2 Quenched central limit theorem for random LSV maps

Throughout this paper, $(\Omega , \mathcal F, \mathbb P)$ is an arbitrary probability space equipped with an invertible $\mathbb P$ -preserving transformation $\sigma \colon \Omega \to \Omega $ . We assume that $\mathbb P$ is ergodic. Moreover, m will denote the Lebesgue measure on the unit interval $[0, 1]$ . We recall that for each $\gamma \in (0, 1)$ , the associated Liverani–Saussol–Vaienti (LSV) map (introduced in [Reference Liverani, Saussol and Vaienti31]) is given by

$$ \begin{align*} T_\gamma(x)=\begin{cases} x(1+2^\gamma x^\gamma), & 0\le x<\frac 1 2, \\ 2x-1, & \frac 1 2 \le x\le 1. \end{cases} \end{align*} $$

Moreover, for $\gamma \in (0, 1)$ and an observable $\phi $ , we set

$$ \begin{align*} N_\gamma\phi:= g_\gamma'\phi\circ g_\gamma, \end{align*} $$

where $g_\gamma $ denotes the inverse of the restriction of $T_\gamma $ to $[0, \frac 1 2]$ . Finally, we will also denote $v_\gamma (x):=\partial _\gamma T_\gamma (x)=2^\gamma x^{1+\gamma }\log (2x)$ and

$$ \begin{align*} X_\gamma:=v_\gamma\circ g_\gamma=2^\gamma g_\gamma^{1+\gamma}\log(2g_\gamma). \end{align*} $$

For each $\omega \in \Omega $ , let $T_\omega \colon [0, 1] \to [0, 1]$ be an LSV map with the parameter $\beta (\omega ) \in (0, 1)$ . As usual, we set

$$ \begin{align*} T_\omega^n:=T_{\sigma^{n-1}\omega}\circ \cdots \circ T_{\sigma \omega}\circ T_\omega, \quad \omega \in \Omega, \ n\in \mathbb N. \end{align*} $$

In the same manner, denoting by $\mathcal L_\omega \colon L^1(m)\to L^1(m)$ the transfer operator associated to $T_\omega $ and m, we set

$$ \begin{align*} \mathcal L_\omega^n:=\mathcal L_{\sigma^{n-1}\omega}\circ \cdots \circ \mathcal L_{\sigma \omega}\circ \mathcal L_\omega, \quad \omega \in \Omega, \ n\in \mathbb N. \end{align*} $$

We also set $\mathcal L_\omega ^0 :=\textrm {Id}$ for $\omega \in \Omega $ . In the following, we will assume that $\omega \mapsto \beta (\omega )$ is measurable and that

$$ \begin{align*} \alpha:=\operatorname{\mathrm{esssup}}_{\omega \in \Omega}\beta(\omega)< 1. \end{align*} $$

It is proved in [Reference Dragičević, Gonzalez-Tokman and Sedro17, Proposition 9] that the cocycle of maps $(T_\omega )_{\omega \in \Omega }$ admits a unique random absolutely continuous invariant measure (a.c.i.m.) $\mu $ on $\Omega \times [0, 1]$ , which can be identified with a family of probability measures $(\mu _\omega )_{\omega \in \Omega }$ on $[0, 1]$ such that

$$ \begin{align*} T_\omega^*\mu_\omega=\mu_{\sigma \omega} \quad \text{for } \mathbb P\text{-almost every (a.e.) } \omega \in \Omega. \end{align*} $$

Moreover, $d\mu _\omega =h_\omega \, dm$ with $h_\omega \in \mathcal C_*\cap \mathcal C_2$ for some $a>1$ , where $\mathcal C_*=\mathcal C_*(a)$ and ${\mathcal C_2=\mathcal C_2(b_1, b_2)}$ are cones as in [Reference Dragičević, Gonzalez-Tokman and Sedro17, §2.2], that is, $\mathcal C_*$ consist of $\phi \in C^0(0, 1]\cap L^1(m)$ such that $\phi \ge 0$ , $\phi $ decreasing, $X^{\alpha +1}\phi $ increasing (where X denotes the identity map), and

$$ \begin{align*} \int_0^x \phi (t)\, dt\le ax^{1-\alpha}\int_0^1 \phi (t)\, dt, \quad x\in (0, 1]. \end{align*} $$

Moreover, $\mathcal C_2$ consists of all $\phi \in C^2(0, 1]$ so that

$$ \begin{align*} \phi (x)\ge 0, \quad |\phi'(x)| \le \frac{b_1}{x}\phi(x) \quad \text{and} \quad |\phi"(x)| \le \frac{b_2}{x^2}\phi(x), \quad x\in (0, 1]. \end{align*} $$

We stress that the parameters a, $b_1$ , and $b_2$ depend only on $\alpha $ . Throughout the paper, we will often use the following simple result.

If $h\in \mathcal C_*$ and $\varphi \in C^1[0, 1]$ , then by [Reference Leppänen and Stenlund30, Lemma 3.3] (see also [Reference Nicol, Török and Vaienti33, Remark 2.5] and [Reference Nicol, Perez Pereira and Török32, Lemma 3.4]),

(1) $$ \begin{align} \varphi h = g_1 - g_2 \end{align} $$

for $g_1,g_2 \in {\mathcal {C}}_*$ with $\Vert g_i \Vert _{L^1(m)} \le C_\alpha \Vert \varphi \Vert _{C^1} \Vert h \Vert _{L^1(m)}$ for $i = 1,2$ . Moreover, if $h \in C(0,1]$ , then $g_i \in C^1(0,1]$ .

Proposition 1. There exists $C_\alpha>0$ such that

(2) $$ \begin{align} \|\mathcal L_\omega^n (\varphi h)\|_{L^1(m)} \le C_\alpha n^{-1/\alpha+1}\|\varphi \|_{C^1} \cdot \|h\|_{L^1(m)} \end{align} $$

for $\mathbb P$ -a.e. $\omega \in \Omega $ , $n\in \mathbb N$ , and $\varphi \in C^1[0, 1]$ , $h\in \mathcal C_*$ with $\int _0^1 \varphi h\, dm=0$ .

Proposition 2. Suppose that $\psi \in C^1(0,1]$ satisfies $\int _0^1 \psi \, d m = 0$ and, for $x \in (0,1]$ ,

$$ \begin{align*} |\psi(x)| \le C_{0} x^{-\gamma} \quad \text{and} \quad |\psi'(x)| \le C_{1} x^{-\gamma - 1}, \end{align*} $$

where $0 < \gamma \le \alpha $ and $C_0,C_1> 0$ . Then, there exists $C=C_{a, \alpha , \gamma }>0$ such that, for ${\mathbb {P}}$ -a.e. $\omega \in \Omega $ ,

(3) $$ \begin{align} \Vert {\mathcal{L}}_\omega^n ( \psi ) \Vert_{L^1(m)} \le C(C_{0} + C_{ 1}) n^{-1/\alpha + \gamma/ \alpha} \end{align} $$

holds for $n \in \mathbb {N}$ . In particular, if $\varphi \in C^1[0, 1]$ satisfies $\int _0^1 \varphi h_\omega \, dm = 0$ for ${\mathbb {P}}$ -a.e. $\omega \in \Omega $ , and

$$ \begin{align*} |\varphi(x)| \le C_\varphi x^{\gamma} \quad \text{ for all } x \in [0,1], \end{align*} $$

where $\gamma \ge 0$ , then there exists a constant $C=C_{a, \alpha , \gamma }> 0$ such that, for ${\mathbb {P}}$ -a.e. $\omega \in \Omega $ ,

(4) $$ \begin{align} \Vert {\mathcal{L}}_\omega^n ( \varphi h_\omega ) \Vert_{L^1(m)} \le C ( C_\varphi + \Vert \varphi \Vert_{C^1} ) n^{ 1 - (1 + \min\{ \gamma, \alpha \} )/\alpha} \end{align} $$

holds for $n \in \mathbb {N}$ .

Proof. Denote by ${\mathcal {C}}_*(a; \gamma )$ the cone obtained by replacing $\alpha $ with $\gamma $ in the definition of $\mathcal C_*$ . It follows from [Reference Dragičević, Gonzalez-Tokman and Sedro17, Lemma 3] or [Reference Leppänen29, Lemma 3.3] that (note that although $F \in C^2(0,1]$ is assumed in [Reference Dragičević, Gonzalez-Tokman and Sedro17, Lemma 3], the proof only uses $F \in C^1(0,1]$ )

$$ \begin{align*} \psi(x) = \psi(x) + \unicode{x3bb} x^{- \gamma} - \unicode{x3bb} x^{- \gamma} = \psi_{1} - \psi_{ 2}, \end{align*} $$

where $\psi _{i} \in {\mathcal {C}}_*(a; \gamma )$ , provided that

$$ \begin{align*} \unicode{x3bb} \ge ( C_{0} + C_{1} ) \max \bigg\{2, \frac{1}{\gamma}, \frac{1}{a - 1}\bigg\}. \end{align*} $$

If $\int _0^1 \psi _{ 1} \, dm = 0$ , then $\psi _1(x) = 0$ for m-a.e. $x \in [0,1]$ and (3) holds trivially. Otherwise, let $\tilde {\nu }_{i}$ denote the probability measure with density $\tilde {\psi }_{ i} = \psi _{ i} / \int _0^1 \psi _{ i} \, dm \in {\mathcal {C}}_*(a; \gamma )$ . Then, $\tilde {\nu }_{i}$ is regular in the sense of [Reference Korepanov and Leppänen26, Definition 3.5]; see [Reference Korepanov and Leppänen26, Proposition 3.14]. Moreover, the first entry time

$$ \begin{align*} \tau_\omega(x) = \inf \{ n \ge 1 \: : \: T_\omega^n(x) \in [1/2,1] \} \end{align*} $$

satisfies

$$ \begin{align*} \tilde{\nu}_{i}( \tau_\omega \ge n ) \le C_{a, \gamma, \alpha} n^{- 1 / \alpha + \gamma / \alpha}. \end{align*} $$

In the terminology of [Reference Korepanov and Leppänen26], this means that $\tilde {\nu }_{i}$ has tail bound $C_{a,\gamma , \alpha } n^{- 1 / \alpha + \gamma / \alpha }$ . Note that

$$ \begin{align*} \Vert {\mathcal{L}}_\omega^n ( \psi ) \Vert_{L^1(m)} = \frac{\unicode{x3bb}}{ 1 -\gamma} \Vert {\mathcal{L}}_\omega^n ( \tilde{\psi}_{ 1} - \tilde{\psi}_{ 2} ) \Vert_{L^1(m)}. \end{align*} $$

Now, (3) follows from [Reference Korepanov and Leppänen26, Theorem 3.8], and (4) is a straightforward consequence of (3) using $h_\omega \in {\mathcal {C}}_* \cap {\mathcal {C}}_2$ .

We are now in a position to state and prove our quenched central limit theorem for random compositions of LSV maps.

Theorem 3. Let $\Phi \colon \Omega \times [0, 1] \to \mathbb R$ be a measurable map such that $\varphi _\omega :=\Phi (\omega , \cdot )\in C^1[0, 1]$ for $\omega \in \Omega $ and that

(5) $$ \begin{align} \omega \mapsto \|\varphi_\omega\|_{C^1} \in L^2(\Omega, \mathcal F, \mathbb P). \end{align} $$

Moreover, assume that one of the following two conditions holds:

1. (i) $\alpha < 1/2$ ;

2. (ii) $\alpha \ge 1/2$ and, for $x \in [0,1]$ and ${\mathbb {P}}$ -a.e. $\omega \in \Omega $ ,

   (6) $$ \begin{align} |\varphi_\omega(x)| \le C_\omega x^{\gamma} \quad \text{and} \quad \int_0^1 \varphi_\omega h_\omega \, dm = 0 \end{align} $$
   hold with $\gamma> 2 \alpha - 1$ , where $\omega \mapsto C_\omega \in L^2(\Omega , \mathcal {F}, {\mathbb {P}})$ .

Then, for $\mathbb P$ -a.e. $\omega \in \Omega $ ,

$$ \begin{align*} \frac{\sum_{i=0}^{n-1}(\varphi_{\sigma^i \omega}\circ T_\omega^i-\int_0^1\varphi_{\sigma^i \omega}\, d\mu_{\sigma^i \omega})}{\sqrt n}\to_d \mathcal N(0, \Sigma^2) \quad \text{on } ([0, 1], \mu_\omega), \end{align*} $$

where

$$ \begin{align*} \Sigma^2:=\int_\Omega \int_0^1 \varphi_{\omega, 0}^2h_\omega \, dm\, d\mathbb P(\omega)+2\sum_{n=1}^\infty\int_\Omega \int_0^1 (\varphi_{\omega, 0}h_\omega) \cdot (\varphi_{\sigma^n \omega, 0}\circ T_\omega^n)\, dm\, d\mathbb P(\omega), \end{align*} $$

and $\varphi _{\omega , 0}:=\varphi _\omega -\int _0^1 \varphi _\omega h_\omega \, dm$ . Here, $\mathcal N(0, \Sigma ^2)$ denotes the normal distribution with mean $0$ and variance $\Sigma ^2$ provided that $\Sigma ^2>0$ , and the unit mass in $0$ in the case $\Sigma ^2=0$ .

