A parameter ASIP for the quadratic family

Consider the quadratic family $T_a(x) = a x (1 - x)$, for $x \in [0, 1]$ and mixing Collet--Eckmann (CE) parameters $a \in (2,4)$. For bounded $\varphi$, set $\tilde \varphi_{a} := \varphi - \int \varphi \, d\mu_a$, with $\mu_a$ the unique acim of $T_a$, and put $(\sigma_a (\varphi))^2 := \int \tilde \varphi_{a}^2 \, d\mu_a + 2 \sum_{i>0} \int \tilde \varphi_{a} (\tilde \varphi_{a} \circ T^i_{a}) \, d\mu_a$. For any transversal mixing Misiurewicz parameter $a_*$, we find a positive measure set $\Omega_*$ of mixing CE parameters, containing $a_*$ as a Lebesgue density point, such that for any H\"older $\varphi$ with $\sigma_{a_*}(\varphi)\ne 0$, there exists $\epsilon_\varphi >0$ such that, for normalised Lebesgue measure on $\Omega_*\cap [a_*-\epsilon_\varphi, a_*+\epsilon_\varphi]$, the functions $\xi_i(a)=\tilde \varphi_a(T_a^{i+1}(1/2))/\sigma_a (\varphi)$ satisfy an almost sure invariance principle (ASIP) for any error exponent $\gamma >2/5$. (In particular, the Birkhoff sums satisfy this ASIP.) Our argument goes along the lines of Schnellmann's proof for piecewise expanding maps. We need to introduce a variant of Benedicks-Carleson parameter exclusion and to exploit fractional response and uniform exponential decay of correlations from a previous work of Baladi, Benedicks, and Schnellmann.