Almost sure central limit theorems

The almost sure central limit theorem (ASCLT) has been stimulated by two works by Brosamler (1988) and Schatte (1988) and extensively studied in the past decade. Let I and F (x) be an indicator function and the standard normal distribution function respectively. Then ASCLT says that for a sequence of independent and identically distributed (i.i.d.) random variables with mean 0 and finite variance, we have limn→infinity 1logn k=1n1kI Sk/k≤x =Fx a.s.; In the dissertation we investigate ASCLT and its extensions to weakly dependent random variables. Its strong approximation is also considered for both independent and dependent random variables. The goal is to prove ASCLT and its invariance principle for weakly dependent random variables.