On the distributions of some statistical estimates of spectral density

Let $X_t$, $t=\dots,-1,0,1,\dots$, be a real Gaussian stationary time series with zero mean and spectral density $f(\lambda)$, $-\pi\le\lambda\le\pi$. In the paper the distribution of estimates (0.1) is considered, where $J_N(x)$ is the periodogram and $W\in L_1(-\pi,\pi)$. The asymptotic expansions of the distribution function and density of r. v. (0.5) are given and the theorem on large deviations is proved. Comparatively exact inequalities for the probabilities
$$ \mathbf P\{|\widehat f(\lambda)-\mathbf E\widehat f(\lambda)|\ge x\},\qquad \mathbf P\{\|\widehat f-\mathbf E\widehat f\|_2\ge x\},\qquad \mathbf P\{\|\widehat f-\mathbf E\widehat f\|_\infty\ge x\} $$
are derived. It is proved also that for some of the estimates (0.1) the inequalities (3.2)–(3.4) hold for all $a>0$.