Almost sure local central limit theorem for the product of some partial sums of negatively associated sequences

Abstract

The almost sure local central limit theorem is a general result which contains the almost sure global central limit theorem. Let {X_{k},kgeq 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 (prod_{i=1}^{k} S_{k,i}/((k-1)^{k}mu^{k}))^{mu/(sigmasqrt{k})}, where mathbb{E}X_{1}=mu, sigma^{2}={mathbb{E}(X_{1}-mu)^{2}}+2sum_{k=2}^{infty}mathbb{E}(X_{1}-mu)(X_{k}-mu), S_{k,i}=sum_{j=1}^{k}X_{j}-X_{i}.