Proof. First, we assume that condition (i) holds. We will verify the assumptions of [Reference Kifer25, Theorem 2.3.] with $Q=\Omega $ (that is, when there is no inducing involved) and with the fiberwise centered observable. Note that $\varphi _{\omega , 0} \in C^1[0, 1]$ for $\omega \in \Omega $ . Since

$$ \begin{align*} \bigg|\!\int_0^1\varphi_\omega \, d\mu_\omega\bigg| = \bigg|\!\int_0^1\varphi_\omega h_\omega \, dm \bigg|\le \int_0^1|\varphi_\omega| h_\omega \, dm\le \|\varphi_\omega \|_{C^1}\int_0^1h_\omega \, dm=\|\varphi_\omega\|_{C^1}, \end{align*} $$

we have that $\|\varphi _{\omega , 0}\|_{C^1}\le 2\|\varphi _\omega \|_{C^1}$ , and thus from (5), we get that

(7) $$ \begin{align} \omega \mapsto \|\varphi_{\omega, 0}\|_{C^1} \in L^2(\Omega, \mathcal F, \mathbb P). \end{align} $$

Observing that $(\mathbb E_{\mu _\omega }|\varphi _{\omega , 0}|^2)^{1/2}\le \|\varphi _{\omega , 0}\|_{C^1}$ , we see that (7) implies [Reference Kifer25, (2.7)].

Next, we aim to verify that [Reference Kifer25, Theorem 2.3(i)] is true. Using (A.2), we have that

(8) $$ \begin{align} \sum_{n=1}^\infty |\mathbb E_{\mu_\omega}(\varphi_{\omega, 0} \cdot (\varphi_{\sigma^n \omega, 0}\circ T_\omega^n))| &=\sum_{n=1}^\infty \bigg|\!\int_0^1 \varphi_{\omega, 0} h_\omega (\varphi_{\sigma^n \omega, 0}\circ T_\omega^n)\, dm \bigg| \nonumber\\ &=\sum_{n=1}^\infty \bigg|\!\int_0^1 \varphi_{\sigma^n \omega, 0} \mathcal L_\omega^n( \varphi_{\omega, 0} h_\omega) \, dm \bigg| \nonumber\\ &\le \sum_{n=1}^\infty \|\mathcal L_\omega^n (\varphi_{\omega, 0} h_\omega)\|_{L^1(m)} \cdot \|\varphi_{\sigma^n \omega, 0}\|_{C^1} \nonumber\\ &\le C_\alpha\sum_{n=1}^\infty n^{-1/\alpha+1}\|\varphi_{\omega, 0} \|_{C^1} \cdot \|h_\omega\|_{L^1(m)}\cdot \|\varphi_{\sigma^n \omega, 0}\|_{C^1} \nonumber\\ &= C_\alpha\sum_{n=1}^\infty n^{-1/\alpha+1}\|\varphi_{\omega, 0}\|_{C^1}\cdot \|\varphi_{\sigma^n \omega, 0}\|_{C^1}. \end{align} $$

Consequently, using Hölder’s inequality and that $\sigma $ preserves $\mathbb P$ , we have that

$$ \begin{align*} \begin{split} \mathbb E_{\mathbb P}&\sum_{n=1}^\infty |\mathbb E_{\mu_\omega}(\varphi_{\omega, 0} \cdot (\varphi_{\sigma^n \omega, 0}\circ T_\omega^n))| \\ & \le C_\alpha\sum_{n=1}^\infty n^{-1/\alpha+1}\mathbb E_{\mathbb P}(\|\varphi_{\omega, 0}\|_{C^1}\cdot \|\varphi_{\sigma^n \omega, 0}\|_{C^1}) \\ &\le C_\alpha\sum_{n=1}^\infty n^{-1/\alpha+1} (\mathbb E_{\mathbb P}(\|\varphi_{\omega, 0}\|_{C^1}^2))^{1/2} \cdot (\mathbb E_{\mathbb P}(\|\varphi_{\sigma^n \omega, 0}\|_{C^1}^2))^{1/2} \\ &\le C_\alpha\mathbb E_{\mathbb P}(\|\varphi_\omega\|_{C^1}^2)\sum_{n=1}^\infty n^{-1/\alpha+1} , \end{split} \end{align*} $$

which is finite due to (7) and $\alpha <1/2$ .

It remains to verify [Reference Kifer25, Theorem 2.3(ii’)]. More precisely, we need to show that

(9) $$ \begin{align} \sum_{n=0}^\infty \mathbb E_{\mu_\omega} |L_{\sigma^{-n}\omega}^n \varphi_{\sigma^{-n}\omega, 0}| \in L^2(\Omega, \mathcal F, \mathbb P), \end{align} $$

where

$$ \begin{align*} L_\omega(\psi):=\frac{\mathcal L_\omega (\psi h_\omega)}{h_{\sigma \omega}} \quad \text{and} \quad L_\omega^n:=L_{\sigma^{n-1}\omega} \circ \cdots \circ L_{\sigma \omega}\circ L_\omega, \end{align*} $$

for $\omega \in \Omega $ and $n\in \mathbb N$ . In addition, $L_\omega ^0:=\textrm {Id}$ . Note that

$$ \begin{align*} L_\omega^n (\psi)=\frac{\mathcal L_\omega^n(\psi h_\omega)}{h_{\sigma^n \omega}} \end{align*} $$

for $\omega \in \Omega $ and $n\in \mathbb N$ . Then, by relying on (A.2) again, we see that

(10) $$ \begin{align} \sum_{n=1}^\infty \mathbb E_{\mu_\omega} |L_{\sigma^{-n}\omega}^n \varphi_{\sigma^{-n}\omega, 0}| &=\sum_{n=1}^\infty \|L_{\sigma^{-n}\omega}^n \varphi_{\sigma^{-n}\omega}\|_{L^1(\mu_\omega)} \nonumber\\ &=\sum_{n=1}^\infty \|\mathcal L_{\sigma^{-n}\omega}^n(\varphi_{\sigma^{-n}\omega, 0}h_{\sigma^{-n}\omega})\|_{L^1(m)} \nonumber\\ &\le C_\alpha\sum_{n=1}^\infty n^{-1/\alpha+1}\|\varphi_{\sigma^{-n}\omega, 0}h_{\sigma^{-n}\omega}\|_{L^1(m)} \nonumber\\ &\le C_\alpha\sum_{n=1}^\infty n^{-1/\alpha+1}\|\varphi_{\sigma^{-n}\omega, 0}\|_{C^1}. \end{align} $$

Therefore, using the fact that $\sigma $ preserves $\mathbb P$ , we have

$$ \begin{align*} \bigg\|\sum_{n=1}^\infty \mathbb E_{\mu_\omega } |L_{\sigma^{-n}\omega}^n \varphi_{\sigma^{-n}\omega, 0}| \bigg\|_{L^2(\Omega, \mathcal F, \mathbb P)}\le C_\alpha \| \|\varphi_{\omega}\|_{C^1}\|_{L^2(\Omega, \mathcal F, \mathbb P)}\sum_{n=1}^\infty n^{-1/\alpha+1}, \end{align*} $$

which is finite due to (7) and $\alpha <\frac 1 2$ . The proof of condition (i) is complete.

The proof in the case of condition (ii) is similar to that of condition (i) and proceeds via an application of [Reference Kifer25, Theorem 2.3] relying on (4). To avoid repetition, we only give an outline of the proof. Note that $\varphi _{\omega , 0}=\varphi _\omega $ . Assuming condition (ii), we obtain [Reference Kifer25, (2.7)] using (5) since $(\mathbb E_{\mu _\omega }|\varphi _{\omega }|^2)^{1/2}\le \|\varphi _\omega \|_{C^1}$ . Arguing as in (8), using (4) and (6), we obtain

$$ \begin{align*} \begin{split} \sum_{n=1}^\infty |\mathbb E_{\mu_\omega}(\varphi_{\omega} \cdot (\varphi_{\sigma^n \omega}\circ T_\omega^n))| &\le \sum_{n=1}^\infty \|\mathcal L_\omega^n (\varphi_{\omega} h_\omega)\|_{L^1(m)} \cdot \|\varphi_{\sigma^n \omega}\|_{C^1} \\ &\le C \sum_{n=1}^\infty ( C_\omega + \Vert \varphi_\omega \Vert_{C^1} ) n^{ 1 - (1 + \min\{\alpha, \gamma \} )/\alpha} \cdot C_{\sigma^n \omega} \end{split} \end{align*} $$

for ${\mathbb {P}}$ -a.e. $\omega \in \Omega $ , where $C> 0$ depends only on $a, \alpha , \gamma $ . Consequently,

$$ \begin{align*} \begin{split} &\mathbb E_{\mathbb P}\sum_{n=1}^\infty |\mathbb E_{\mu_\omega}(\varphi_{\omega} \cdot (\varphi_{\sigma^n \omega}\circ T_\omega^n))| \\ &\quad \le C \sum_{n=1}^\infty \mathbb E_{\mathbb P} [ ( C_\omega + \Vert \varphi_\omega \Vert_{C^1} ) C_{\sigma^n \omega} ] n^{ 1 - (1 + \min\{\alpha, \gamma \} )/\alpha} \\ &\quad\le C \{((\mathbb E_{\mathbb P} (C_{ \omega}^2 ))^{1/2} + (\mathbb E_{\mathbb P} ( \Vert \varphi_\omega \Vert_{C^1}^2 ) )^{1/2} ) ( \mathbb E_{\mathbb P} ( C_{\omega}^2 ))^{1/2}\} \sum_{n=1}^\infty n^{ 1 - (1 + \min\{\alpha, \gamma \} )/\alpha} < \infty, \end{split} \end{align*} $$

since $\alpha < (1 + \gamma ) / 2$ . Hence, [Reference Kifer25, Theorem 2.3(i)] holds. Finally, [Reference Kifer25, Theorem 2.3(ii’)] follows by a computation similar to (10), but using (4) instead of (A.2) to obtain

$$ \begin{align*} \sum_{n=1}^\infty \mathbb E_{\mu_\omega } |L_{\sigma^{-n}\omega}^n \varphi_{\sigma^{-n}\omega}| &\le C \sum_{n=1}^\infty ( C_{ \sigma^{-n} \omega }+ \|\varphi_{\sigma^{-n}\omega}\|_{C^1} ) n^{ 1 - (1 + \min\{\alpha,\gamma\} )/\alpha}. \end{align*} $$

3 Continuity of the variance

We are interested in studying the regularity of the variance $\Sigma ^2$ in Theorem 3 under suitable perturbations of random dynamics.

Let us start by introducing the appropriate setup. We take $\varepsilon _0>0$ , and measurable maps $\beta \colon \Omega \to (0, 1)$ and $\delta \colon \Omega \to (0, 1)$ such that

$$ \begin{align*} \alpha:=\operatorname{\mathrm{esssup}}_{\omega}\beta (\omega)+\varepsilon_0 <\tfrac 1 2 \quad \text{and} \quad \operatorname{\mathrm{essinf}}_{\omega}\beta(\omega)-\varepsilon_0>0. \end{align*} $$

For $\omega \in \Omega $ and $\varepsilon \in (-\varepsilon _0, \varepsilon _0)$ , let $T_{\omega , \varepsilon }$ be an LSV map with parameter $\beta (\omega )+\varepsilon \delta (\omega )$ . We will write $T_{\omega }$ instead of $T_{\omega , 0}$ . By $\mathcal L_{\omega , \varepsilon }$ , we will denote the transfer operator associated with $T_{\omega , \varepsilon }$ , again writing $\mathcal L_\omega $ instead of $\mathcal L_{\omega , 0}$ . By $(\mu _{\omega , \varepsilon })_{\omega \in \Omega }$ , we will denote the random a.c.i.m. associated with the cocycle $(T_{\omega , \varepsilon })_{\omega \in \Omega }$ . We will write $\mu _\omega $ instead of $\mu _{\omega , 0}$ . By $h_{\omega , \varepsilon }\in \mathcal C_*\cap \mathcal C_2$ , we denote the density of $\mu _{\omega , \varepsilon }$ (again writing $h_\omega $ instead of $h_{\omega , 0}$ ).

Let $F\colon \Omega \times [0, 1] \to \mathbb R$ be a measurable map such that $F(\omega , \cdot )\in C^1[0, 1]$ for $\omega \in \Omega $ and

(11) $$ \begin{align} \operatorname{\mathrm{esssup}}_{\omega \in \Omega}\|F(\omega, \cdot)\|_{C^1}<+\infty. \end{align} $$

In the following, we will denote $F_\omega :=F(\omega , \cdot )$ . Set

$$ \begin{align*} f_{\omega, \varepsilon}:=F_\omega-\int_0^1F_\omega \, d_{\mu_{\omega, \varepsilon}}, \quad \omega \in \Omega, \ \varepsilon \in (-\varepsilon_0, \varepsilon_0). \end{align*} $$

Moreover, let

(12) $$ \begin{align} \Sigma_\varepsilon^2:=\int_\Omega \int_0^1f_{\omega, \varepsilon}^2 \, d_{\mu_{\omega, \varepsilon}} \, d\mathbb P(\omega)+2\sum_{n=1}^\infty \int_\Omega \int_0^1 f_{\omega, \varepsilon}\cdot (f_{\sigma^n \omega, \varepsilon}\circ T_{\omega, \varepsilon}^n)\, d\mu_{\omega, \varepsilon}\, d\mathbb P(\omega). \end{align} $$

By Theorem 3, $\Sigma _\varepsilon ^2$ is finite for $\varepsilon \in (-\varepsilon _0, \varepsilon _0)$ . Then, we have the following theorem.

