L S Penrose's limit theorem: Tests by simulation

Cases where the Penrose limit theorem does not hold

Penrose's limit theorem (PLT, really a conjecture) states that the relative power measure of two voters tends asymptotically to their relative voting weight (number of votes). This property approximately holds in most of real life and in randomly generated WVGs for various measures of voting power. Lindner and Machover prove it for some special cases; amongst others they give a condition for th...

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...

Fixed point theorem for non-self mappings and its applications in the modular ‎space

‎In this paper, based on [A. Razani, V. Rako$check{c}$evi$acute{c}$ and Z. Goodarzi, Nonself mappings in modular spaces and common fixed point theorems, Cent. Eur. J. Math. 2 (2010) 357-366.] a fixed point theorem for non-self contraction mapping $T$ in the modular space $X_rho$ is presented. Moreover, we study a new version of Krasnoseleskii's fixed point theorem for $S+T$, where $T$ is a cont...

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