Diffusion constants and martingales for senile random walks

Abstract

We derive diffusion constants and martingales for senile random walks with the help of a time-change. We provide direct computations of the diffusion constants for the time-changed walks. Alternatively, the values of these constants can be derived from martingales associated with the time-changed walks. Using an inverse time-change, the diffusion constants for senile random walks are then obtained via these martingales. When the walks are diffusive, weak convergence to Brownian motion can be shown using a martingale functional limit theorem.Comment: 17 pages, LaTeX; the proof of Proposition 2.3 has been simplified, and an error in the proof of Theorem 2.4 has been correcte