Proof. We will first show that

(13) $$ \begin{align} \lim_{\varepsilon \to 0}\int_\Omega\int_0^1 f_{\omega, \varepsilon}^2\, d \mu_{\omega, \varepsilon}\, d\mathbb P(\omega)=\int_\Omega\int_0^1 f_{\omega, 0}^2\, d\mu_\omega\, d\mathbb P(\omega). \end{align} $$

To this end, observe that

$$ \begin{align*} \begin{split} & \int_\Omega\int_0^1 f_{\omega, \varepsilon}^2\, d\mu_{\omega, \varepsilon}\, d\mathbb P(\omega)-\int_\Omega\int_0^1 f_{\omega, 0}^2\, d\mu_\omega\, d\mathbb P(\omega) \\ &\quad=\int_\Omega \int_0^1 (f_{\omega, \varepsilon}^2-f_{\omega, 0}^2)\, d\mu_{\omega, \varepsilon}\, d\mathbb P(\omega)\\ &\qquad +\int_\Omega \int_0^1 f_{\omega, 0}^2(h_{\omega, \varepsilon}-h_{\omega})\,dm\, d\mathbb P(\omega). \end{split} \end{align*} $$

Observe that for $\mathbb P$ -a.e. $\omega \in \Omega $ ,

(14) $$ \begin{align} f_{\omega, \varepsilon}-f_{\omega, 0}=\int_0^1F_\omega (h_\omega-h_{\omega, \varepsilon})\, dm, \end{align} $$

and consequently,

$$ \begin{align*} |f_{\omega, \varepsilon}-f_{\omega, 0}|\le \|F_\omega\|_{C^0}\|h_{\omega, \varepsilon}-h_{\omega, 0}\|_{L^1(m)} \le C_{\alpha, F} |\varepsilon|, \end{align*} $$

where in the second inequality, we used Proposition A.3 and (11). However,

$$ \begin{align*} \|f_{\omega, \varepsilon}+f_{\omega, 0}\|_{C^0}\le 4\|F_\omega \|_{C^0}. \end{align*} $$

Consequently,

$$ \begin{align*} \|f_{\omega, \varepsilon}^2-f_{\omega, 0}^2\|_{C^0}\le C_{\alpha, F}|\varepsilon| \end{align*} $$

for $\mathbb P$ -a.e. $\omega \in \Omega $ . This implies that

$$ \begin{align*} \begin{split} \bigg|\!\int_\Omega \int_0^1 (f_{\omega, \varepsilon}^2-f_{\omega, 0}^2)\, d\mu_{\omega, \varepsilon}\, d\mathbb P(\omega)\bigg| &\le \int_\Omega \|f_{\omega, \varepsilon}^2-f_{\omega, 0}^2\|_{C^0}\, d\mathbb P(\omega) \le C_{\alpha, F}|\varepsilon|, \end{split} \end{align*} $$

yielding

$$ \begin{align*} \lim_{\varepsilon \to 0}\int_\Omega \int_0^1 (f_{\omega, \varepsilon}^2-f_{\omega, 0}^2)\, d\mu_{\omega, \varepsilon}\, d\mathbb P(\omega)=0. \end{align*} $$

Similarly, $|f_{\omega , 0}^2|\le 4\|F_\omega \|_{C^0}^2$ and thus, using Proposition A.3 and (11), we have

$$ \begin{align*} \lim_{\varepsilon \to 0}\int_\Omega \int_0^1 f_{\omega, 0}^2(h_{\omega, \varepsilon}-h_{\omega})\,dm\, d\mathbb P(\omega)=0, \end{align*} $$

since

$$ \begin{align*} \begin{split} & \bigg|\!\int_\Omega \int_0^1 f_{\omega, 0}^2(h_{\omega, \varepsilon}-h_{\omega})\,dm\, d\mathbb P(\omega)\bigg| \\ &\quad \le \int_\Omega \|f_{\omega, 0}^2\|_{C^0}\cdot \|h_{\omega, \varepsilon}-h_\omega \|_{L^1(m)}\, d\mathbb P(\omega)\le C_{\alpha, F} |\varepsilon|. \end{split} \end{align*} $$

Therefore, we conclude that (13) holds.

Next, we observe that

$$ \begin{align*} \begin{split} & \bigg|\!\int_\Omega \int_0^1 f_{\omega, \varepsilon}\cdot (f_{\sigma^n \omega, \varepsilon}\circ T_{\omega, \varepsilon}^n)\, d\mu_{\omega, \varepsilon}\, d\mathbb P(\omega) \bigg| \\ &\quad=\bigg|\!\int_\Omega\int_0^1 \mathcal L_{\omega, \varepsilon}^n(f_{\omega, \varepsilon}h_{\omega, \varepsilon})f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega) \bigg| \\ &\quad\le \int_\Omega \|\mathcal L_{\omega, \varepsilon}^n(f_{\omega, \varepsilon}h_{\omega, \varepsilon})\|_{L^1(m)}\cdot \|f_{\sigma^n \omega, \varepsilon}\|_{C^0}\, d\mathbb P(\omega) \\ &\quad\le C_\alpha n^{-1/\alpha+1}(\operatorname{\mathrm{esssup}}\|F_\omega \|_{C^1})^2, \end{split} \end{align*} $$

where we used (A.2) and (11). Since the series $\sum _{n=1}^\infty n^{-1/\alpha +1}$ is summable, to prove our theorem, it remains to show that for each $n\in \mathbb N$ , the map

$$ \begin{align*} \begin{split} & \varepsilon \mapsto \int_\Omega \int_0^1 f_{\omega, \varepsilon}\cdot (f_{\sigma^n \omega, \varepsilon}\circ T_{\omega, \varepsilon}^n)\, d\mu_{\omega, \varepsilon}\, d\mathbb P(\omega)\\ &\quad =\int_\Omega\int_0^1 \mathcal L_{\omega, \varepsilon}^n(f_{\omega, \varepsilon}h_{\omega, \varepsilon})f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega) \end{split} \end{align*} $$

is continuous in $0$ . To this end, we first observe that

$$ \begin{align*} \begin{split} &\int_\Omega\int_0^1 \mathcal L_{\omega, \varepsilon}^n(f_{\omega, \varepsilon}h_{\omega, \varepsilon})f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega)-\int_\Omega\int_0^1 \mathcal L_{\omega}^n(f_{\omega, 0}h_{\omega})f_{\sigma^n \omega, 0}\, dm\, d\mathbb P(\omega)\\ &\quad=\int_\Omega \int_0^1 \mathcal L_\omega^n(f_{\omega, 0}h_\omega)(f_{\sigma^n \omega, \varepsilon}-f_{\sigma^n \omega, 0})\, dm\, d\mathbb P(\omega)\\ &\qquad+\int_\Omega \int_0^1 (\mathcal L_{\omega, \varepsilon}^n-\mathcal L_\omega^n)(f_{\omega, 0}h_\omega) f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega)\\ &\qquad+\int_\Omega \int_0^1 \mathcal L_{\omega, \varepsilon}^n(f_{\omega, \varepsilon}h_{\omega, \varepsilon}-f_{\omega, 0}h_\omega) f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega) \\ &\quad=\int_\Omega \int_0^1 (\mathcal L_{\omega, \varepsilon}^n-\mathcal L_\omega^n)(f_{\omega, 0}h_\omega) f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega)\\ &\qquad +\int_\Omega \int_0^1 \mathcal L_{\omega, \varepsilon}^n(f_{\omega, \varepsilon}h_{\omega, \varepsilon}-f_{\omega, 0}h_\omega) f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega), \end{split} \end{align*} $$

since $f_{\sigma ^n \omega , \varepsilon }-f_{\sigma ^n \omega , 0}$ depends only on $\varepsilon $ and $\omega $ (see (14)) and

$$ \begin{align*} \int_0^1 \mathcal L_{\omega}^n (f_{\omega, 0}h_\omega)\, dm=\int_0^1 f_{\omega, 0}h_\omega \, dm=0. \end{align*} $$

Next,

$$ \begin{align*} \begin{split} & \bigg|\!\int_\Omega \int_0^1 \mathcal L_{\omega, \varepsilon}^n(f_{\omega, \varepsilon}h_{\omega, \varepsilon}-f_{\omega, 0}h_\omega) f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega) \bigg| \\ &\quad\le \int_\Omega \|\mathcal L_{\omega, \varepsilon}^n(f_{\omega, \varepsilon}h_{\omega, \varepsilon}-f_{\omega, 0}h_\omega)\|_{L^1(m)}\cdot \|f_{\sigma^n \omega, \varepsilon}\|_{C^0}\, d\mathbb P(\omega) \\ &\quad\le \int_\Omega \|f_{\omega, \varepsilon}h_{\omega, \varepsilon}-f_{\omega, 0}h_\omega\|_{L^1(m)}\cdot \|f_{\sigma^n \omega, \varepsilon}\|_{C^0}\, d\mathbb P(\omega) \\ &\quad\le 2\operatorname{\mathrm{esssup}}_{\omega \in \Omega}\|F_\omega\|_{C^0}\int_\Omega \|f_{\omega, \varepsilon}h_{\omega, \varepsilon}-f_{\omega, 0}h_\omega\|_{L^1(m)}\, d\mathbb P(\omega). \end{split} \end{align*} $$

Moreover,

$$ \begin{align*} \begin{split} \|f_{\omega, \varepsilon}h_{\omega, \varepsilon}-f_{\omega, 0}h_\omega\|_{L^1(m)} &\le \|f_{\omega, \varepsilon}\|_{C^0}\|h_{\omega, \varepsilon}-h_\omega\|_{L^1(m)}+|f_{\omega, \varepsilon}-f_{\omega, 0}| \\ &\le C_{\alpha, F}|\varepsilon| , \end{split} \end{align*} $$

where we used Proposition A.3. This together with (11) implies that

$$ \begin{align*} \lim_{\varepsilon \to 0}\int_\Omega \int_0^1 \mathcal L_{\omega, \varepsilon}^n(f_{\omega, \varepsilon}h_{\omega, \varepsilon}-f_{\omega, 0}h_\omega) f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega)=0. \end{align*} $$

Finally, we have that

$$ \begin{align*} \begin{split} &\int_\Omega \int_0^1 (\mathcal L_{\omega, \varepsilon}^n-\mathcal L_\omega^n)(f_{\omega, 0}h_\omega) f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega) \\ &\quad=\sum_{j=1}^n \int_\Omega \int_0^1 \mathcal L_{\sigma^j \omega, \varepsilon}^{n-j}(\mathcal L_{\sigma^{j-1}\omega, \varepsilon}-\mathcal L_{\sigma^{j-1}\omega}) \mathcal L_\omega^{j-1}(f_{\omega, 0}h_\omega)f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega). \end{split} \end{align*} $$

Therefore,

$$ \begin{align*} \begin{split} &\bigg|\!\int_\Omega \int_0^1 (\mathcal L_{\omega, \varepsilon}^n-\mathcal L_\omega^n)(f_{\omega, 0}h_\omega) f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega) \bigg| \\ &\quad\le \sum_{j=1}^n \int_\Omega \|\mathcal L_{\sigma^j \omega, \varepsilon}^{n-j}(\mathcal L_{\sigma^{j-1}\omega, \varepsilon}-\mathcal L_{\sigma^{j-1}\omega}) \mathcal L_\omega^{j-1}(f_{\omega, 0}h_\omega)\|_{L^1(m)}\cdot \|f_{\sigma^n \omega, \varepsilon}\|_{C^0}\, d\mathbb P(\omega) \\ &\quad\le \sum_{j=1}^n \int_\Omega \|(\mathcal L_{\sigma^{j-1}\omega, \varepsilon}-\mathcal L_{\sigma^{j-1}\omega}) \mathcal L_\omega^{j-1}(f_{\omega, 0}h_\omega)\|_{L^1(m)}\cdot \|f_{\sigma^n \omega, \varepsilon}\|_{C^0}\, d\mathbb P(\omega) \\ &\quad\le 2 (\operatorname{\mathrm{esssup}}_{\omega \in \Omega}\|F_\omega\|_{C^0})\sum_{j=1}^n \int_\Omega \|(\mathcal L_{\sigma^{j-1}\omega, \varepsilon}-\mathcal L_{\sigma^{j-1}\omega}) \mathcal L_\omega^{j-1}(f_{\omega, 0}h_\omega)\|_{L^1(m)}\, d\mathbb P(\omega). \end{split} \end{align*} $$

In addition, we have

$$ \begin{align*} \begin{split} & (\mathcal L_{\sigma^{j-1}\omega, \varepsilon}-\mathcal L_{\sigma^{j-1}\omega}) \mathcal L_\omega^{j-1}(f_{\omega, 0}h_\omega)\\ &\quad =-\int_{\beta(\sigma^{j-1}\omega)}^{\beta(\sigma^{j-1}\omega)+\varepsilon \delta (\sigma^{j-1}\omega)} (X_\gamma N_\gamma (\mathcal L_\omega^{j-1}(f_{\omega, 0}h_\omega)))'\, d\gamma, \end{split} \end{align*} $$

since $\partial _\gamma \mathcal L_\gamma (\phi )=-(X_\gamma N_\gamma (\phi ))'$ (see [Reference Baladi and Todd9, p. 865]). It follows from $h_\omega \in \mathcal C_*\cap \mathcal C_2$ and (1) that, for $\mathbb P$ -a.e. $\omega \in \Omega $ , we have $f_{\omega , 0}h_\omega =\psi _1-\psi _2$ for some $\psi _i=\psi _i(\omega )\in \mathcal C_*\cap C^1(0,1]$ . Moreover, $\|\psi _i\|_{L^1}\le K$ for $i=1, 2$ , where K does not depend on $\omega $ . Hence,

$$ \begin{align*} \mathcal L_\omega^{j-1}(f_{\omega, 0}h_\omega)=\mathcal L_\omega^{j-1}(\psi_1)-\mathcal L_\omega^{j-1}(\psi_2), \quad \mathcal L_\omega^{j-1}(\psi_i)\in \mathcal C_*\cap C^1(0,1], \ i=1, 2. \end{align*} $$

