Convergence rates in the central limit theorem for means of autoregressive and moving average sequences

AbstractLet X̄ denote the mean of a consecutive sequence of length n from an autoregression or moving average process. Suppose the covariance function of the process is regularly varying with exponent −α, where α ⩾ 0. We show that the rate of convergence in a central limit theorem for X̄ is identical to that in the central limit theorem for the mean of n independent innovations, if and only if α ⩾ 0. Strikingly, the convergence rate when α = 0 can be faster than in the case of the independent sequence; it can never be slower. Furthermore, the convergence rate is fastest in the case of strongest dependence. This result is established in two ways: firstly by developing an Edgeworth expansion under the condition of finite third moment of innovations, and secondly by deriving the precise convergence rate in the central limit theorem without an assumption of finite third moment