## Abstract

Let f be a measurable function satisfying f(x + 1) = f(x),| 1 0 f(x) dx = 0, Var[0,1]f < +∞,and let (nk)k≥1 be a sequence of integers satisfying nk+1/nk ≥ q > 1 (k = 1, 2, ⋯). By the classical theory of lacunary series, under suitable Diophantine conditions on nk, (f(nkx))k≥1 satisfies the central limit theorem and the law of the iterated logarithm. These results extend for a class of subexponentially growing sequences (nk)k≥1 as well, but as Fukuyama showed, the behavior of f(nkx) is generally not permutation-invariant; e.g. a rearrangement of the sequence can ruin the CLT and LIL. In this paper we construct an infinite order Diophantine condition implying the permutation-invariant CLT and LIL without any growth conditions on (nk)k≥1 and show that the known finite order Diophantine conditions in the theory do not imply permutation-invariance even if f(x) = sin2πx and (nk)k≤1 grows almost exponentially. Finally, we prove that in a suitable statistical sense, for almost all sequences (nk)k≤1 growing faster than polynomially, (f(nkx))κ=1 has permutation-invariant behavior.

Original language | English |
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Pages (from-to) | 2505-2517 |

Number of pages | 13 |

Journal | Proceedings of the American Mathematical Society |

Volume | 139 |

Issue number | 7 |

DOIs | |

Publication status | Published - 1 Jul 2011 |

## Keywords

- Central limit theorem
- Diophantine equations
- Lacunary series
- Law of the iterated logarithm
- Permutation-invariance

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics