Fisher information and the central limit theorem

The Local Limit Theorem: A Historical Perspective

The local limit theorem describes how the density of a sum of random variables follows the normal curve. However the local limit theorem is often seen as a curiosity of no particular importance when compared with the central limit theorem. Nevertheless the local limit theorem came first and is in fact associated with the foundation of probability theory by Blaise Pascal and Pierre de Fer...

J ul 2 00 3 Fisher Information inequalities and the Central Limit Theorem Oliver

We give conditions for an O(1/n) rate of convergence of Fisher information and relative entropy in the Central Limit Theorem. We use the theory of projections in L2 spaces and Poincaré inequalities, to provide a better understanding of the decrease in Fisher Information implied by results of Barron and Brown. We show that if the standardized Fisher Information ever becomes finite then it conver...

N ov 2 00 1 Fisher Information inequalities and the Central Limit Theorem Oliver

We give conditions for an O(1/n) rate of convergence of Fisher information and relative entropy in the Central Limit Theorem. We use the theory of projections in L2 spaces and Poincaré inequalities, to provide a better understanding of the decrease in Fisher information implied by results of Barron and Brown. We show that if the standardized Fisher information ever becomes finite then it conver...

Central and Local Limit Theories in Markov Dependent Random Variables

The fractional Fisher information and the central limit theorem for stable laws

A new information-theoretic approach to the central limit theorem for stable laws is presented. The main novelty is the concept of relative fractional Fisher information, which shares most of the properties of the classical one, included Blachman-Stam type inequalities. These inequalities relate the fractional Fisher information of the sum of n independent random variables to the information co...