Sojourns and extremes of Fourier sums and series with random coefficients

Let X(t) be the trigonometric polynomial Σ^k_j=0a_j(U_tcosjt+V_jsinjt), –∞< t<∞, where the coefficients U_t and V_t are random variables and a_j is real. Suppose that these random variables have a joint distribution which is invariant under all orthogonal transformations of R^2k–2. Then X(t) is stationary but not necessarily Gaussian. Put L_t(u) = Lebesgue measure {s: 0?s?t, X(s) > u}, and M(t) = max{X(s): 0?s?t}. Limit theorems for L_t(u) and $\text{P}\text{(M(t) > u)}$ for u→∞ are obtained under the hypothesis that the distribution of the random norm (Σ^k_j=0(U²_j+V²_j))^{1 2} belongs to the domain of attraction of the extreme value distribution exp{ e^–2}. The results are also extended to the random Fourier series (k=∞).