A Conditioned Limit Theorem for Random Walk and Brownian Local Time on Square Root Boundaries

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.

Brownian Excursion Conditioned on Its Local Time

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...

A Local Limit Theorem for random walks conditioned to stay positive

We consider a real random walk Sn = X1 + . . . + Xn attracted (without centering) to the normal law: this means that for a suitable norming sequence an we have the weak convergence Sn/an ⇒ φ(x)dx, φ(x) being the standard normal density. A local refinement of this convergence is provided by Gnedenko’s and Stone’s Local Limit Theorems, in the lattice and nonlattice case respectively. Now let Cn d...

Central Limit Theorem for the Excited Random Walk in Dimension

We prove that a law of large numbers and a central limit theorem hold for the excited random walk model in every dimension d ≥ 2.