×

On the asymptotic distributions of maxima of trigonometric polynomials with random coefficients. (English) Zbl 0562.62014

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.

MSC:

62E20 Asymptotic distribution theory in statistics
62M15 Inference from stochastic processes and spectral analysis
60F99 Limit theorems in probability theory
60E99 Distribution theory
PDFBibTeX XMLCite
Full Text: DOI