Can we further improve the central limit theorem for stationary $ρ$-mixing sequences $?$

Abstract: Through the seminal works of Peligrad and Bradley, the existing sufficient conditions for the central limit theorem for $ρ$-mixing sequences are known to be very close to necessary. Nevertheless, we show that further improvements to these conditions are achievable when the $ρ$-mixing coefficients are non-summable, bridging a long-standing gap in the borderline case. The key idea is an iterative block-size technique that enables precise evaluations of the slowly varying variance components. Finally, known examples and new extensions--including iterated logarithmic mixing rates--are investigated to demonstrate the sharpness of our criteria, revealing a clear parameter structure and posing related open problems.