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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1963, Volume 8, Issue 4, Pages 391–430 (Mi tvp4689)

On Estimation of the Spectral Function of a Stationary Gaussian Process

I. A. Ibragimov

Abstract: Let $x_1,x_2,…,x_N$ be a sample time series drawn from the real stationary Gaussian process $\{x_n\},{\mathbf E}x_n\equiv0$, with unknown spectral distribution function (s.d.f.) and spectral density function $f(\lambda)$. The problem of estimating of s.d.f. $F(\lambda)$ is discussed and the estimate $F_N^*(\lambda)=\int_0^\lambda{I_N} (\lambda )d\lambda$ of s.d.f. $F(\lambda)$ is considered, where
$$I_N(\lambda)=\frac{1}{{2\pi N}}| {\sum\limits_1^N{x_j e^{i\lambda j}}}|^2.$$
In §1–§2 the asymptotic properties of expressions like
$${\mathbf E}\int_{-\pi}^\pi{\varphi(\lambda)I_N(\lambda) d\lambda},\quad{\mathbf E}\int_{-\pi}^\pi{T_1(\lambda)I_N(\lambda ) d\lambda}\int_{-\pi}^\pi{T_2(\mu )I_N(\mu) d\mu}$$
are investigated. The main section of this paper is §5. Let
$$\zeta_N(\lambda)=\sqrt N[{F_N^*(\lambda)-F\lambda}],$$
and let $\zeta(\lambda)$ be a Gaussian stochastic process with
$$\zeta(0)=0, \mathbf E\zeta(\lambda)\equiv0, {\mathbf E}\zeta(\lambda)\zeta(\mu)=2\pi\displaystyle\int_0^{\min(\lambda,\mu)}f^2(\lambda) d\lambda,\quad0\leq\lambda, \mu\leq\pi.$$
We denote by $P_N$ the probability measure induced in $C[0,\pi]$ by $\zeta_N(\lambda)$, and by $P$ the probability measure induced in $C[0,\pi]$ by $\zeta(\lambda)$. The following is proved in §5:
Theorem 5.1 Let
$$1.\int_a^b{f(\lambda) d\lambda}>0\qquad{for every}\quad[a,b]\subset[-\pi,\pi]; 2.\int_{-\pi}^\pi{(f(\lambda))^{2+\delta} d\lambda}<\infty\qquad{for some}\quad\delta>0,$$
then $\mathop{P_N\Rightarrow P}\limits_{N\to\infty}$, where the sign $\Rightarrow$ denotes weak convergence of the measures.
In §8 some estimates are given for probabilities of large deviations $F_N^*(\lambda)$ from $F(\lambda)$.
In §9 it is shown that all results of §§$1$$8$ are valid for continuous time.

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English version:
Theory of Probability and its Applications, 1963, 8:4, 366–401

Citation: I. A. Ibragimov, “On Estimation of the Spectral Function of a Stationary Gaussian Process”, Teor. Veroyatnost. i Primenen., 8:4 (1963), 391–430; Theory Probab. Appl., 8:4 (1963), 366–401

Citation in format AMSBIB
\Bibitem{Ibr63} \by I.~A.~Ibragimov \paper On Estimation of the Spectral Function of a Stationary Gaussian Process \jour Teor. Veroyatnost. i Primenen. \yr 1963 \vol 8 \issue 4 \pages 391--430 \mathnet{http://mi.mathnet.ru/tvp4689} \transl \jour Theory Probab. Appl. \yr 1963 \vol 8 \issue 4 \pages 366--401 \crossref{https://doi.org/10.1137/1108044} 

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1. A. A. Sahakian, M. S. Ginovyan, “On the central limit theorem for Toeplitz quadratic forms of stationary sequences”, Theory Probab. Appl., 49:4 (2005), 612–628
2. V. N. Solev, L. Gerville-Reache, “Large Toeplitz operators and quadratic form generated by stationary Gaussian sequence”, J. Math. Sci. (N. Y.), 139:3 (2006), 6625–6630
3. Theory Probab. Appl., 56:1 (2012), 57–71
4. Pipiras V. Taqqu M., “Long-Range Dependence and Self-Similarity”, Long-Range Dependence and Self-Similarity, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge Univ Press, 2017, 1–668