Aging arcsine law in Brownian motion and its generalization.

Classical arcsine law states that the fraction of occupation time on the positive or the negative side in Brownian motion does not converge to a constant but converges in distribution to the arcsine distribution. Here we consider how a preparation of the system affects the arcsine law, i.e., aging of the arcsine law. We derive an aging distributional theorem for occupation time statistics in Brownian motion, where the ratio of time when measurements start to the measurement time plays an important role in determining the shape of the distribution. Furthermore, we show that this result can be generalized as an aging distributional limit theorem in renewal processes.