Consequently,

$$ \begin{align*} \Vert (X_\gamma N_\gamma (\mathcal L_\omega^{j-1}(f_{\omega, 0}h_\omega)))' \Vert_{L^1(m)} \le \sum_{i=1,2} \Vert (X_\gamma N_\gamma (\mathcal L_\omega^{j-1}( \psi_i )))' \Vert_{L^1(m)} \le \tilde{K}, \end{align*} $$

where the last inequality is a consequence of bounds on $X_\gamma $ and $X_\gamma '$ in [Reference Baladi and Todd9, (2.3) and (2.4)], the invariance of ${\mathcal {C}}_* \cap C^1(0,1]$ with respect to $N_\gamma $ and ${\mathcal {L}}_\gamma $ (see [Reference Dragičević, Gonzalez-Tokman and Sedro17, Proposition 5]), and the observation (see [Reference Dragičević, Gonzalez-Tokman and Sedro17, Remark 1]) that

(15) $$ \begin{align} \phi \in {\mathcal{C}}_* \cap C^1(0,1] \implies |\phi'(x)| \le (\alpha + 1) x^{-1} \phi(x) \quad \text{ for all } x \in (0,1]. \end{align} $$

This gives that

$$ \begin{align*} \int_\Omega \|(\mathcal L_{\sigma^{j-1}\omega, \varepsilon}-\mathcal L_{\sigma^{j-1}\omega}) \mathcal L_\omega^{j-1}(f_{\omega, 0}h_\omega)\|_{L^1(m)}\, d\mathbb P(\omega) \le C|\varepsilon| \end{align*} $$

for some $C>0$ independent on $\varepsilon $ . Thus,

$$ \begin{align*} \lim_{\varepsilon \to 0}\int_\Omega \int_0^1 (\mathcal L_{\omega, \varepsilon}^n-\mathcal L_\omega^n)(f_{\omega, 0}h_\omega) f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega)=0. \end{align*} $$

This completes the proof of the theorem.

4 Differentiability of the variance

Take $\beta \colon \Omega \to (0, 1)$ , $\delta \colon \Omega \to [0, 1)$ to be measurable maps and $\underline {\alpha }>0$ such that $0<\underline {\alpha }<\alpha <\frac 1 2$ ,

$$ \begin{align*} 0<\underline{\alpha}\le \operatorname{\mathrm{essinf}}_{\omega \in \Omega}\beta(\omega)-\varepsilon_0 \quad \text{and} \quad \operatorname{\mathrm{esssup}}_{\omega \in \Omega}\beta(\omega)+\varepsilon_0\le \alpha. \end{align*} $$

Let $T_{\omega , \varepsilon }$ be the LSV map with parameter $\beta (\omega )+\varepsilon \delta (\omega )$ for every $\omega \in \Omega $ and $\varepsilon \in (-\varepsilon _0, \varepsilon _0)$ . By $(\mu _{\omega , \varepsilon })_{\omega \in \Omega }$ , we will denote the random a.c.i.m. that corresponds to the cocycle $(T_{\omega , \varepsilon })_{\omega \in \Omega }$ , where $h_{\omega , \varepsilon }$ is the density of $\mu _{\omega , \varepsilon }$ . As in the previous section, we will drop the subscript $\varepsilon $ when $\varepsilon =0$ .

We will use the following result, which is a sharper version of [Reference Dragičević, Gonzalez-Tokman and Sedro17, Theorem 14]. Its proof is given in Appendix C.

Theorem 6. There exists $C=C_{\alpha , \underline {\alpha }}>0$ such that for $\varepsilon $ sufficiently close to $0$ ,

(16) $$ \begin{align} \|\varepsilon^{-1}(h_{\omega, \varepsilon}-h_\omega)-\hat h_\omega \|_{L^1(m)}\le C|\varepsilon|^{1-2\alpha} \end{align} $$

for $\mathbb P$ -a.e. $\omega \in \Omega $ , where

(17) $$ \begin{align} \hat h_\omega:=-\sum_{i=0}^\infty \delta (\sigma^{-(i+1)}\omega) \mathcal L_{\sigma^{-i}\omega}^i (X_{\beta(\sigma^{-(i+1)}\omega)} N_{\beta(\sigma^{-(i+1)}\omega)} (h_{\sigma^{-(i+1)}\omega}))' \in L^1(m). \end{align} $$

Moreover,

(18) $$ \begin{align} \operatorname{\mathrm{esssup}}_{\omega \in \Omega}\|\hat h_\omega \|_{L^1(m)}<+\infty. \end{align} $$

Let $F\colon \Omega \times [0, 1] \to \mathbb R$ be a measurable map such that $F(\omega , \cdot )\in C^2[0, 1]$ for $\omega \in \Omega $ and that

(19) $$ \begin{align} \operatorname{\mathrm{esssup}}_{\omega \in \Omega}\|F(\omega, \cdot)\|_{C^2}<+\infty. \end{align} $$

Let $f_{\omega , \varepsilon }$ and $\Sigma _\varepsilon ^2$ be as in the previous section. Then, we have the following theorem.

Remark 10. We note that the proof of Theorem 9 yields

$$ \begin{align*} \begin{split} & \frac{d}{d\varepsilon}\Sigma_\varepsilon^2 \rvert_{\varepsilon=0}\\ &\quad=\int_\Omega \int_0^1 f_{\omega, 0}^2 \hat h_\omega \, dm\, d\mathbb P(\omega)-2\int_\Omega \bigg(\!\int_0^1 F_\omega \hat h_\omega \, dm\bigg)\int_0^1 f_{\omega, 0}\, d\mu_\omega \, d\mathbb P(\omega) \\ &\qquad+\sum_{n=1}^\infty \int_\Omega \int_0^1 \mathcal L_\omega^n (-L(\omega, F_\omega)h_\omega+f_{\omega, 0}\hat h_\omega)f_{\sigma^n \omega, 0}\, dm\, d\mathbb P(\omega) \\ &\qquad+\sum_{n=1}^\infty \sum_{j=0}^{n-1}\int_\Omega \delta (\sigma^j \omega)\int_0^1\mathcal L_{\sigma^{j+1}\omega}^{n-j-1} (\partial_\gamma \mathcal L_{\beta(\sigma^j \omega)}(\mathcal L_\omega^j(f_{\omega, 0}h_\omega)))f_{\sigma^n \omega, 0}\, dm\, d\mathbb P(\omega), \end{split} \end{align*} $$

where $L(\omega , F_\omega ):=\int _0^1 F_\omega \hat h_\omega \, dm$ . The proof of Theorem 9 shows that this formula for the derivative of $\Sigma _\varepsilon ^2$ is well defined under the assumption $\alpha < 1/5$ . In particular, Lemma 18 ensures the summability

(20) $$ \begin{align} \sum_{n=1}^\infty \sum_{j=0}^{n-1}\int_\Omega \int_0^1 |\mathcal L_{\sigma^{j+1}\omega}^{n-j-1} (\partial_\gamma \mathcal L_{\beta(\sigma^j \omega)}(\mathcal L_\omega^j(f_{\omega, 0}h_\omega))) | \, dm\, d\mathbb P(\omega) < \infty. \end{align} $$

We do not know whether (20) remains valid beyond the range $\alpha < 1/5$ .

The proof of Theorem 9 will be established in a series of auxiliary results.

4.1 Differentiability of the first term in (12)

Lemma 12. The map

(21) $$ \begin{align} \varepsilon \to \int_\Omega \int_0^1f_{\omega, \varepsilon}^2 \, d\mu_{\omega, \varepsilon}\, d\mathbb P(\omega) \end{align} $$

is differentiable in $\varepsilon =0$ .

Proof. Writing

$$ \begin{align*} \begin{split} &\varepsilon^{-1}\bigg(\!\int_\Omega \int_0^1f_{\omega, \varepsilon}^2 \, d\mu_{\omega, \varepsilon}\, d\mathbb P(\omega)-\int_\Omega \int_0^1f_{\omega, 0}^2 \, d\mu_{\omega}\, d\mathbb P(\omega)\bigg) \\ &\quad=\varepsilon^{-1}\int_\Omega\int_0^1 (f_{\omega, \varepsilon}^2-f_{\omega, 0}^2)\, d\mu_{\omega, \varepsilon}\, d\mathbb P(\omega)+\varepsilon^{-1}\int_\Omega\int_0^1f_{\omega, 0}^2(h_{\omega, \varepsilon}-h_{\omega})\, dm\, d\mathbb P(\omega), \end{split} \end{align*} $$

it follows from (16) that

$$ \begin{align*} \lim_{\varepsilon \to 0}\frac{1}{\varepsilon}\int_\Omega \int_0^1f_{\omega, 0}^2(h_{\omega, \varepsilon}-h_{\omega, 0})\, dm\, d\mathbb P(\omega)=\int_\Omega \int_0^1 f_{\omega, 0}^2\hat h_\omega\, dm\, d\mathbb P(\omega), \end{align*} $$

since

$$ \begin{align*} \begin{split} &\bigg|\!\int_\Omega \int_0^1f_{\omega, 0}^2(\varepsilon^{-1}(h_{\omega, \varepsilon}-h_{\omega, 0}))\, dm\, d\mathbb P(\omega)-\int_\Omega \int_0^1 f_{\omega, 0}^2\hat h_\omega\, dm\, d\mathbb P(\omega)\bigg| \\ &\quad \le \int_\Omega \|f_{\omega, 0}^2\|_{C^0}\cdot \|\varepsilon^{-1}(h_{\omega, \varepsilon}-h_{\omega, 0})-\hat h_\omega\|_{L^1(m)}\, d\mathbb P(\omega) \le C_{\alpha, \underline{\alpha}, F}|\varepsilon|^{1-2\alpha} \end{split} \end{align*} $$

for $\varepsilon $ sufficiently close to $0$ . Next, we note that

$$ \begin{align*} \lim_{\varepsilon\to 0}\frac{1}{\varepsilon}\bigg(\!\int_\Omega\int_0^1 (f_{\omega, \varepsilon}^2-f_{\omega, 0}^2)\, d\mu_{\omega, \varepsilon}\, d\mathbb P(\omega)-\int_\Omega\int_0^1 (f_{\omega, \varepsilon}^2-f_{\omega, 0}^2)\, d\mu_{\omega}\, d\mathbb P(\omega)\bigg)=0, \end{align*} $$

as

$$ \begin{align*} \begin{split} &\bigg|\!\int_\Omega\int_0^1 (f_{\omega, \varepsilon}^2-f_{\omega, 0}^2)\, d\mu_{\omega, \varepsilon}\, d\mathbb P(\omega)-\int_\Omega\int_0^1 (f_{\omega, \varepsilon}^2-f_{\omega, 0}^2)\, d\mu_{\omega}\, d\mathbb P(\omega)\bigg| \\ &\quad\le \int_\Omega \|f_{\omega, \varepsilon}^2-f_{\omega, 0}^2\|_{C^0}\|h_{\omega, \varepsilon}-h_\omega \|_{L^1(m)}\, d\mathbb P(\omega) \\ &\quad \le C_{\alpha, F}|\varepsilon|^2 \end{split} \end{align*} $$

for some $C_{\alpha , F}>0$ , where we used (14) and Proposition A.3. Finally,

$$ \begin{align*} \lim_{\varepsilon \to 0}\frac{1}{\varepsilon}\int_\Omega\int_0^1 (f_{\omega, \varepsilon}^2-f_{\omega, 0}^2)\, d\mu_{\omega}\, d\mathbb P(\omega)=-2\!\int_\Omega \bigg(\!\int_0^1F_\omega \hat h_\omega\, dm\bigg)\int_0^1 f_{\omega, 0}\, d\mu_\omega\, d\mathbb P(\omega), \end{align*} $$

since

$$ \begin{align*} \begin{split} &\bigg\|\varepsilon^{-1}(f_{\omega, \varepsilon}^2-f_{\omega, 0}^2)+2\bigg(\!\int_0^1F_\omega \hat h_\omega\, dm\bigg)f_{\omega, 0} \bigg\|_{C^0} \\ &\quad\le \bigg|\varepsilon^{-1}(f_{\omega, \varepsilon}-f_{\omega, 0})+\int_0^1 F_\omega \hat h_\omega\, dm\bigg| \cdot \|f_{\omega, \varepsilon}+\!f_{\omega, 0}\|_{C^0}+\bigg|\!\int_0^1 F_\omega \hat h_\omega \, dm\bigg| \cdot |f_{\omega, \varepsilon}\! -f_{\omega, 0}| \\ &\quad\le C_{\alpha, \underline{\alpha}, F}|\varepsilon|^{1-2\alpha}+C_{\alpha, F} |\varepsilon| \end{split} \end{align*} $$

for $\varepsilon $ sufficiently close to $0$ , where we used (14), Proposition A.3, and Theorem 6. Thus, the map in (21) is differentiable in $\varepsilon =0$ .

