A quenched limit theorem for the local time of random walks on Z

A PRELUDE TO THE THEORY OF RANDOM WALKS IN RANDOM ENVIRONMENTS

A random walk on a lattice is one of the most fundamental models in probability theory. When the random walk is inhomogenous and its inhomogeniety comes from an ergodic stationary process, the walk is called a random walk in a random environment (RWRE). The basic questions such as the law of large numbers (LLN), the central limit theorem (CLT), and the large deviation principle (LDP) are ...

Local Limit Theorems for Random Walks in a 1D Random Environment

We consider random walks (RW) in a one-dimensional i.i.d. random environment with jumps to the nearest neighbours. For almost all environments, we prove a quenched Local Limit Theorem (LLT) for the position of the walk if the diffusivity condition is satisfied. As a corollary, we obtain the annealed version of the LLT and a new proof of the theorem of Lalley which states that the distribution o...

A Local Limit Theorem for Random Walks in Balanced Environments

Central limit theorems for random walks in quenched random environments have attracted plenty of attention in the past years. More recently still, finer local limit theorems — yielding a Gaussian density multiplied by a highly oscillatory modulating factor — for such models have been obtained. In the one-dimensional nearest-neighbor case with i.i.d. transition probabilities, local limits of uni...

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...