On the asymptotic behavior of weakly lacunary series

Abstract:

Let $f$ be a measurable function satisfying \[ f(x+1)=f(x), \quad \int _0^1 f(x) dx=0, \quad \textrm {Var}_{[0,1]} f < + \infty , \] and let $(n_k)_{k\ge 1}$ be a sequence of integers satisfying $n_{k+1}/n_k \ge q >1$ $(k=1, 2, \ldots )$. By the classical theory of lacunary series, under suitable Diophantine conditions on $n_k$, $(f(n_kx))_{k\ge 1}$ satisfies the central limit theorem and the law of the iterated logarithm. These results extend for a class of subexponentially growing sequences $(n_k)_{k\ge 1}$ as well, but as Fukuyama showed, the behavior of $f(n_kx)$ 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 $(n_k)_{k\ge 1}$ and show that the known finite order Diophantine conditions in the theory do not imply permutation-invariance even if $f(x)=\sin 2\pi x$ and $(n_k)_{k\ge 1}$ grows almost exponentially. Finally, we prove that in a suitable statistical sense, for almost all sequences $(n_k)_{k\ge 1}$ growing faster than polynomially, $(f(n_kx))_{k\ge 1}$ has permutation-invariant behavior.