Turkman, K. F.; Walker, A. M. On the asymptotic distributions of maxima of trigonometric polynomials with random coefficients. (English) Zbl 0562.62014 Adv. Appl. Probab. 16, 819-842 (1984). Let \(X_ n(\lambda)\) and \(Y_ n(\lambda)\) be defined as \(X_ n(\lambda)+iY_ n(\lambda)=\sqrt{2/n}\sum^{n}_{1}\epsilon_ je^{i\lambda j}\) where the \(\{\epsilon_ j\}\) are independent N(0,1) random variables. The authors show that \(\max_{0\leq \lambda \leq \pi}X_ n(\lambda)\), \(\max_{0\leq \lambda \leq \pi}Y_ n(\lambda)\) and \(\max_{0\leq \lambda \leq \pi}I_ n(\lambda)\), where \(I_ n(\lambda)=X^ 2_ n(\lambda)+Y^ 2_ n(\lambda)\), conveniently reduced have asymptotically a Gumbel distribution. The result is extended to the case where the \(\{\epsilon_ j\}\), with mean value zero and variance, have a moving-average representation. Reviewer: J.Tiago de Oliveira Cited in 11 Documents MSC: 62E20 Asymptotic distribution theory in statistics 62M15 Inference from stochastic processes and spectral analysis 60F99 Limit theorems in probability theory 60E99 Distribution theory Keywords:asymptotic distributions of maxima; trigonometric polynomials with random coefficients; stationary Gaussian sequence; periodogram; extreme-value distributions; Gumbel distribution; moving-average representation PDFBibTeX XMLCite \textit{K. F. Turkman} and \textit{A. M. Walker}, Adv. Appl. Probab. 16, 819--842 (1984; Zbl 0562.62014) Full Text: DOI