4.2 Differentiability of the second term in (12)

Next, we consider the second term in (12):

$$ \begin{align*} \begin{split} & \sum_{n=1}^\infty \int_\Omega \int_0^1 (f_{\sigma^n \omega, \varepsilon}\circ T_{\omega, \varepsilon}^n)f_{\omega, \varepsilon}\, d\mu_{\omega, \varepsilon}\, d\mathbb P(\omega)\\ &\quad =\sum_{n=1}^\infty \int_\Omega \int_0^1 \mathcal L_{\omega, \varepsilon}^n (f_{\omega, \varepsilon}h_{\omega, \varepsilon})f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega). \end{split} \end{align*} $$

Let

$$ \begin{align*} C_n(\varepsilon):=\int_\Omega \int_0^1 \mathcal L_{\omega, \varepsilon}^n (f_{\omega, \varepsilon}h_{\omega, \varepsilon})f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega) \end{align*} $$

and

$$ \begin{align*} D_n(\varepsilon):=\frac{C_n(\varepsilon)-C_n(0)}{\varepsilon}. \end{align*} $$

Observe that

$$ \begin{align*} |C_n(\varepsilon)| \le \int_\Omega \|\mathcal L_{\omega, \varepsilon}^n(f_{\omega, \varepsilon}h_{\omega, \varepsilon} )\|_{L^1(m)} \cdot \|f_{\sigma^n \omega, \varepsilon}\|_{C^0}\, d\mathbb P(\omega) \le C_{\alpha, F} n^{-1/\alpha+1}. \end{align*} $$

Consequently,

$$ \begin{align*} |D_n(\varepsilon)| \le C_{\alpha, F} |\varepsilon|^{-1}n^{-1/\alpha+1}. \end{align*} $$

Let us now take $\gamma \in (0, \frac 1 2)$ such that

(22) $$ \begin{align} \gamma (1/\alpha-2)>1. \end{align} $$

We note that this is possible since $\alpha <\frac 1 4$ . Then, we have

$$ \begin{align*} \bigg|\sum_{n\ge |\varepsilon|^{-\gamma}}D_n(\varepsilon)\bigg|\le C|\varepsilon|^{\gamma (1/\alpha-2)-1}, \end{align*} $$

where $C>0$ is not dependent on $\varepsilon $ . Hence, due to (22), we have

(23) $$ \begin{align} \lim_{\varepsilon \to 0}\sum_{n\ge |\varepsilon|^{-\gamma}}D_n(\varepsilon)=0. \end{align} $$

Next, note that

(24) $$ \begin{align} D_n(\varepsilon) &=\int_\Omega \int_0^1 \mathcal L_\omega^n(f_{\omega, 0}h_\omega)(\varepsilon^{-1}(f_{\sigma^n \omega, \varepsilon}-f_{\sigma^n \omega, 0}))\, dm\, d\mathbb P(\omega) \nonumber\\ &\phantom{=}+\frac{1}{\varepsilon}\int_\Omega \int_0^1 (\mathcal L_{\omega, \varepsilon}^n-\mathcal L_\omega^n)(f_{\omega, 0}h_\omega) f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega) \nonumber\\ &\phantom{=}+\int_\Omega \int_0^1 \mathcal L_{\omega, \varepsilon}^n(\varepsilon^{-1}(f_{\omega, \varepsilon}h_{\omega, \varepsilon}-f_{\omega, 0}h_\omega)) f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega) \nonumber\\ &=\frac{1}{\varepsilon}\int_\Omega \int_0^1 (\mathcal L_{\omega, \varepsilon}^n-\mathcal L_\omega^n)(f_{\omega, 0}h_\omega) f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega) \nonumber\\ &\phantom{=}+\int_\Omega \int_0^1 \mathcal L_{\omega, \varepsilon}^n(\varepsilon^{-1}(f_{\omega, \varepsilon}h_{\omega, \varepsilon}-f_{\omega, 0}h_\omega)) f_{\sigma^n \omega, \varepsilon}\, dm\, d\mathbb P(\omega) \nonumber\\ &=:D_n^1(\varepsilon)+D_n^2(\varepsilon), \end{align} $$

since as in the proof of Theorem 5, we have

$$ \begin{align*} \int_\Omega \int_0^1 \mathcal L_\omega^n(f_{\omega, 0}h_\omega)(\varepsilon^{-1}(f_{\sigma^n \omega, \varepsilon}-f_{\sigma^n \omega, 0}))\, dm\, d\mathbb P(\omega)=0. \end{align*} $$

We first focus on $D_n^2(\varepsilon )$ . Taking into account (14) and

$$ \begin{align*} \int_0^1 \mathcal L_{\omega, \varepsilon}^n (f_{\omega, \varepsilon}h_{\omega, \varepsilon}-f_{\omega, 0}h_\omega)\, dm=\int_0^1 (f_{\omega, \varepsilon}h_{\omega, \varepsilon}-f_{\omega, 0}h_\omega)\, dm=0, \end{align*} $$

we have

$$ \begin{align*} D_n^2(\varepsilon)=\int_\Omega \int_0^1 \mathcal L_{\omega, \varepsilon}^n(\varepsilon^{-1}(f_{\omega, \varepsilon}h_{\omega, \varepsilon}-f_{\omega, 0}h_\omega)) f_{\sigma^n \omega, 0}\, dm\, d\mathbb P(\omega). \end{align*} $$

Let

$$ \begin{align*} \bar D_n^2(\varepsilon):=\int_\Omega \int_0^1\mathcal L_{\omega, \varepsilon}^n(-L(\omega, F_\omega)h_\omega+f_{\omega,0}\hat h_\omega)f_{\sigma^n \omega, 0}\, dm\, d\mathbb P(\omega), \end{align*} $$

where $L(\omega , F_\omega ):=\int _0^1 F_\omega \hat h_\omega \, dm$ .

Lemma 13. We have

$$ \begin{align*} |D_n^2(\varepsilon)-\bar D_n^2(\varepsilon)|\le C_{\alpha, \underline{\alpha}, F}|\varepsilon|^{1-2\alpha} \end{align*} $$

for $\varepsilon $ sufficiently close to $0$ .

Proof. Note that

$$ \begin{align*} \begin{split} &|D_n^2(\varepsilon)-\bar D_n^2(\varepsilon)|\\ &\quad\le \int_\Omega \|\varepsilon^{-1}(f_{\omega, \varepsilon}-f_{\omega, 0})h_{\omega, \varepsilon}+L(\omega, F_\omega)h_\omega\\ &\qquad +f_{\omega, 0}(\varepsilon^{-1}(h_{\omega, \varepsilon}-h_\omega)-\hat h_\omega)\|_{L^1(m)}\cdot \|f_{\sigma^n \omega, 0}\|_{C^0}\, d\mathbb P(\omega)\\ &\quad\le C_F\int_\Omega \|\varepsilon^{-1}(f_{\omega, \varepsilon}-f_{\omega, 0})h_{\omega, \varepsilon}\\ &\qquad+L(\omega, F_\omega)h_\omega+f_{\omega, 0}(\varepsilon^{-1}(h_{\omega, \varepsilon}-h_\omega)-\hat h_\omega)\|_{L^1(m)}\, d\mathbb P(\omega). \end{split} \end{align*} $$

Moreover,

$$ \begin{align*} \begin{split} & \|\varepsilon^{-1}(f_{\omega, \varepsilon}-f_{\omega, 0})h_{\omega, \varepsilon}+L(\omega, F_\omega)h_\omega\|_{L^1(m)}\\ &\quad\le \|\varepsilon^{-1}(f_{\omega, \varepsilon}-f_{\omega, 0})h_{\omega, \varepsilon}+L(\omega, F_\omega)h_{\omega, \varepsilon}\|_{L^1(m)} \\ &\qquad+|L(\omega, F_\omega)|\|h_{\omega, \varepsilon}-h_\omega\|_{L^1(m)}\\ &\quad= |\varepsilon^{-1}(f_{\omega, \varepsilon}-f_{\omega, 0})+L(\omega, F_\omega)|\\ &\qquad+|L(\omega, F_\omega)|\|h_{\omega, \varepsilon}-h_\omega\|_{L^1(m)}\\ &\quad\le C_{\alpha, \underline{\alpha}, F}|\varepsilon|^{1- 2\alpha}+C_{\alpha, F} |\varepsilon| \\ &\quad\le C_{\alpha, \underline{\alpha}, F}|\varepsilon|^{1-2\alpha} \end{split} \end{align*} $$

for $\mathbb P$ -a.e. $\omega \in \Omega $ and $\varepsilon $ sufficiently close to $0$ , where we used Proposition A.3 and Theorem 6. Similarly,

$$ \begin{align*} \begin{split} \|f_{\omega, 0}(\varepsilon^{-1}(h_{\omega, \varepsilon}-h_\omega)-\hat h_\omega)\|_{L^1(m)}&\le \|f_{\omega, 0}\|_{C^0} \cdot \|\varepsilon^{-1}(h_{\omega, \varepsilon}-h_\omega)-\hat h_\omega\|_{L^1(m)}\\ &\le C_{\alpha, \underline{\alpha}, F}|\varepsilon|^{1-2\alpha} \end{split} \end{align*} $$

for $\mathbb P$ -a.e. $\omega \in \Omega $ and $\varepsilon $ sufficiently close to $0$ . The conclusion of the lemma follows readily from the above estimates.

By Lemma 13, we have

$$ \begin{align*} \sum_{n<|\varepsilon|^{-\gamma}}| D_n^2(\varepsilon)-\bar D_n^2(\varepsilon)| \le C_{\alpha, \underline{\alpha}, F}|\varepsilon|^{1-2\alpha-\gamma}. \end{align*} $$

Since $ 1-2\alpha>\gamma $ , we have

(25) $$ \begin{align} \lim_{\varepsilon \to 0}\sum_{n<|\varepsilon|^{-\gamma}}| D_n^2(\varepsilon)-\bar D_n^2(\varepsilon)|=0. \end{align} $$

Taking into account (23), (24), and (25), to conclude the differentiability of $\varepsilon \mapsto \Sigma ^2_\varepsilon $ , it remains to establish the following.

• • The limits $\lim _{\varepsilon \to 0} D_n^1(\varepsilon )$ and $\lim _{\varepsilon \to 0}\bar D_n^2(\varepsilon )$ exist for each $n \ge 1$ .

• • There exist $\bar {\varepsilon }_0> 0$ and $u_n$ such that for all $n < |\varepsilon |^{-\gamma }$ and all $\varepsilon \in [-\bar {\varepsilon }_0, \bar {\varepsilon }_0]\setminus \{0\}$ ,

  $$ \begin{align*} |D_n^1(\varepsilon)| \le u_n, \quad | \bar D_n^2(\varepsilon)|\le u_n \quad \text{and} \quad \sum_{n=1}^\infty u_n < \infty. \end{align*} $$

The first point above is established in Lemma 17 for $\bar D_n^2$ and in Lemma 19 for $D_n^1$ . The second point is proved in Lemma 16 for $\bar D_n^2$ and in Lemma 18 for $D_n^1$ .

4.3 A series of auxiliary lemmas

We write

(26) $$ \begin{align} \bar D_n^2(\varepsilon) &=\int_\Omega \int_0^1\mathcal L_{\omega, \varepsilon}^n(-L(\omega, F_\omega)h_\omega+f_{\omega,0}\hat h_\omega)f_{\sigma^n \omega, 0}\, dm\, d\mathbb P(\omega) \nonumber\\ &=\int_\Omega \int_0^1 \mathcal L_{\omega, \varepsilon}^n (f_{\omega, 0}\hat h_\omega)f_{\sigma^n \omega, 0}\, dm\, d\mathbb P(\omega) \nonumber\\ &\quad-\int_\Omega L(\omega, F_\omega) \int_0^1 \mathcal L_{\omega, \varepsilon}^n (h_\omega)f_{\sigma^n \omega, 0}\, dm\, d\mathbb P(\omega) \nonumber\\ &=: \bar D_{n, 1}^2(\varepsilon)-\bar D_{n, 2}^2(\varepsilon). \end{align} $$

Lemma 14. We have

$$ \begin{align*} |\bar D_{n, 2}^2(\varepsilon)|\le C_{\alpha, F}(n^{-1/\gamma}+n^{-1/\alpha+1}), \quad n<|\varepsilon|^{-\gamma}. \end{align*} $$

Proof. Note that

$$ \begin{align*} \|\mathcal L_{\omega, \varepsilon}^n h_\omega -h_{\sigma^n \omega, \varepsilon}\|_{L^1(m)}=\|\mathcal L_{\omega, \varepsilon}^n (h_\omega-h_{\omega, \varepsilon})\|_{L^1(m)}\le C_\alpha n^{-1/\alpha+1}. \end{align*} $$

Then, using that

$$ \begin{align*} \int_0^1 \mathcal L_{\omega, \varepsilon}^n (h_{\omega, \varepsilon})f_{\sigma^n \omega, \varepsilon}\, dm=\int_0^1 h_{\sigma^n \omega, \varepsilon}f_{\sigma^n \omega, \varepsilon}\, dm=0, \end{align*} $$

we have

$$ \begin{align*} \begin{split} &\bigg|\!\int_0^1 \mathcal L_{\omega, \varepsilon}^n(h_\omega)f_{\sigma^n \omega, 0}\, dm\bigg| \\ &\le \bigg|\!\int_0^1 \mathcal L_{\omega, \varepsilon}^n(h_\omega)(f_{\sigma^n \omega, \varepsilon}-f_{\sigma^n \omega, 0})\, dm\bigg|+\bigg|\!\int_0^1 \mathcal L_{\omega, \varepsilon}^n (h_\omega)f_{\sigma^n \omega, \varepsilon}\, dm\bigg| \\ &=\bigg|\!\int_0^1 \mathcal L_{\omega, \varepsilon}^n(h_\omega)(f_{\sigma^n \omega, \varepsilon}-f_{\sigma^n \omega, 0})\, dm\bigg|+\bigg|\!\int_0^1 \mathcal L_{\omega, \varepsilon}^n (h_\omega-h_{\omega, \varepsilon})f_{\sigma^n \omega, \varepsilon}\, dm\bigg| \\ &\le \|\mathcal L_{\omega, \varepsilon}^n h_\omega \|_{L^1(m)}\cdot |f_{\sigma^n \omega, \varepsilon}-f_{\sigma^n \omega, 0}|+\|\mathcal L_{\omega, \varepsilon}^n(h_\omega-h_{\omega, \varepsilon})\|_{L^1(m)}\cdot \|f_{\sigma^n \omega, \varepsilon}\|_{C^0} \\ &\le |f_{\sigma^n \omega, \varepsilon}-f_{\sigma^n \omega, 0}|+\|\mathcal L_{\omega, \varepsilon}^n(h_\omega-h_{\omega, \varepsilon})\|_{L^1(m)}\cdot \|f_{\sigma^n \omega, \varepsilon}\|_{C^0} \\ &\le C_{\alpha, F}|\varepsilon|+C_{\alpha, F}n^{-1/\alpha+1}, \end{split} \end{align*} $$

which yields the desired conclusion.

