bigg) \bigg(\frac{\mathbb{E}\big[\|\vartheta_1\|^a\big]}{\gamma^a} + \frac{(2A_\varepsilon\widetilde{\kappa}_a)^a + (2\widetilde{M}_a)^a}{\xi^a}\bigg). \end{align*}

The conclusion of Proposition 5.1 follows from this and (D.1).