Some Optimal Conditions for the ASCLT

Abstract

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