Some asymptotic properties of a kernel spectrum estimate with different multitapers

Let $X(t)$, $t=0,\pm 1,\ldots,$ be a zero mean real-valued stationary time series with spectrum $f_{XX}(\lambda )$, $-\pi\le\lambda\le\pi$. Given the realization $X(1),X(2),\dots,X(N)$, we construct $L$ different multitapered periodograms $I_{XX}^{(mt)_{j}}(\lambda)$, $j=1,2,\dots,L$, on non-overlapped and overlapped segments $X^{(j)}(t)$, $1\le t<N$. Also, we give asymptotic expressions of the mean and variance of the average of these different multitapered periodograms. We obtain an estimate of $f_{XX}(\lambda)$ via $I_{XX}^{(mt)_{j}}(\lambda )$ and different kernels $W_{\beta}^{(j)}(\alpha)$, $j=1,2,\dots,L$; $-\pi<\alpha\le\pi$; $\beta$ is a bandwidth. We find asymptotic expressions of the first and second-order moments of this estimate. Moreover, we propose a choice of the considered bandwidth. An asymptotic expression of the integrated relative mean squared error (IMSE) of the estimate is formulated.