Convergence rates in the central limit theorem for means of autoregressive and moving average sequences
Let an autoregression be given by \(\sum^ \infty_{k=0} b_ k X_{j- k}=\xi_ j\), \(-\infty<j<\infty\), where the innovations \(\xi_ j\) are i.i.d. random variables with zero mean and unit variance (the \(b_ k\)'s being constants). Assume that the function \(B(\theta)=\sum^ \infty_{k=0} b_ k\cos(k\theta)\), \(0<\theta<2\pi\), is well defined and satisfies \(\int^{2\pi}_ 0 B(\theta)^{-2} d\theta<\infty\). Then \(X_ j=\sum^ \infty_{k=0} a_ k\xi_{j-k}\), \(-\infty<j<\infty\). Suppose that the covariance function is regularly varying with exponent \(-\alpha\) \((\alpha\geq 0)\). It is shown that the rate of convergence in a CLT for \(\overline X={1\over n}(X_ 1+\cdots+X_ n)\) is identical to that in the CLT for \(\overline\xi={1\over n}(\xi_ 1+\cdots+\xi_ n)\) iff \(\alpha>0\). Strikingly, when \(\alpha=0\), the corresponding rate for \(\overline X\) can be faster than that for \(\overline\xi\); it can, however, never be slower.
central limit theorem for means of autoregressive and moving average sequences
scientific article; zbMATH DE number 98447
scientific article