A Central Limit Theorem for Convex Chains in the Square

A Central Limit Theorem for Convex

Points P1; : : : ; Pn in the unit square de ne a convex n-chain if they are below y = x and, together with P0 = (0; 0) and Pn+1 = (1; 1), they are in convex position. Under uniform probability, we prove an almost sure limit theorem for these chains that uses only probabilistic arguments, and which strengthens similar limit shape statements established by other authors. An interesting feature is...

Density Estimators for Truncated Dependent Data

In some long term studies, a series of dependent and possibly truncated lifetime data may be observed. Suppose that the lifetimes have a common continuous distribution function F. A popular stochastic measure of the distance between the density function f of the lifetimes and its kernel estimate fn is the integrated square error (ISE). In this paper, we derive a central limit theorem for t...

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.

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