The Central Limit Theorem : studying “ small ” deviations

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 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.

Central and Local Limit Theories in Markov Dependent Random Variables

Large Deviations of Combinatorial Distributions I: Central Limit Theorems

We prove a general central limit theorem for probabilities of large deviations for sequences of random variables satisfying certain natural analytic conditions. This theorem has wide applications to combinatorial structures and to the distribution of additive arithmetical functions. The method of proof is an extension of Kubilius’ version of Cramér’s classical method based on analytic moment ge...

Moderate deviations and functional LIL for super-Brownian motion

A moderate deviation principle and a Strassen type law of the iterated logarithm for the small-time propagation of super-Brownian motion are derived. Moderate deviation estimates which are uniform with respect to the starting point are developed in order to prove the law of the iterated logarithm. Our method also yields a functional central limit theorem.