The decay-of-correlations summability claim for non-uniformly expanding maps

Let $f:\mathbb{T}^n\to\mathbb{T}^n$ be a non-uniformly expanding map with first hyperbolic time map $h$, and suppose that its correlations decay sufficiently fast. The summability condition is

$$\sum_{n\geq 1}\sum_{j=0}^{n-1}\lambda\bigl(f^{j}(h^{-1}(n))\bigr)<\infty.$$

Decay-of-correlations summability claim. A non-uniformly expanding map having a sufficiently fast rate of decay of correlations satisfies the summability condition above. This condition is used in the paper to obtain stochastic stability for adapted random perturbations in higher dimensions. The source does not specify what rate of decay is sufficiently fast or provide a resolution of the claim.