Some properties of estimators of the spectrum of a stationary process

T. L. Malevich

Teoriâ veroâtnostej i ee primeneniâ, Tome 10 (1965) no. 3, pp. 500-509

Résumé

Let $x_n$ ($n=0,\pm1,\pm2,\dots$) be a real Gaussian stationary process with $\mathbf Ex_n=0$ and with the spectral function $F(\lambda)$ which is unknown and is supposed to be continuous. The statistic $$ F_N(\lambda)=\frac1{2\pi N}\int_0^\lambda\biggl|\sum_{n=1}^Nx_ne^{-iny}\biggr|^2\,dy $$ is used as an estimator of $F(\lambda)$. In § 1 estimations of the moments $\mathbf E\max\limits_{0\le\lambda\le\pi}|F_N(\lambda)-F(\lambda)|^k$ are obtained. For example the following theorem holds true. Theorem 1.3. For the process $x_n$ $$ \mathbf E\max_{0\le\lambda\le\pi}|F_N(\lambda)-F(\lambda)|^k\le C^kk!\biggl[\omega_F\biggl(\frac1N\biggr)\biggr]^{\frac k2}, $$ where $\omega_F(\cdot)$ is the modulus of continuity of $F(\lambda)$. In § 2 the probability of large deviations of $F_N(\lambda)$ from $F(\lambda)$ is studied. The obtained results are also generalized for a certain class of estimators of $F(\lambda)$.