A Central Limit Theorem for Punctuated Equilibrium

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

What is punctuated equilibrium? What is macroevolution? A response to Pennell et al.

What is punctuated equilibrium [or ‘equilibria’ as termed by one of us (N.E.)]? What is macroevolution? These are questions central to evolutionary biology. Yet, recently in an article in Trends Ecol. Evol. by Pennell et al. [1] the former is mischaracterized and the latter is not even defined. Here, we rectify these oversights by providing a definition of macroevolution, an accurate rendition ...

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...