Conditional Limit Theorems for Renewal Random Walks

We consider the trajectories of a renewal random walk, that is, a random walk on the two-dimensional integer lattice whose jumps have positive horizontal component. In a contrast to the usual limit theorems for random walks, we do not require the jumps to have width 1, and we consider models in which the height of each jump may depend on its width. We prove a Functional Central Limit Theorem for these trajectories: the distribution of their fluctuations around a limiting profile converges weakly to that of Brownian motion. We also derive a conditional version of this theorem, under large-deviations conditions on the terminal height and the integral of the trajectories. We find the shape of the corresponding limiting profile, and prove the convergence of the distribution of the fluctuations around this profile to that of a conditioned Gaussian process.