$\ell$ is a measurable slowly varying function (so $\lim_{x\to\infty}\ell(\lambda x)/\ell(x)=1$ for all $\lambda>0$).

    Let $d_\beta=\frac{1}{\pi}\sin\beta\pi$ for $\beta<1$ and $d_\beta=1$ for $\beta=1$. Define $\tilde\ell(t)=\ell(t)$ for $\beta<1$ and $\tilde\ell(t)=\int_1^t \ell(s)s^{-1}\,ds$ for $\beta=1$.

    Then there exist $\theta\in(0,1)$, $m\ge1$ such that $$\tilde\ell(t)t^{1-\beta}\rho_{v,w}(t)\to d_\beta\bar v\bar w\enspace\text{as $t\to\infty$},$$ for all $v\in F_\theta(\tilde Y)$, $w\in L^{\infty,m}(\tilde Y)$.

-   Suppose moreover that $\mu(\varphi>t)=ct^{-\beta}+O(t^{-q})$ where $\beta\in(\frac12,1)$, $q>1$ and $c>0$. There exist constants $d_1=c^{-1}d_\beta$, $d_2,d_3,\ldots \in{{\mathbb R}}$ (depending only on $F$ and $\varphi$, and typically nonzero), and for any $\epsilon>0$, there exist $\theta\in(0,1)$, $m\ge1$, such that $$\rho_{v,w}(t)=
    \sum_j d_jt^{-j(1-\beta)}
    \bar v \bar w
    +O(t^{-(\frac12-\epsilon)}),$$ for all $v\in F_\theta(\tilde Y)$, $w\in L^{\infty,m}(\tilde Y)$, $t>0$. Here, the sum is over those $j\ge1$ with $j(1-\beta)\le \frac12-\epsilon$.

In particular, if $\beta>\frac34$ and $d_2\neq0$, we obtain second order asymptotics.

In the finite case, define $$\begin{aligned}
 \label{eq-xi}
\gamma(t) & =\int_t^\infty \mu(\varphi>\tau)\,d\tau, 
\quad
\xi_{\beta,\epsilon}(t) = 
\begin{cases}  t^{-(\beta-\epsilon)}, &  \beta\ge 2 \\
t^{-(2\beta-2)}, & 1<\beta<2 \end{cases}.\end{aligned}$$

\[thm-finite\] Assume that $F:Y\to Y$ is Gibbs-Markov with roof function $\phi:Y\to{{\mathbb R}}^+$ satisfying conditions (A1) and (A2).

-   Suppose that $\mu(\varphi>t)=O(t^{-\beta})$ where $\beta\ge1$. Then for any $\epsilon>0$, there exists $\theta\in(0,1)$, $m\ge1$, such that $$\rho_{v,w}(t)-\bar v\bar w =(1/{\bar\varphi})\bar v\bar w\,\gamma(t)
    +O(\|v\|_\theta|w|_{\infty,m}\,\xi_{\beta,\epsilon}(t)),$$ for all $v\in F_\theta(\tilde Y)$, $w\in L^{\infty,m}(\tilde Y)$, $t>0$.

-   Suppose further 