An Invariance for the Large-sample Empirical Distribution of Waiting Time between Successive Extremes

For a given set of observations, we consider the waiting times between successive returns to extreme values. Our main result is an invariance theorem that says that, as the size of the data set gets large, the empirical distribution of the waiting time converges with probability one to a geometric distribution, whenever the observations are i.i.d. or, more generally, a realization of any exchangeable joint distribution. We also prove a central limit theorem for these empirical distributions (properly normalized), which leads in natural way to a Donsker-type theorem on D[0,∞). Our study was motivated by empirical and simulation results, which indicate the existence of an empirical invariance, obtained in connection with our investigation on stock price data. In many of the accepted models for stock prices, our results would apply and suggest such invariance to hold. It may be pointed out that the empirical invariance from the real stock price data was quite far away from the theoretical invariance. Received March 30, 2012. AMS Subject Classification: 60F05, 60F15, 60F17, 62E20, 91B24, 91B28, 91G99.