Lemma 15. We have

$$ \begin{align*} |\bar D_{n, 1}^2(\varepsilon)| \le C_{\alpha, \underline{\alpha}, F}( n^{1-{1}/{2\alpha}} +n^{-1/\gamma}), \quad n<|\varepsilon|^{-\gamma}. \end{align*} $$

Proof. By using Theorem 6, we have

(27) $$ \begin{align} & \bigg|\!\int_0^1 \mathcal L_{\omega, \varepsilon}^n (f_{\omega, 0}\hat h_\omega)f_{\sigma^n \omega, 0}\, dm\bigg| \nonumber\\ &\quad\le \bigg|\!\int_0^1 \mathcal L_{\omega, \varepsilon}^n (f_{\omega, 0}(\hat h_\omega-r^{-1}(h_{\omega, r}-h_\omega))) f_{\sigma^n \omega, 0}\, dm\bigg| \nonumber\\ &\qquad+\bigg|\!\int_0^1 \mathcal L_{\omega, \varepsilon}^n (f_{\omega, 0}r^{-1}(h_{\omega, r}-h_\omega))f_{\sigma^n \omega, 0}\, dm\bigg| \nonumber\\ &\quad\le \|\mathcal L_{\omega, \varepsilon}^n (f_{\omega, 0}(\hat h_\omega-r^{-1}(h_{\omega, r}-h_\omega)))\|_{L^1(m)} \cdot \|f_{\sigma^n \omega, 0}\|_{C^0}\nonumber\\ &\qquad+\bigg|\!\int_0^1 \mathcal L_{\omega, \varepsilon}^n (f_{\omega, 0}r^{-1}(h_{\omega, r}-h_\omega))f_{\sigma^n \omega, 0}\, dm\bigg| \nonumber\\ &\quad\le C_{\alpha, \underline{\alpha}, F}r^{1-2\alpha}+ \bigg|\!\int_0^1 \mathcal L_{\omega, \varepsilon}^n (f_{\omega, 0}r^{-1}(h_{\omega, r}-h_\omega))f_{\sigma^n \omega, 0}\, dm\bigg| \end{align} $$

for $r\in (0, 1)$ that will be fixed later on. Next,

$$ \begin{align*} \begin{split} & \int_0^1 \mathcal L_{\omega, \varepsilon}^n (f_{\omega, 0}r^{-1}(h_{\omega, r}-h_\omega))f_{\sigma^n \omega, 0}\, dm \\ &\quad=\int_0^1 \mathcal L_{\omega, \varepsilon}^n \bigg(r^{-1}\bigg(f_{\omega, 0}-\int_0^1f_{\omega, 0}h_{\omega, r}\, dm\bigg)h_{\omega, r}\bigg)f_{\sigma^n \omega, 0}\, dm \\ &\qquad+\int_0^1 \mathcal L_{\omega, \varepsilon}^n \bigg(r^{-1}\bigg(\!\int_0^1 f_{\omega, 0}h_{\omega, r}\, dm\bigg)h_{\omega, r}\bigg)f_{\sigma^n \omega, 0}\, dm \\ &\qquad -\int_0^1 \mathcal L_{\omega, \varepsilon}^n \bigg(r^{-1} \bigg(f_{\omega, 0}-\int_0^1 f_{\omega, 0}h_\omega\, dm\bigg)h_\omega \bigg)f_{\sigma^n \omega, 0}\, dm \\ &\qquad-\int_0^1 \mathcal L_{\omega, \varepsilon}^n \bigg(r^{-1} \bigg(\!\int_0^1 f_{\omega, 0}h_\omega\, dm\bigg)h_\omega \bigg)f_{\sigma^n \omega, 0}\, dm. \end{split} \end{align*} $$

Since

$$ \begin{align*} \int_0^1 \bigg(f_{\omega, 0}-\int_0^1f_{\omega, 0}h_{\omega, r}\, dm\bigg)h_{\omega, r}\, dm=0, \end{align*} $$

we have using (A.2) that

(28) $$ \begin{align} &\bigg|\!\int_0^1 \mathcal L_{\omega, \varepsilon}^n \bigg(r^{-1}\bigg(f_{\omega, 0}-\int_0^1f_{\omega, 0}h_{\omega, r}\, dm\bigg)h_{\omega, r}\bigg)f_{\sigma^n \omega, 0}\, dm\bigg| \nonumber\\ &\quad\le C_F r^{-1} \bigg\|\mathcal L_{\omega, \varepsilon}^n \bigg(\bigg(f_{\omega, 0}-\int_0^1f_{\omega, 0}h_{\omega, r}\, dm\bigg)h_{\omega, r} \bigg)\bigg\|_{L^1(m)} \nonumber\\ &\quad\le C_{\alpha, F} r^{-1}n^{-1/\alpha+1} \bigg\|f_{\omega, 0}-\int_0^1f_{\omega, 0}h_{\omega, r}\, dm \bigg\|_{C^1} \nonumber\\ &\quad\le C_{\alpha, F} r^{-1}n^{-1/\alpha+1}. \end{align} $$

In the same manner,

(29) $$ \begin{align} \bigg|\!\int_0^1 \mathcal L_{\omega, \varepsilon}^n \bigg(r^{-1} \bigg(f_{\omega, 0}-\int_0^1 f_{\omega, 0}h_\omega\, dm\bigg)h_\omega \bigg)f_{\sigma^n \omega, 0}\, dm\bigg|\le C_{\alpha, F} r^{-1}n^{-1/\alpha+1}. \end{align} $$

Moreover,

$$ \begin{align*} \begin{split} &\int_0^1 \mathcal L_{\omega, \varepsilon}^n \bigg(r^{-1}\bigg(\!\int_0^1 f_{\omega, 0}h_{\omega, r}\, dm\bigg)h_{\omega, r}\bigg)f_{\sigma^n \omega, 0}\, dm\\ &\qquad-\int_0^1 \mathcal L_{\omega, \varepsilon}^n \bigg(r^{-1} \bigg(\!\int_0^1 f_{\omega, 0}h_\omega\, dm\bigg)h_\omega \bigg)f_{\sigma^n \omega, 0}\, dm \\ &\quad =\int_0^1 \mathcal L_{\omega, \varepsilon}^n \bigg(r^{-1} \bigg(\!\int_0^1f_{\omega, 0}h_{\omega, r}\, dm\bigg) (h_{\omega, r}-h_{\omega}) \bigg)f_{\sigma^n \omega, 0}\, dm \\ &\qquad+\int_0^1 \mathcal L_{\omega, \varepsilon}^n \bigg(r^{-1}\bigg(\!\int_0^1 f_{\omega, 0}h_{\omega, r}\, dm-\int_0^1f_{\omega, 0}h_\omega \, dm\bigg)h_\omega \bigg)f_{\sigma^n \omega, 0}\, dm. \end{split} \end{align*} $$

Observe that

(30) $$ \begin{align} & \bigg|\!\int_0^1 \mathcal L_{\omega, \varepsilon}^n \bigg(r^{-1} \bigg(\!\int_0^1f_{\omega, 0}h_{\omega, r}\, dm\bigg) (h_{\omega, r}-h_{\omega}) \bigg)f_{\sigma^n \omega, 0}\, dm \bigg| \nonumber\\ &\quad\le C_F r^{-1}\bigg|\!\int_0^1 f_{\omega, 0}h_{\omega, r}\, dm\bigg| \cdot \|\mathcal L_{\omega, \varepsilon}^n (h_{\omega, r}-h_\omega)\|_{L^1(m)} \nonumber\\ &\quad\le C_{\alpha, F} r^{-1}n^{-1/\alpha+1}. \end{align} $$

In addition

$$ \begin{align*} \begin{split} &\int_0^1 \mathcal L_{\omega, \varepsilon}^n \bigg(r^{-1}\bigg(\!\int_0^1 f_{\omega, 0}h_{\omega, r}\, dm-\int_0^1f_{\omega, 0}h_\omega \, dm\bigg)h_\omega \bigg)f_{\sigma^n \omega, 0}\, dm \\ &\quad=\int_0^1\! \mathcal L_{\omega, \varepsilon}^n \bigg(\kern-1.3pt \bigg(r^{-1}\bigg(\!\int_0^1 f_{\omega, 0}h_{\omega, r}\, dm-\int_0^1f_{\omega, 0}h_\omega \, dm\bigg )-\! \int_0^1f_{\omega, 0}\hat h_\omega \, dm\bigg)h_\omega \bigg)f_{\sigma^n \omega, 0}\, dm \\ &\qquad+\int_0^1 f_{\omega, 0}\hat h_\omega \, dm \int_0^1 \mathcal L_{\omega, \varepsilon}^n (h_\omega) f_{\sigma^n \omega, 0}\, dm. \end{split} \end{align*} $$

By Theorem 6,

(31) $$ \begin{align} & \bigg|\!\int_0^1\! \mathcal L_{\omega, \varepsilon}^n \bigg( \bigg(r^{-1}\bigg(\!\int_0^1 f_{\omega, 0}h_{\omega, r}\, dm-\int_0^1f_{\omega, 0}h_\omega \, dm\bigg )-\! \int_0^1f_{\omega, 0}\hat h_\omega \, dm\bigg)h_\omega \bigg)f_{\sigma^n \omega, 0}\, dm\bigg| \nonumber\\ &\quad\le C_F \bigg|r^{-1}\bigg(\!\int_0^1 f_{\omega, 0}h_{\omega, r}\, dm-\int_0^1f_{\omega, 0}h_\omega \, dm\bigg)-\int_0^1f_{\omega, 0}\hat h_\omega \, dm\bigg| \nonumber\\ &\quad\le C_{\alpha, \underline{\alpha}, F} r^{1-2\alpha}. \end{align} $$

Furthermore,

(32) $$ \begin{align} & \bigg|\!\int_0^1 f_{\omega, 0}\hat h_\omega \, dm \int_0^1 \mathcal L_{\omega, \varepsilon}^n (h_\omega) f_{\sigma^n \omega, 0}\, dm-\int_0^1 f_{\omega, 0}\hat h_\omega \, dm \int_0^1 h_{\sigma^n \omega, \varepsilon} f_{\sigma^n \omega, 0}\, dm \bigg| \nonumber\\ &\quad\le C_F \|\mathcal L_{\omega, \varepsilon}^n (h_\omega-h_{\omega, \varepsilon})\|_{L^1(m)} \nonumber\\ &\quad\le C_{\alpha, F} n^{-1/\alpha+1}, \end{align} $$

and thus (taking into account (27)–(32)),

$$ \begin{align*} \begin{split} &\bigg|\bar D_{n, 1}^2(\varepsilon)-\int_\Omega \int_0^1 f_{\omega, 0}\hat h_\omega \, dm \int_0^1 h_{\sigma^n \omega, \varepsilon} f_{\sigma^n \omega, 0}\, dm \, d\mathbb P(\omega)\bigg|\\ &\quad \le C_{\alpha, \underline{\alpha}, F} (r^{1-2\alpha}+r^{-1}n^{-1/\alpha+1}) \end{split} \end{align*} $$

for r close to $0$ . In addition, since $\int _0^1 f_{\omega , \varepsilon }h_{\omega , \varepsilon }\, dm=0$ ,

$$ \begin{align*} \begin{split} & \int_\Omega \int_0^1 f_{\omega, 0}\hat h_\omega \, dm \int_0^1 h_{\sigma^n \omega, \varepsilon} f_{\sigma^n \omega, 0}\, dm \, d\mathbb P(\omega)\\ &\quad=\int_\Omega \int_0^1 f_{\omega, 0}\hat h_\omega \, dm \int_0^1 h_{\sigma^n \omega, \varepsilon} (f_{\sigma^n \omega, 0}-f_{\sigma^n \omega, \varepsilon})\, dm \, d\mathbb P(\omega), \end{split} \end{align*} $$

which gives that

$$ \begin{align*} \bigg|\!\int_\Omega \int_0^1 f_{\omega, 0}\hat h_\omega \, dm \int_0^1 h_{\sigma^n \omega, \varepsilon} f_{\sigma^n \omega, 0}\, dm \, d\mathbb P(\omega)\bigg| \le C_{\alpha, F}|\varepsilon| \le C_{\alpha, F}n^{-1/\gamma} \end{align*} $$

for $n<|\varepsilon |^{-\gamma }$ . By choosing $r=n^{-({1-\alpha }/{\alpha (2-2\alpha )})}$ (so that $r^{1-2\alpha }=r^{-1}n^{-1/\alpha +1}$ ), we obtain the desired conclusion.

