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

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.