Diffusion constants and martingales for senile random walks

TL;DR

This paper investigates senile random walks, deriving their diffusion constants and associated martingales through time-change techniques, and establishes their convergence to Brownian motion under certain conditions.

Contribution

It introduces a novel method using time-change and martingales to compute diffusion constants for senile random walks and proves their weak convergence to Brownian motion.

Findings

Diffusion constants for senile random walks are explicitly computed.

Martingales associated with the walks are constructed and used for analysis.

Weak convergence to Brownian motion is established under diffusive conditions.

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.