Berry–Esseen Type Estimates for Standardized Martingales

In this paper, we establish a non-uniform Berry–Esseen type bound for standardized martingales. While classical bounds primarily address unstandardized partial sums, standardizing by the conditional quadratic variation is highly relevant for practical statistical inference. Our main result demonstrates that the standardized martingale achieves the same optimal polynomial decay rate as its unstandardized counterpart, extending existing classical results. Furthermore, we apply our theoretical framework to derive convergence rates for the least-squares estimator of a first-order autoregressive (AR(1)) process, explicitly verifying the underlying hypotheses and highlighting the practical utility of our bounds in time series analysis.