Local Limit Theorem for the Lorentz Process and Its Recurrence in the Plane

Central and Local Limit Theories in Markov Dependent Random Variables

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...

Recurrence properties of a special type of Heavy-Tailed Random Walk

In the proof of the invariance principle for locally perturbed periodic Lorentz process with finite horizon, a lot of delicate results were needed concerning the recurrence properties of its unperturbed version. These were analogous to the similar properties of Simple Symmetric Random Walk. However, in the case of Lorentz process with infinite horizon, the analogous results for the correspondin...

Markov Towers and Stochastic Properties of Billiards

Markov partitions work most efficiently for Anosov systems or for Axiom A systems. However, for hyperbolic dynamical systems which are either singular or whose hyperbolicity is nonuniform, the construction of a Markov partition, which in these cases is necessarily countable, is a rather delicate issue even when such a construction exists. An additional problem is the use of a countable Markov p...

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.