A Note on a Random Walk on the L-Lattice and Relative First-Passage-Time Problems

We analyze a discrete-time random walk on the vertices of an unbounded two-dimensional L-lattice. We determine the probability generating function, and we prove the independence of the coordinates. In particular, we find a relation of each component with a one-dimensional biased random walk with time changing. Therefore, the transition probabilities and the main moments of the random walk can be obtained. The asymptotic behavior of the process is studied, both in the classical sense and involving the large deviations theory. We investigate first-passage-time problems of the random walk through certain straight lines, and we determine the related probabilities in closed form and other features of interest. Finally, we develop a simulation approach to study the first-exit problem of the process thought ellipses.