Central limit theorem for weakly lacunary Walsh series
Let \(x\in [0,1]\), \(r_ k(x)=sgn(\sin 2^ k\pi x)\), k-1,2,..., be the Rademacher functions. For each natural number n, \(n=\sum_{k}\epsilon_ k2^ k\) \((\epsilon_ k=0\) or \(\epsilon_ k=1)\), set \(\omega_ n(x)=\prod_{k}\{r_{k+1}(x)\}^{\epsilon_ k}.\) Moreover, let \(\omega_ 0(x)\equiv 1\). Then \(\{\omega_ n(x)\}\) is a Walsh system in Paley's ordering. In 1975 A. Földes proved the following central limit theorem: Theorem A. Let us be given a sequence of natural numbers \(\{n_ k\}\) and a sequence of non-negative numbers \(\{a_ k\}\), satisfying the conditions (1.1) \(n_{k+1}>n_ k(1+c\cdot k^{-\alpha})\), \(k=1,2,...\), (1.2) \(A^ 2_ N=\sum^{N}_{k=1}a^ 2_ k\to \infty\) as \(N\to \infty\), (1.3) \(a_ N=O(A_ N\cdot N^{-\alpha})\) as \(N\to \infty\), where c and \(\alpha\) are constants such that \(c>0\) and \(0\leq \alpha \leq\). Then for all t \[ (1.4)\quad \lim_{m\to \infty}| \{x:\quad x\in [0,1],\quad \sum^{N}_{k=1}a_ k\omega_{n_ k}(x)\leq t\cdot A_ N\}| =\frac{1}{\sqrt{2\pi}}\int^{t}\quad_{-\infty}\exp (- \frac{z^ 2}{2})dz, \] where the sign \(| \cdot |\) denotes the Lebesgue measure of a set. The aim of the present paper is to prove that estimate (1.3) is the best possible. More precisely the author proves that: Theorem B. For any fixed constants c and \(\alpha\) such that \(c>0\) and \(0\leq \alpha \leq\), there exists sequences \(\{n_ k\}\) and \(\{a_ k\}\), satisfying conditions (1.1), (1.2) and \((1.5)\quad a_ N=O(A_ N\cdot N^{-\alpha})\) as \(N\to \infty\), such that relation (1.4) is not satisfied.
scientific article; zbMATH DE number 3960225