The central limit theorem for sums of functions of independent random variables and for sums of the form $\sum f(2^kt)$

I. A. Ibragimov

Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 4, pp. 655-665

Résumé

Let $\varepsilon_1,\varepsilon_2,\dots$ be a sequence of independent random variables and let a random variable $f=f(\varepsilon_1,\varepsilon_2,\dots)$. Consider a sequence of random variables $\{f_j\}$ where $f_j=f(\varepsilon_j,\varepsilon_{j+1},\dots)$. The main result of this paper is Theorem 2. {\it If $1)\ \mathbf E|f|^{2+\delta}=\rho_\delta\infty$ for some $\delta$, $0\delta\le1$; $2)\ \mathbf E^{\frac1{2+\delta}}|f-\mathbf E\{f\mid\varepsilon_1,\dots,\varepsilon_n\}|^{2+\delta}\le A2^{-n\alpha}$ where $A$, $\alpha$ are positive constants; $3)\ \sigma^2=\mathbf Ef_1^2+2\sum_2^\infty\mathbf E\{f_1f_j\}\ne0$ then $$ \biggl|\mathbf P\biggl\{\frac1{\sigma\sqrt n}\sum_1^nf_j\biggr\}-\Phi(z)\biggr|\le C\biggl(\frac{\ln n}{n}\biggr)^{\delta/2}, $$ where $\Phi(z)=\frac1{\sqrt{2\pi}}\int_{-\infty}^ze^{-\frac{x^2}{2}}\,dx$ and $C$ depends on $A$, $\alpha$, $\sigma$, $\rho_\delta$ only}. This theorem is applied to find distributions of sums $\sum_1^\infty f(2^kt)$.