  We extend the results of [Ullah, A., Vilarinho, H,.: Statistical properties
of dynamical systems via induced weak Gibbs Markov maps. arXiv:2311.17531
(2023)] by considering larger classes of observables. More precisely, we obtain
estimates on the decay of correlations, Central Limit Theorem and Large
Deviations for dynamical systems having an induced weak Gibbs Markov map, for
larger classes of observables with weaker regularity than H\"{o}lder.
