Large Deviations for Independent Random Walks on the Line

Abstract

For a system of infinitely many independent symmetric random walks on $\mathbb{Z}$ let $K_n(x)$ be the number of visits to $x \in \mathbb{Z}$ from time 0 to $n - 1$. The probabilities of some rare events involving $(K_n(0), K_n(1))$ are estimated as $n \rightarrow \infty$ and the corresponding large deviation rate functions are derived for both deterministic and invariant initial distributions. The dependence on the initial distributions is discussed. A simple method is used for guessing at the rate functions. This method is effective for independent random walks on the line and is worth exploring in more general settings.