Some Optimal Conditions for the ASCLT

Abstract Let $$X_{1},X_{2},\ldots $$ X 1 , 2 … be independent random variables with $${E}X_{k}=0$$ E k = 0 and $$\sigma _{k}^{\,2}:={E}X_{k}^2<\infty σ : < ∞ $$(k\ge 1)$$ ( ≥ ) . Set $$S_k=X_1+\cdots +X_k$$ S + ⋯ assume that $$s_{k}^{\,2}:={E}S_k^2\rightarrow \infty s → We prove under the Kolmogorov condition $$\begin{aligned} |X_n|\le L_n, \quad L_n =o(s_n/(\log \log s_n)^{1/2}) \end{aligned}$$ | n ≤ L o / log we have \frac{1}{\log s_{n}^{\,2}}\sum _{k=1}^{n}\frac{\sigma _{k+1}^{\,2}}{s_{k}^{\,2}}f\left( \frac{S_{k}}{s_{k}}\right) \rightarrow \frac{1}{\sqrt{2\pi }}\int _{\mathbb {R}}f(x)e^{-x^2/2}\,\textrm{d}x\quad \mathrm {a.s.}\end{aligned}$$ ∑ f π ∫ R x e - d a . for any almost everywhere continuous function $$f: {\mathbb R} R}$$ satisfying $$|f(x)|\le e^{\gamma x^2}$$ γ , $$\gamma <1/2$$ also show replacing o in (1) by O relation (2) becomes generally false. Finally, case when is not assumed, give an optimal terms of remainder term Wiener approximation partial sum process $$\{S_n, \, n\ge 1\}$$ { } a process.