Berry–Esseen Bounds of Residual Density Estimators in the First-Order Autoregressive Model with the α-Mixing Errors

This study establishes explicit Berry–Esseen bounds for residual kernel density estimators in AR(1) models with α-mixing errors. Since the true innovations are unobservable, we introduce a residual-based estimator f^n(x) and establish its normal approximation under stationarity. By imposing conditions on the bandwidth, mixing coefficients, and moments, we obtain Kolmogorov distance bounds between the standardized estimator and its Gaussian limit. These bounds explicitly depend on the bandwidth, block parameters, and mixing coefficients. A key corollary quantifies the convergence rate as O(n(2c−2b+a)/4). Our results generalize prior work, advancing theoretical foundations for nonparametric inference in high-dimensional time series.