The theorem of Shannon-McMillan-Breiman states that for every generating partition on an ergodic system of finite entropy the exponential decay rate of the measure of cylinder sets equals the metric entropy almost everywhere. We will then give conditions under which the measure of n-cylinders is log-normally distributed in the limit (i.e. satisfy a Central Limit Theorem). We then also provide conditions under which the logarithm of the measure of n-cylinder, the information function, satisfies the almost sure invariance principle (ASIP). For this we have to require that the measure is β-mixing. This extends previous results due to Philipp and Stout who deduced the ASIP when the measure is strong mixing and satisfies an L1-type Gibbs condition. We can then also prove that the recurrence time satisfies the ASIP if the state space has a finite alphabet.