The Maximum Deviation of Sample Spectral Densities

Abstract

The present paper gives sufficient conditions on a linear process $\{X_t\}$ and its spectral density $f(\cdot)$ for the following limit relation to hold: \begin{equation*}\tag{0.1}\parallel W \parallel^{-1}_2(N/2m_N \log m_N)^{\frac{1}{2}} \max_{-\pi\leqq\lambda\leqq\pi} \lbrack |f_N (\lambda) - f(\lambda)| / f(\lambda)\rbrack \rightarrow 1\end{equation*} in probability as $N \rightarrow \infty$ where $f_N(\cdot)$ is the usual windowed sample spectral density, $m_NW(m_N\cdot)$ is the (varying) window, and $m_N \uparrow \infty$ as $N \rightarrow \infty$ at a suitable rate. Under the same conditions it is shown that \begin{align}\tag{0.2}P(a_N^{-1}\lbrack N^{\frac{1}{2}}m_N^{-\frac{1}{2}} \parallel W\parallel^{-1}_2 \max_{|i|\leqq m_N} \lbrack |f_N (\lambda^\ast_{N,i}) - f(\lambda^\ast_{N,i})|/f(\lambda^\ast_{N,i})\rbrack \\ - b_N\rbrack \leqq x) \rightarrow \exp (-\exp(-x))\end{align} as $N \rightarrow \infty \text{for} - \infty < x < \infty$ where $\lambda^\ast_{N,i}, a_N$, and $b_N$ are defined by (2.1) and (2.2). Observe that the difference between the maximum deviation and the deviation at a single $\lambda$ point [5] manifests itself in the factor $(\log m_N)^{-\frac{1}{2}}$. Thus in practice a confidence band for all $\lambda$ is $O((\log m_N)^{\frac{1}{2}})$ times that for a finite set.