A multiparameter variant of the Salem–Zygmund central limit theorem on lacunary trigonometric series

Mordechay B. Levin¹

¹ Department of Mathematics Bar-Ilan University Ramat-Gan, 5290002, Israel

Colloquium Mathematicum, Tome 131 (2013) no. 1, pp. 13-27

Résumé

We prove the central limit theorem for the multisequence $$ \sum_{1 \leq n_1 \leq N_1} \cdots \sum_{1 \leq n_d \leq N_d} a_{n_1, \ldots ,n_d} \cos (\langle 2\pi \mathbf{m}, A_1^{n_1} \dots A_d^{n_d} \mathbf{x} \rangle) $$ where $\mathbf{m} \in \mathbb Z^s$, $a_{n_1, \ldots ,n_d}$ are reals, $A_1, \ldots ,A_d$ are partially hyperbolic commuting $s\times s$ matrices, and $\mathbf{x}$ is a uniformly distributed random variable in $[0,1]^s$. The main tool is the S-unit theorem.

Détail

DOI : 10.4064/cm131-1-2

Keywords: prove central limit theorem multisequence sum leq leq cdots sum leq leq ldots cos langle mathbf dots mathbf rangle where mathbf mathbb ldots reals ldots partially hyperbolic commuting times matrices mathbf uniformly distributed random variable main tool s unit theorem