Journal of Inequalities and Applications (Dec 2019)
Almost sure local central limit theorem for the product of some partial sums of negatively associated sequences
Abstract
Abstract The almost sure local central limit theorem is a general result which contains the almost sure global central limit theorem. Let {Xk,k≥1} $\{X_{k},k\geq 1\}$ be a strictly stationary negatively associated sequence of positive random variables. Under the regular conditions, we discuss an almost sure local central limit theorem for the product of some partial sums (∏i=1kSk,i/((k−1)kμk))μ/(σk) $(\prod_{i=1}^{k} S_{k,i}/((k-1)^{k}\mu^{k}))^{\mu/(\sigma\sqrt{k})}$, where EX1=μ $\mathbb{E}X_{1}=\mu$, σ2=E(X1−μ)2+2∑k=2∞E(X1−μ)(Xk−μ) $\sigma^{2}={\mathbb{E}(X_{1}-\mu)^{2}}+2\sum_{k=2}^{\infty}\mathbb{E}(X_{1}-\mu)(X_{k}-\mu)$, Sk,i=∑j=1kXj−Xi $S_{k,i}=\sum_{j=1}^{k}X_{j}-X_{i}$.
Keywords