On trigonometric sums with random frequencies

There is a wide and nearly complete theory of trigonometric series with random coefficients; on the other hand, much less is known on trigonometric series with random frequencies. In this paper we study the asymptotic behavior of SN = ∑N k=1 sinnkx for random sequences (nk)k≥1, independent and identically distributed over disjoint intervals Ik ⊂ (0,∞) of the same length. As it turns out, the behavior of SN depends on the size of the gaps ∆k between the intervals Ik: for small gaps the limit distribution of SN/ √ N is mixed Gaussian, for large gaps it is pure Gaussian and for intermediate gaps it is the convolution of a mixed Gaussian distribution and the contribution of an associated nonrandom trigonometric sum which can be nongaussian.