A central limit theorem for trigonometric series with bounded gaps

Abstract

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 (cos 2π n 1 x + · · · + cos 2π n N )/ √ 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.

On evil Kronecker sequences and lacunary trigonometric products

“…In the simplest model, when for every number n ≥ 1 we decide independently and with fair probability whether it should be contained in (r k ) k≥1 or not, then (13) holds almost surely (with respect to the probability space over which the r k 's are defined) for almost all α. In a similar fashion both results (13) and (14) essentially remain valid when the random sequence (r k ) k≥1 is constructed in a more complicated fashion (see for example [19,20,34]).…”

“…To a certain degree this almost-independence property extends to systems (f (s ℓ α)) ℓ≥1 for a function f which is periodic with period one and satisfies certain regularity properties; however, in this case the number-theoretic properties of (s ℓ ) ℓ≥1 play an important role, and the almost-independent behavior generally fails when (19) is relaxed to a weaker growth condition. The case which has been investigated in the greatest detail is that when f has bounded variation on [0, 1], since this case is (by Koksma's inequality) closely connected to the discrepancy of the sequence of fractional parts ({s ℓ α}) ℓ≥1 , which in turn can be interpreted as the (one-sided) Kolmogorov-Smirnov statistic adopted to the case of the uniform measure on [0, 1].…”

“…There exist a few results concerning lacunary series when f is neither required to have bounded variation, nor to be Lipschitz-or Hölder-continuous, nor to have a modulus of continuity of a certain regularity; see for example [31]. However, for these results the growth requirements for (s ℓ ) ℓ≥1 are much stronger than (19), which means that they are not applicable in our case, since by (20) and (21) we have to deal with lacunary sequences growing exactly with the speed presumed in (19), and not faster. and that (s ℓ ) ℓ≥1 is a sequence satisfying (19).…”

Pure Gaussian limit distributions of trigonometric series with bounded gaps

“…Since the right hand side is summable in N if we replace N by N 3/2 , by applying Beppo Levi's theorem we have (6) and hence (4). Now we prove (2).…”

“…As to this problem, our previous paper [6] proved that there exists a sequence of bounded gaps obeying the central limit theorem with σ 2 = 1/4. It was also proved that the class of limit distributions of bounded gap sequences contains huge class of mixtures of Gaussian distributions.…”