A local limit theorem for perturbed random walks

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 ...

Central and Local Limit Theories in Markov Dependent Random Variables

Dynamic Quantum Bernoulli Random Walks

Quantum Bernoulli random walks can be realized as random walks on the dual of SU(2). We use this realization in order to study a model of dynamic quantum Bernoulli random walk with time dependent transitions. For the corresponding dynamic random walk on the dual of SU(2), we prove several limit theorems (local limit theorem, central limit theorem, law of large numbers, large deviations principl...

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...

Random differential inequalities and comparison principles for nonlinear hybrid random differential equations

 In this paper, some basic results concerning strict, nonstrict inequalities, local existence theorem and differential inequalities  have been proved for an IVP of first order hybrid  random differential equations with the linear perturbation of second type. A comparison theorem is proved and  applied to prove the uniqueness of random solution for the considered perturbed random differential eq...