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On the asymptotic distributions of maxima of trigonometric polynomials with random coefficients

Published online by Cambridge University Press:  01 July 2016

K. F. Turkman  and
A. M. Walker
K. F. Turkman*
Affiliation:
University of Sheffield
A. M. Walker*
Affiliation:
University of Sheffield
*
∗Present address: Faculdade de Ciencias de Universidade de Lisboa, Departamento da Estatistica, Investigaçao Operacional e Computacao, 58 Rua da Escola Politecnica, 1294 Lisboa Codex, Portugal.
∗∗ Postal address: Department of Probability and Statistics, The University, Sheffield S3 7RH, UK.
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Abstract

Let {ε t, t = 1, 2, ···, n} be a sequence of mutually independent standard normal random variables. Let Xn(λ) and Yn(λ) be respectively the real and imaginary parts of exp iλ t, and let . It is shown that as n tends to∞, the distribution functions of the normalized maxima of the processes {Xn(λ)}, (Yn(λ)}, {In(λ)} over the interval λ∈ [0,π] each converge to the extremal distribution function exp [–e–x], —∞ < x <∞.

It is also shown that these results can be extended to the case where {ε t} is a stationary Gaussian sequence with a moving-average representation.

Keywords

PERIODOGRAMEXTREME-VALUE DISTRIBUTIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 4 , December 1984 , pp. 819 - 842
Copyright
Copyright © Applied Probability Trust 1984 

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