Limiting behaviour for the theorem of Shannon-McMillan-Breiman

CDSNS Colloquium
Monday, February 4, 2013 - 4:05pm
1 hour (actually 50 minutes)
Skiles 005
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). We show 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. We also show that it satisfies the almost sure invariance principle. 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.