Take a one-dimensional random walk with zero mean increments, and consider the sizes of its overshoots over the zero level. It turns out that this sequence, which forms a Markov chain, always has a stationary distribution of a simple explicit form. The questions of uniqueness of this stationary distribution and convergence towards it are surprisingly hard. We were able to prove only the total variation convergence, which holds for lattice random walks and for the ones whose distribution, essentially, has density. We also obtained the rate of this convergence under additional mild assumptions. We will also discuss connections to related topics: local times of random walks, ergodic theory and renewal theory. This is joint work with Vlad Vysotsky.
- Stochastic Analysis Seminar