As a direct consequence of the previous two lemmas and (26), we obtain the following lemmas.

Lemma 16. We have

$$ \begin{align*} |\bar{D}_n^2(\varepsilon)| \le C_{\alpha, \underline{\alpha}, F} (n^{1-{1}/{2\alpha}}+n^{-1/\gamma}+n^{-1/\alpha+1}), \quad n<|\varepsilon|^{-\gamma}. \end{align*} $$

Lemma 17. For each $n\in \mathbb N$ ,

$$ \begin{align*} \lim_{\varepsilon \to 0}\bar D_n^2(\varepsilon)=\bar D_n^2(0). \end{align*} $$

Proof. We first show that

(33) $$ \begin{align}\lim_{\varepsilon \to 0} \bar{D}_{n,1}^2(\varepsilon) = \bar{D}_{n,1}^2(0).\end{align} $$

We start by decomposing

$$ \begin{align*} \hat{h}_\omega &= - \sum_{i=0}^\infty \delta( \sigma^{ - ( i + 1 ) } \omega ) {\mathcal{L}}_{ \sigma^{-i} \omega }^i ( X_{ \beta( \sigma^{ - ( i + 1) } \omega ) } N_{\beta( \sigma^{ - ( i + 1) } \omega )}( h_{ \sigma^{-(i+1)} \omega } ))' \\ &= - \sum_{i=0}^{k} \delta( \sigma^{ - ( i + 1 ) } \omega ) {\mathcal{L}}_{ \sigma^{-i} \omega }^i ( X_{ \beta( \sigma^{ - ( i + 1) } \omega ) } N_{\beta( \sigma^{ - ( i + 1) } \omega )}( h_{ \sigma^{-(i+1)} \omega } ))' \\ &\quad - \sum_{i= k + 1 }^\infty \delta( \sigma^{ - ( i + 1 ) } \omega ) {\mathcal{L}}_{ \sigma^{-i} \omega }^i ( X_{ \beta( \sigma^{ - ( i + 1) } \omega ) } N_{\beta( \sigma^{ - ( i + 1) } \omega )}( h_{ \sigma^{-(i+1)} \omega } ))'\\ &=: \hat{h}_{\omega,k} + R_{\omega, k}, \end{align*} $$

where $k\in \mathbb N$ . It follows from Theorem A.1 and Lemma A.2 that for $\mathbb P$ -a.e. $\omega \in \Omega $ ,

$$ \begin{align*} \|R_{\omega, k}\|_{L^1(m)} \le C_\alpha \sum_{i = k + 1}^\infty i^{ - 1 / \alpha + 1 } \le C_{\alpha} k^{ - 1 / \alpha + 2 }\quad \text{for } k\in \mathbb N. \end{align*} $$

Thus, for $k\in \mathbb N$ ,

$$ \begin{align*} |\bar{D}_{n,1}^2(\varepsilon) - \bar{D}_{n,1}^2(0)| &\le |\int_\Omega \int_0^1 [ {\mathcal{L}}_{\omega, \varepsilon}^n ( f_{\omega, 0} \hat{h}_{\omega, k} ) - {\mathcal{L}}_{\omega, 0}^n ( f_{\omega, 0} \hat{h}_{\omega, k} ) ] f_{\sigma^n\omega, 0} \, d m \, d {\mathbb{P}}(\omega)| \\ &\phantom{\le}+ C_{\alpha, F} k^{ - 1 / \alpha + 2}. \end{align*} $$

By $h_\omega \in {\mathcal {C}}_* \cap {\mathcal {C}}_2$ and Lemma A.2,

$$ \begin{align*} ( X_{ \beta( \sigma^{ - ( i + 1) } \omega ) } N_{\beta( \sigma^{ - ( i + 1) } \omega )}( h_{ \sigma^{-(i+1)} \omega } ))' = \psi_1 - \psi_2, \end{align*} $$

where $\psi _1, \psi _2 \in {\mathcal {C}}_* \cap C^1(0,1]$ . Hence, [Reference Leppänen and Stenlund30, Lemma 3.3] implies that

$$ \begin{align*} f_{\omega, 0} {\mathcal{L}}_{\sigma^{-i} \omega }^i ( X_{ \beta( \sigma^{ - ( i + 1) } \omega ) } N_{\beta( \sigma^{ - ( i + 1) } \omega )}( h_{ \sigma^{-(i+1)} \omega } ))' = \tilde{\psi}_{i,1} - \tilde{\psi}_{i,2}, \end{align*} $$

where $\tilde {\psi }_{i,j} \in C^1(0,1] \cap {\mathcal {C}}_*$ . Consequently,

(34) $$ \begin{align} f_{\omega, 0} \hat{h}_{\omega, k} = - \sum_{i=0}^k \sum_{j=1,2} \delta( \sigma^{-(i+1)} \omega ) (-1)^{j - 1 } \tilde{\psi}_{i,j}. \end{align} $$

Substituting (34) into the remaining expression and using $\varphi (x + \varepsilon ) - \varphi (x) = \varepsilon \int _{0}^1 \varphi '( x + t \varepsilon ) \, dt$ , we obtain

$$ \begin{align*} &\int_\Omega \int_0^1 [ {\mathcal{L}}_{\omega, \varepsilon}^n ( f_{\omega, 0} \hat{h}_{\omega, k} ) - {\mathcal{L}}_{\omega, 0}^n ( f_{\omega, 0} \hat{h}_{\omega, k} ) ] f_{\sigma^n\omega, 0} \, d m \, d {\mathbb{P}}(\omega) \\ &\quad = - \sum_{i=0}^k \!\sum_{j=1,2} \delta( \sigma^{ - ( i + 1 ) } \omega ) (-1)^{j - 1 } \int_\Omega \int_0^1 [ {\mathcal{L}}_{\omega, \varepsilon}^n ( \tilde{\psi}_{i,j} ) - {\mathcal{L}}_{\omega, 0}^n ( \tilde{\psi}_{i,j} ) ] f_{\sigma^n\omega, 0} \, d m \, d {\mathbb{P}}(\omega) \\ &\quad = - \sum_{i=0}^k \sum_{j=1,2} \sum_{\ell = 0}^{ n - 1} \delta( \sigma^{ - ( i + 1 ) } \omega ) (-1)^{j - 1 } \int_\Omega \int_0^1 {\mathcal{L}}_{ \sigma^{n - \ell } \omega, \varepsilon }^\ell\\ &\qquad \times [ {\mathcal{L}}_{ \beta( \sigma^{n-1-\ell} \omega ) + \varepsilon \delta(\sigma^{n-1-\ell} \omega ) } - {\mathcal{L}}_{ \beta( \sigma^{n-1-\ell} \omega ) } ] \\ &\qquad \times {\mathcal{L}}_{ \omega }^{n-1-\ell} ( \tilde{\psi}_{i,j} ) f_{\sigma^n\omega, 0} \, d m \, d {\mathbb{P}}(\omega) \\ &\quad= - \sum_{i=0}^k \sum_{j=1,2} \sum_{\ell = 0}^{ n - 1} \delta( \sigma^{ - ( i + 1 ) } \omega ) (-1)^{j - 1 } \varepsilon \delta(\sigma^{n-1-\ell} \omega ) \\ &\qquad \times \int_\Omega \int_0^1\! \int_0^1 \!{\mathcal{L}}_{ \sigma^{n - \ell } \omega, \varepsilon }^\ell \partial_\gamma {\mathcal{L}}_{ \beta( \sigma^{n-1-\ell} \omega ) + t \varepsilon \delta(\sigma^{n-1-\ell} \omega ) } {\mathcal{L}}_{ \omega }^{n-1-\ell} ( \tilde{\psi}_{i,j} ) f_{\sigma^n\omega, 0} \, dt \, d m \, d {\mathbb{P}}(\omega). \end{align*} $$

Moreover,

$$ \begin{align*} &\partial_\gamma {\mathcal{L}}_{ \beta( \sigma^{n-1-\ell} \omega ) + t \varepsilon \delta(\sigma^{n-1-\ell} \omega ) } {\mathcal{L}}_{ \omega }^{n-1-\ell} ( \tilde{\psi}_{i,j} ) \\ &\quad= ( X_{ \beta( \sigma^{n-1-\ell} \omega ) + t \varepsilon \delta(\sigma^{n-1-\ell} \omega ) } N_{ \beta( \sigma^{n-1-\ell} \omega ) + t \varepsilon \delta(\sigma^{n-1-\ell} \omega ) } ( {\mathcal{L}}_{ \omega }^{n-1-\ell} ( \tilde{\psi}_{i,j} ) ) )'. \end{align*} $$

It follows from (15), bounds on $X_\gamma $ and $X_\gamma '$ in [Reference Baladi and Todd9, (2.3) and (2.4)], and invariance properties of the cones in (A.1) that

$$ \begin{align*} \Vert ( X_{ \beta( \sigma^{n-1-\ell} \omega ) + t \varepsilon \delta(\sigma^{n-1-\ell} \omega ) } N_{ \beta( \sigma^{n-1-\ell} \omega ) + t \varepsilon \delta(\sigma^{n-1-\ell} \omega ) } ( {\mathcal{L}}_{ \omega }^{n-1-\ell} ( \tilde{\psi}_{i,j} ) ) )' \Vert_{L^1(m)} \le C_{\alpha, \underline{\alpha}, F}. \end{align*} $$

Consequently,

$$ \begin{align*} &\bigg|\!\int_\Omega \int_0^1 [ {\mathcal{L}}_{\omega, \varepsilon}^n ( f_{\omega, 0} \hat{h}_{\omega, k} ) - {\mathcal{L}}_{\omega, 0}^n ( f_{\omega, 0} \hat{h}_{\omega, k})] f_{\sigma^n\omega, 0} \, d m \, d {\mathbb{P}}(\omega) \bigg| \le C_{\alpha, \underline{\alpha}, F} |\varepsilon| kn. \end{align*} $$

We choose $k = \lfloor |\varepsilon |^{ - \alpha / ( 1 - \alpha ) } \rfloor $ so that $|\varepsilon | k$ and $k^{- 1 / \alpha + 2}$ are approximately equal. Since $\alpha < 1/2$ ,

$$ \begin{align*} |\bar{D}_{n,1}^2(\varepsilon) - \bar{D}_{n,1}^2(0)| \le C_{\alpha, \underline{\alpha}, F} (|\varepsilon| kn + k^{- 1 / \alpha + 2} )\le C_{\alpha, \underline{\alpha}, F, n}|\varepsilon|^{ 1 - \alpha / ( 1 - \alpha ) } \to 0 \quad \text{as } \varepsilon \to 0. \end{align*} $$

Hence, (33) holds.

We now claim that

(35) $$ \begin{align} \lim_{\varepsilon \to 0}\bar{D}_{n, 2}^2(\varepsilon)=\bar{D}_{n, 2}(0). \end{align} $$

Note that

$$ \begin{align*} \begin{split} \mathcal L_{\omega, \varepsilon}^n (h_\omega)-\mathcal L_\omega^n (h_\omega) &=\sum_{j=0}^{n-1} \mathcal L_{\sigma^{j+1} \omega, \varepsilon}^{n-j-1}(\mathcal L_{\sigma^{j}\omega, \varepsilon}-\mathcal L_{\sigma^{j}\omega})\mathcal L_\omega^{j}(h_\omega) \\ &=\sum_{j=0}^{n-1} \int_{\beta(\sigma^{j}\omega)}^{\beta(\sigma^{j}\omega)+\varepsilon \delta(\sigma^{j}\omega)}\mathcal L_{\sigma^{j +1}\omega, \varepsilon}^{n-j-1}[\partial_\gamma \mathcal L_\gamma (h_{\sigma^{j}\omega})]\, d\gamma \\ &=-\sum_{j=0}^{n-1} \int_{\beta(\sigma^{j}\omega)}^{\beta(\sigma^{j}\omega)+\varepsilon \delta(\sigma^{j}\omega)} \mathcal L_{\sigma^{j+1} \omega, \varepsilon}^{n-j-1}[(X_\gamma N_\gamma(h_{\sigma^{j}\omega}))']\, d\gamma. \end{split} \end{align*} $$

By applying Lemma A.2, we conclude (using the convention $0^{-1/\alpha +1}:=1$ ) that

$$ \begin{align*} \|\mathcal L_{\omega, \varepsilon}^n (h_\omega)-\mathcal L_\omega^n (h_\omega)\|_{L^1(m)}\le C_\alpha |\varepsilon|\sum_{j=0}^{n-1} (n-j-1)^{-1/\alpha+1}\le C_\alpha|\varepsilon|, \end{align*} $$

which easily implies (35). Finally, it remains to note that the conclusion of the lemma follows readily from (33) and (35).

