  The theorem of Shannon-McMillan-Breiman states that for every generating
partition on an ergodic system, the exponential decay rate of the measure of
cylinder sets equals the metric entropy almost everywhere (provided the entropy
is finite). In this paper we prove that the measure of cylinder sets are
lognormally distributed for strongly mixing systems and infinite partitions and
show that the rate of convergence is polynomial provided the fourth moment of
the information function is finite. Also, unlike previous results by Ibragimov
and others which only apply to finite partitions, here we do not require any
regularity of the conditional entropy function. We also obtain the law of the
iterated logarithm and the weak invariance principle for the information
function.
