A central limit theorem for trigonometric series with bounded gaps

摘要

In this paper it is proved that there exists a sequence {n _k } of integers with 1 ≤ n _k+1 − n _k ≤ 5 such that the distribution of (cos2pn₁ x + ... + cos2pn_N) / ÖN{(cos 2pi n_1 x + dots + cos 2pi n_{N}) / sqrt N} on ([ 0, 1 ], B, dx) converges to a Gaussian distribution. It gives an affirmative answer to the long standing problem on lacunary trigonometric series which ask the existence of series with bounded gaps satisfying a central limit theorem.