We now consider $D_n^1(\varepsilon )$ and note that

$$ \begin{align*} D_n^1(\varepsilon)=\frac{1}{\varepsilon}\int_\Omega \int_0^1 (\mathcal L_{\omega, \varepsilon}^n-\mathcal L_\omega^n)(f_{\omega, 0}h_\omega) f_{\sigma^n \omega, 0}\, dm\, d\mathbb P(\omega). \end{align*} $$

Lemma 18. We have

(36) $$ \begin{align} |D_n^1(\varepsilon)| \le C_{\alpha, \underline{\alpha}, F}(n^{ 1 - 1 / \gamma } + n^{1-{(1-\alpha)}/({2\alpha})}), \quad n<|\varepsilon|^{-\gamma}. \end{align} $$

Proof. Observe that

(37) $$ \begin{align} D_n^1(\varepsilon)&=\int_\Omega \int_0^1 \sum_{j=0}^{n-1}\mathcal L_{\sigma^{j+1}\omega, \varepsilon}^{n-j-1}(\varepsilon^{-1}(\mathcal L_{\sigma^j \omega, \varepsilon}-\mathcal L_{\sigma^j \omega})\mathcal L_{\omega}^j(f_{\omega, 0}h_\omega))f_{\sigma^n \omega, 0}\, dm\, d\mathbb P(\omega) \nonumber\\ &\quad=\int_\Omega\int_0^1 \sum_{j=0}^{n-1}\delta (\sigma^j \omega)\mathcal L_{\sigma^{j+1}\omega, \varepsilon}^{n-j-1}(\partial_\gamma \mathcal L_{\beta(\sigma^j \omega)}(\mathcal L_\omega^j(f_{\omega, 0}h_\omega)))f_{\sigma^n \omega,0}\, dm\, d\mathbb P(\omega) \nonumber\\ &\qquad+\varepsilon\int_\Omega \sum_{j=0}^{n-1}(\delta (\sigma^j \omega))^2\int_0^1 \int_0^1 (1-t) f_{\sigma^n \omega, 0}\mathcal L_{\sigma^{j+1}\omega, \varepsilon}^{n-j-1} \nonumber\\ & \qquad \times (\partial^2_\gamma \mathcal L_{\beta(\sigma^j \omega)+t\varepsilon \delta (\sigma^j \omega)}(\mathcal L_\omega^j(f_{\omega, 0}h_\omega)))\, dm\, dt\, d\mathbb P(\omega) \nonumber\\ &\quad=:D_{n,1}^1(\varepsilon)+D_{n, 2}^1(\varepsilon). \end{align} $$

It follows from Lemmas B.2 and B.3 that

(38) $$ \begin{align} |D_{n,2}^1(\varepsilon)| \le C_{\alpha, \underline{\alpha}, F} n|\varepsilon| \le C_{\alpha, \underline{\alpha}, F}n^{1-1/\gamma}. \end{align} $$

We now turn to $D_{n,1}^1(\varepsilon )$ . By definition,

$$ \begin{align*} \begin{split} D_{n,1}^1(\varepsilon) =\sum_{j=0}^{n-1}I_j, \end{split} \end{align*} $$

where

$$ \begin{align*} I_j:= \int_\Omega \delta (\sigma^j \omega) \int_0^1 \mathcal L_{\sigma^{j+1}\omega, \varepsilon}^{n-j-1}(\partial_\gamma \mathcal L_{\beta(\sigma^j \omega)}(\mathcal L_\omega^j(f_{\omega, 0}h_\omega)))f_{\sigma^n \omega,0}\, dm\, d\mathbb P(\omega). \end{align*} $$

For j such that $n-j-1\ge \lfloor n/2\rfloor -1$ , we have (using Lemma B.2 and Theorem A.1) that

(39) $$ \begin{align} |I_j| \le C_{\alpha, F} (n-j-1)^{-1/\alpha+1}\le C_{\alpha, F} n^{-1/\alpha+1}. \end{align} $$

We now consider j so that $j\ge \lfloor n/2\rfloor $ . For $r>0$ to be fixed later, we have by writing

$$ \begin{align*} r^{-1}(\mathcal L_{\beta(\sigma^j \omega)+r}-\mathcal L_{\beta (\sigma^j \omega)})(\varphi)=\partial_\gamma \mathcal L_{\beta(\sigma^j \omega)}(\varphi)+r\int_0^1(1-t)\partial_\gamma^2 \mathcal L_{\beta(\sigma^j \omega)+tr}(\varphi)\, dt, \end{align*} $$

and using Lemmas B.2 and B.3 that

$$ \begin{align*} \| \partial_\gamma \mathcal L_{\beta(\sigma^j \omega)}(\mathcal L_\omega^j(f_{\omega, 0}h_\omega))-r^{-1}(\mathcal L_{\beta(\sigma^j \omega)+r}-\mathcal L_{\beta (\sigma^j \omega)})(\mathcal L_\omega^j (f_{\omega, 0}h_\omega))\|_{L^1(m)} \le C_{\alpha, \underline{\alpha}, F}r. \end{align*} $$

Writing $I_j$ as

$$ \begin{align*} I_j&=\\ &=\int_\Omega \delta(\sigma^j \omega)\int_0^1 \mathcal L_{\sigma^{j+1}\omega, \varepsilon }^{n-j-1}( (\partial_\gamma \mathcal L_{\beta( \sigma^j \omega)}\\ &\quad -r^{-1}(\mathcal L_{\beta(\sigma^j \omega)+r}-\mathcal L_{\beta (\sigma^j \omega)}))(\mathcal L_\omega^j (f_{\omega, 0}h_\omega))) f_{\sigma^n \omega, 0}\, dm\, d\mathbb P(\omega)\\ &\quad+r^{-1}\int_\Omega \delta(\sigma^j \omega)\int_0^1 \mathcal L_{\sigma^{j+1}\omega, \varepsilon}^{n-j-1}\\ &\quad \times ((\mathcal L_{\beta(\sigma^j \omega)+r}-\mathcal L_{\beta (\sigma^j \omega)})(\mathcal L_\omega^j (f_{\omega, 0}h_\omega))) f_{\sigma^n \omega, 0}\, dm\, d\mathbb P(\omega), \end{align*} $$

we have that

(40) $$ \begin{align} |I_j| \le C_{\alpha, \underline{\alpha}, F}r+C_{\alpha, F}r^{-1}j^{-1/\alpha+1}\le C_{\alpha, \underline{\alpha}, F}(r+r^{-1}n^{-1/\alpha+1}), \end{align} $$

where we used that

$$ \begin{align*} \begin{split} &\|\mathcal L_{\sigma^{j+1}\omega, \varepsilon}^{n-j-1} ((\mathcal{L}_{\beta(\sigma^j \omega)+r}-\mathcal L_{\beta (\sigma^j \omega)})(\mathcal L_\omega^j (f_{\omega, 0}h_\omega)))\\ &\quad \|_{L^1(m)}\le 2\|\mathcal L_\omega^j (f_{\omega, 0}h_\omega)\|_{L^1(m)}\le 2C_{\alpha, F}j^{-1/\alpha+1}. \end{split} \end{align*} $$

Setting $r=n^{-({1-\alpha })/{2\alpha }}$ , it follows that

(41) $$ \begin{align} |I_j| \le C_{\alpha, \underline{\alpha}, F} n^{-({1-\alpha})/{2\alpha}}, \quad j\ge \lfloor n/2\rfloor. \end{align} $$

From (39) and (41), we conclude that

(42) $$ \begin{align} |D_{n,1}^1(\varepsilon)|\le \sum_{j=0}^{n-1}|I_j| \le C_{\alpha, \underline{\alpha}, F}n^{1-({1-\alpha})/{2\alpha}}. \end{align} $$

Estimate (36) now follows readily from (38) and (42).

Observe that the upper bound in Lemma 16 is summable if $\alpha < 1/4$ , and the upper bound in Lemma 18 is summable if $\alpha < 1/5$ . To complete the proof of Theorem 9, it remains to prove the following lemma.

Lemma 19. We have

$$ \begin{align*} \lim_{\varepsilon \to 0}D_n^1(\varepsilon)=\sum_{j=0}^{n-1}\int_\Omega \delta (\sigma^j \omega)\int_0^1\mathcal L_{\sigma^{j+1}\omega}^{n-j-1} (\partial_\gamma \mathcal L_{\beta(\sigma^j \omega)}(\mathcal L_\omega^j(f_{\omega, 0}h_\omega)))f_{\sigma^n \omega, 0}\, dm\, d\mathbb P(\omega). \end{align*} $$

Proof. We decompose as in (37):

$$ \begin{align*} D_n^1(\varepsilon) = D_{n,1}^1(\varepsilon)+D_{n, 2}^1(\varepsilon), \end{align*} $$

where $\lim _{\varepsilon \to 0} D_{n, 2}^1(\varepsilon ) = 0$ (see (38)). Next,

$$ \begin{align*} D_{n,1}^1(\varepsilon) &=\int_\Omega\int_0^1 \sum_{j=0}^{n-1}\delta (\sigma^j \omega)\mathcal L_{\sigma^{j+1}\omega}^{n-j-1}(\partial_\gamma \mathcal L_{\beta(\sigma^j \omega)}(\mathcal L_\omega^j(f_{\omega, 0}h_\omega)))f_{\sigma^n \omega,0}\, dm\, d\mathbb P(\omega)\\ &\quad+\int_\Omega\int_0^1 \sum_{j=0}^{n-1}\delta (\sigma^j \omega)(\mathcal L_{\sigma^{j+1}\omega, \varepsilon}^{n-j-1}-\mathcal L_{\sigma^{j+1}\omega}^{n-j-1})\\ &\quad \times (\partial_\gamma \mathcal L_{\beta(\sigma^j \omega)}(\mathcal L_\omega^j(f_{\omega, 0}h_\omega)))f_{\sigma^n \omega,0}\, dm\, d\mathbb P(\omega)\\ &=:\mathcal D_{n, 1}^1+\mathcal D_{n, 2}^1(\varepsilon). \end{align*} $$

Furthermore,

$$ \begin{align*} \begin{split} \mathcal D_{n, 2}^1(\varepsilon) &= \sum_{j=0}^{n-1}\sum_{k=0}^{n-j-2}\int_\Omega \delta(\sigma^j \omega) \int_0^1 \mathcal L_{\sigma^{j+k+2}\omega, \varepsilon}^{n-j-k-2}\int_{\beta(\sigma^{j+1}\omega)}^{\beta(\sigma^{j+1}\omega+\varepsilon \delta(\sigma^{(j+1)}\omega)}\\ & \quad\times \partial_\gamma \mathcal L_\gamma (\mathcal L_{\sigma^{j+1}\omega}^k (\partial_\gamma \mathcal L_{\beta(\sigma^j \omega)}(\mathcal L_{\omega}^j(f_{\omega, 0}h_\omega))) \\ &\quad\times f_{\sigma^n \omega, 0}\, d\gamma \, dm\, d\mathbb P(\omega). \end{split} \end{align*} $$

Using Lemma A.2 together with Lemmas B.2 and B.4, we have

$$ \begin{align*} |\mathcal D_{n, 2}^1(\varepsilon)| \le C_{\alpha, \underline{\alpha}, F}n^2|\varepsilon|. \end{align*} $$

Hence, $\lim _{\varepsilon \to 0} \mathcal D_{n, 2}^1(\varepsilon ) = 0$ . This immediately implies the desired conclusion.

4.4 An improvement of Theorem 9 for a special class of observables

Here, we indicate how we can relax the assumption $\alpha < 1/5$ in the statement of Theorem 9 provided that we impose additional assumptions on F. We first note that the restriction $\alpha < 1/5$ is used only in Lemma 18; the rest of the proof remains valid under the weaker condition $\alpha < 1/4$ .

We continue to assume that $F\colon \Omega \times [0, 1]\to \mathbb R$ is measurable with $F(\omega , \cdot )\in C^2[0, 1]$ for $\omega \in \Omega $ and that (19) is valid. We need the following version of Lemma 18.

Lemma 20. Suppose that there exist $0 \le \eta \le \alpha $ and $C>0$ such that, for ${\mathbb {P}}$ -a.e. $\omega \in \Omega $ and $x \in [0,1]$ ,

(43) $$ \begin{align} |F_\omega(x)| \le C x^\eta \quad \text{and} \quad \int_0^1 F_\omega h_\omega \, d m = 0. \end{align} $$

Then,

(44) $$ \begin{align} |D_n^1(\varepsilon)| \le C_{\alpha, \underline{\alpha}, F}(n^{ 1 - 1 / \gamma }+ n^{ 1 - ({1 + \eta - \alpha })/{ 2 \alpha } } ), \quad n<|\varepsilon|^{-\gamma}. \end{align} $$

Proof. The proof proceeds as the proof of Lemma 18 up to (40). It follows from (4) that, for ${\mathbb {P}}$ -a.e. $\omega \in \Omega $ ,

$$ \begin{align*} \|\mathcal L_\omega^j (f_{\omega, 0}h_\omega)\|_{L^1(m)} \le C_{\alpha, \underline{\alpha}, F} n^{ 1 - (1 + \eta)/\alpha}, \quad j\ge \lfloor n/2\rfloor. \end{align*} $$

Thus, choosing $r = n^{ \tfrac 12 ( 1 - ({1 + \eta })/{\alpha }) }$ , we obtain

$$ \begin{align*} |I_j| \le C_{\alpha, \underline{\alpha}, F}(r+r^{-1}n^{ 1 - (1 + \eta)/\alpha }) \le C_{\alpha, \underline{\alpha}, F} n^{ - ({1 + \eta - \alpha })/{ 2 \alpha } } , \quad j\ge \lfloor n/2\rfloor, \end{align*} $$

which yields (44).

Theorem 21. Suppose that (43) holds with $0\le \eta \le \alpha $ and $C>0$ . Provided that ${\alpha <({1+\eta })/{5}}$ , we have that $\varepsilon \mapsto \Sigma _\varepsilon ^2$ is differentiable in $\varepsilon =0$ .

Proof. We proceed exactly as in the proof of Theorem 9 with the only difference being that we use Lemma 20 instead of Lemma 18. The conclusion follows since the upper bound (44) is summable if α < (1 + η)/5.