Functional correlation decay and multivariate normal approximation for non-uniformly expanding maps

Given a unit vector $\newcommand{\bR}{{{\mathbb R}}} v\in\bR^d$, we say that f is a coboundary in the direction v if there exists a function $\newcommand{\bR}{{{\mathbb R}}} g_v:[0, 1]\to\bR$ in $L^2(\hat{\mu}_{\beta_*})$ such that $v\cdot f = g_v - g_v\circ T_{\beta_*}$.

Strictly speaking, the authors of [1] considered a slightly modified version of the map $T_{\alpha}$, but they pointed out that their results hold for more general maps and in particular for the map $T_{\alpha}$. See [1, 14] for details.

We denote by $\newcommand{\round}[1]{\lfloor#1\rfloor} \round{x}$ the greatest non-negative integer n with $n \leqslant x$.

This condition was not stipulated in the main result of [15], but we have added it to ensure that the covariance Σ is positive definite. This is not necessary, if a more general definition of normal distribution, such as the one given in [15], is used.