Almost sure central limit theorems under minimal conditions

Let X1, X2,... be independent, identically distributed random variables with EX1 = 0, EX12 = 1 and let Sn = [summation operator]k[less-than-or-equals, slant]n Xk. We give nearly optimal criteria for an (unbounded) measurable function f to satisfy the a.s. central limit theorem, i.e., a.s., where [phi] is the standard normal density function.Logarithmic average Wiener process a.s. central limit theorem