On non-uniform Berry–Esseen bounds for time series

Abstrakt

Given a stationary sequence {Xk}_kεZ, non-uniform bounds for the normal approximation in the Kolmogorov metric are established. The underlying weak dependence assumption includes many popular linear and nonlinear time series from the literature, such as ARMA or GARCH models. Depending on the number of moments p, typical bounds in this context are of the size Om^p−1n^−p/2+1, where we often find that m = m_n = log n. In our setup, we can essentially improve upon this rate by the factorm m^−p/2, yielding a bound of Om^p/2−1n^−p/2+1. Among other things, this allows us to recover a result from the literature, which is due to Ibragimov.