The central limit theorem and Poisson approximation

Entropy and the ‘compound’ Law of Small Numbers

An information-theoretic foundation for compound Poisson approximation limit theorems is presented, in analogy to the corresponding developments for the central limit theorem and for simple Poisson approximation. It is shown that the compound Poisson distributions satisfy a natural maximum entropy property within a natural class of distributions. Simple compound Poisson approximation bounds are...

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...

Central and Local Limit Theories in Markov Dependent Random Variables

Central Limit Theorem in Multitype Branching Random Walk

A discrete time multitype (p-type) branching random walk on the real line R is considered. The positions of the j-type individuals in the n-th generation form a point process. The asymptotic behavior of these point processes, when the generation size tends to infinity, is studied. The central limit theorem is proved.

RAPPORT Central limit theorem for a class of random measures associated with germ - grain models

The paper introduces a family of stationary random measures in R d generated by so-called germ-grain models. The germ-grain model is deened as the union of i.i.d. compact random sets (grains) shifted by points (germs) of a point process. This model gives rise to random measures deened by the sum of contributions of non-overlapping parts of the individual grains. The corresponding moment measure...