Arithmetic sensitivity of cumulant growth in lacunary sums: transcendental versus algebraic ratio limits

Abstract

We study the asymptotic behavior of cumulants of lacunary trigonometric sums $S_n(\omega ):={\sum }_{k=1}^n\cos (2\pi a_k\omega )$, $\omega \in [0,1]$, and show that cumulant growth is highly sensitive to the arithmetic structure of the sequence $(a_k{)}_{k\ge 1}$ of positive integers. In particular, if ${\lim }_{k\to \infty }{a}_{k+1}/a_k=\eta >1$ for some transcendental number $\eta$, we prove that for every $m\in \mathbb{N}$ the $m$-th cumulant of $S_n$ is asymptotically equivalent to the $m$-th cumulant of the ``independent model'' ${\tilde{S}}_n:={\sum }_{k=1}^n\cos (2\pi a_k{U}_k)$, where $U_1,U_2,\dots$ are independent random variables having uniform distribution on $[0,1]$. In particular, the order of growth of the cumulants as $n\to \infty$ is linear in this case. We also show that the transcendence condition for ${\lim }_{k\to \infty }{a}_{k+1}/a_k$ is in general necessary: when the ratio limit $\eta$ is algebraic, the cumulants of $S_n$ may have a different asymptotic order from those of ${\tilde{S}}_n$. For instance, for $a_k=2^k+1$ (with $\eta =2$), the sixth cumulant of $S_n$ grows quadratically in $n$. In contrast, for $a_k=2^k$ (again $\eta =2$) or when $(a_k{)}_{k\ge 1}$ is the Fibonacci sequence (with $\eta =(1+\sqrt{5})/2$), the $m$-th cumulant of $S_n$ grows linearly as $n\to \infty$, but with a growth rate that differs from the one of the independent model ${\tilde{S}}_n$. Overall, our results show that the asymptotic behavior of the cumulants of lacunary trigonometric sums depends on arithmetic effects in a very delicate way. This is particularly remarkable since many other probabilistic limit theorems, such as the Central Limit Theorem, hold for lacunary trigonometric sums in a universal way without any such sensitivity towards arithmetic effects.

30 pages